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Dynamical Analysis of a Stochastic Multispecies Turbidostat Model

Mu Yu Li Zuxiong Xiang Huili Wang Hailing

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Published 01012019

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Published version (DOI)10115520194681205

Publication detailsMu Y Li Z Xiang H amp Wang H (2019) Dynamical Analysis of a Stochastic Multispecies Turbidostat ModelComplexity 2019 [4681205] httpsdoiorg10115520194681205

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Research ArticleDynamical Analysis of a StochasticMultispecies Turbidostat Model

Yu Mu12 Zuxiong Li 13 Huili Xiang 1 and Hailing Wang 1

1Department of Mathematics Hubei University for Nationalities Enshi Hubei 445000 China2Department of Mathematics City University of Hong Kong Hong Kong China3School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan Hubei 430074 China

Correspondence should be addressed to Zuxiong Li lizx0427126com

Received 31 July 2018 Accepted 4 December 2018 Published 10 January 2019

Academic Editor Marcelo Messias

Copyright copy 2019 Yu Mu et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A stochastic turbidostat system in which the dilution rate is subject to white noise is investigated in this paper First of all sufficientconditions of the competitive exclusion among microorganisms are obtained by employing the techniques of stochastic analysisFurthermore the results demonstrate that the competition among microorganisms and stochastic disturbance will affect thedynamical behaviors of microorganisms Finally the theoretical results obtained in this contribution are illustrated by numericalsimulations

1 Introduction

The chemostat and turbidostat two types of devices forcontinuous cultivation of microorganism have been utilizedto analyze population dynamics Novick et al [1] first pro-posed the mechanism of chemostat in 1950 and then anincreasing number of researchers have devoted themselvesto investigate chemostat systems [2ndash7] However there existsome drawbacks in chemostat model with a constant dilutionrate such as thewaste of substrate and higher viscosity causedby the mass transfer efficiency Models with the dilution raterelated to the state ofmicroorganism which can be called tur-bidostat (see [8]) can overcome the above drawbacks and behelpful to design optimal strategies to improve the substrateutilization Flegr [9] investigated turbidostat system with twospecies from numerical simulation and De Leenheer et al[4] analyzed the system theoretically Li [10] systematicallyinvestigated the turbidostat model and established sufficientconditions of coexistence of two species Subsequently manyresearchers investigated turbidostat and chemostat systemsMany valuable and interesting results were obtained [11ndash16]

According to May [17] the parameters of the systemsuch as birth rate death rate and the input concentration ofnutrient are inevitably disturbed by environmental factors

In recent years more and more stochastic ecological mod-els were chosen to describe the dynamics of populationsGrasmanet et al [18] established a stochastic chemostatmodel with three trophic levels Using singular perturbationmethods they obtained the expected breakdown time andanalyzed the influence of stochastic factors on dynamicbehaviors of the system in detail Considering the stochasticfactors in chemostat Campillo et al [19] proposed stochasticsystems with different population scales and they inves-tigated these systems and derived the field of validity fordifferent population scales Collet et al [5] pointed out thatthe population size in the device was determined not only bythe concentration of the nutrient but also by the stochasticbirth and death rates By analyzing the long time behaviorsof microbes they derived the conditions for the globalexistence of the solutions of the system and the existenceof quasi-stationary distribution Meng et al [6] constructedan impulsive stochastic chemostat model and investigatedthe extinction and permanence of the microorganisms Theyalso pointed out that small perturbation from white noisecould cause the extinction of microorganisms Xu et al [7]investigated a chemostat model containing telegraph noisewith the help of Markov chain and derived the break-evenconcentrationwhich determines the persistence or extinction

HindawiComplexityVolume 2019 Article ID 4681205 18 pageshttpsdoiorg10115520194681205

2 Complexity

of microorganisms One can also find a large number ofstochastic models on cultivation of microorganisms [14 1520ndash23] as well as in other fields [24ndash28]

Competition is all around environment because of thelimitation of natural resource Ghoul et al [29] and Kaye et al[30] pointed out that microbes expressed many competitivebehaviors with their neighbours or plants for scarce nutrientsand limited spaces Many practical results were obtained byinvestigating the corresponding competitive models Butleret al [16] established a system containing two competitorsand a growth-limiting nutrient They derived conditions foruniform persistence and local stability of the food web andalso obtained the conditions for predator-mediated coexis-tenceWolkowicz et al [31] proposed a competitive chemostatmodel with distributed delay to describe the process ofnutrient consumption They proved that there existed onlyone survivor under any conditions and also pointed outthat the theoretical results in their paper were valid for allsystems with monotone growth response functions Liu et al[32] established a stochastic competitive model and obtainedsufficient conditions of extinction and persistence (includingweak persistence and strong persistence) They stated thatonly one species survived under certain stochastic noiseperturbation Xu et al [33] analyzed a competitive chemostatmodel and derived the critical value of the noise Zhao etal [34] discussed on a stochastic competition system andderived sufficient conditions of persistence and stationarydistribution for each population Actually a great number ofstochastic competition models have been studied [35ndash40]

In this paper considering the dilution rate of microor-ganisms related to the feedback control and the influenceof stochastic factors from the environment we establish thefollowing stochastic turbidostat system based on Zhang etal [21] who constructed a stochastic chemostat model andobtained the conditions of competitive exclusion

d119878 (119905) = [(1198780 minus 119878 (119905)) (119889 + 119899sum119894=1

119896119894119909119894 (119905))

minus 119899sum119894=1

119886119894119878 (119905)1 + 119887119894119878 (119905)119909119894 (119905)] d119905 + 1205901119878 (119905) d1198611 (119905)

d119909119894 (119905) = [minus(119889119894 + 119899sum119894=1

119896119894119909119894 (119905)) 119909119894 (119905)

+ 119886119894119878 (119905)1 + 119887119894119878 (119905)119909119894 (119905)] d119905 + 120590119894+1119909119894 (119905) d119861119894+1 (119905) 119894 = 1 2 119899

(1)

where 119878(119905) and 119909119894(119905) (119894 = 1 2 119899) represent the con-centration of nutrient and microorganisms at the time 119905separately 1198780 expresses the input concentration of nutri-tion 119861119894(119905) (119894 = 1 2 119899 + 1) is Brownian motiondefined on a complete probability space (ΩF 119875) (F119905 =120590(119878(119905) 1199091(119905) 1199092(119905) 119909119899(119905)) 0 le 119905 le 120591119890) and 120590119894 gt 0 (119894 =1 2 119899 + 1) stands for the density of the white noise119889 + sum119899119894=1 119896119894119909119894 (119889 gt 0 and 119896119894 gt 0 are constants) denotes the

dilution rate of the turbidostat and 119889119894 + sum119899119894=1 119896119894119909119894 (119889119894 gt 0is constant) is the sum of the dilution rate and death rateof the microorganisms in the system 119886119894119878(1 + 119887119894119878) is theHolling 2 functional response function (119886119894 gt 0 and 119887119894 ge 0are constants)

This paper is organized as follows In Section 2 theexistence and uniqueness of the positive solution of system(1) are verified Section 3 demonstrates the main results Adiscussion is given and a numerical example is offered toverify our theoretical results in Section 4

2 Existence and Uniqueness ofthe Positive Solution

We in this section demonstrate the existence and uniquenessof global positive solution of system (1)

Theorem 1 If 119889119894119896119894 gt 1198780 + 1 (119894 = 1 2 119899) then stochasticsystem (1) has a unique positive solution (119878(119905) 1199091(119905) 119909119899(119905))almost surely with initial value (119878(0) 1199091(0) 119909119899(0)) isin 119877119899+1+

Proof For 119905 ge 0 let119906 (119905) = ln 119878 (119905) V119894 (119905) = ln 119909119894 (119905) (119894 = 1 2 119899) (2)

According to the Ito formula we have

d119906 (119905) = [(1198780 minus e119906(119905)) (119889 + sum119899119894=1 119896119894eV119894(119905))e119906(119905)

minus 119899sum119894=1

119886119894eV119894(119905)1 + 119887119894e119906(119905) minus120590212 ] d119905 + 1205901d1198611 (119905)

dV119894 (119905) = [minus(119889119894 + 119899sum119894=1

119896119894eV119894(119905)) + 119886119894e119906(119905)1 + 119887119894e119906(119905) minus1205902119894+12 ] d119905

+ 120590119894+1d119861119894+1 (119905) 119894 = 1 2 119899

(3)

with initial value (119906(0) V1(0) V119899(0)) = (ln 119878(0) ln1199091(0) ln119909119899(0)) It is obvious that e119906 and eV119894 are continuousand differentiable functions Hence (3) satisfies local Lip-schitz condition which means that there exists a uniquelocal solution (119906(119905) V1(119905) V119899(119905)) for 119905 isin [0 120591119890) Then(119878(119905) 1199091(119905) 119909119899(119905)) = (e119906(119905) eV1(119905) eV119899(119905)) is the uniquepositive local solution for system (1) with the initial value(119878(0) 1199091(0) 119909119899(0)) isin 119877119899+1+ Next we prove that thepositive solution of system (1) is global that is 120591119890 = infinWe first define the filtration F119905 as F119905 = 119883119904 0 le119904 le 120591119890 and choose an enough large 1198960 gt 0 such that(119878(0) 1199091(0) 119909119899(0)) isin [11198960 1198960] Then for all 119896 ge 1198960 wedefine a stopping time by

120591119896 = inf 119905 isin [0 120591119890) min 119878 (119905) 1199091 (119905) 119909119899 (119905)le 1119896 or max 119878 (119905) 1199091 (119905) 119909119899 (119905) ge 119896

(4)

Complexity 3

Next we further show 120591119890 997888rarr infin According to thedefinition of the stopping time

min 119878 (119905) 1199091 (119905) 119909119899 (119905) le 1119896 997888rarr 0max 119878 (119905) 1199091 (119905) 119909119899 (119905) ge 119896 997888rarr infin

as 119896 997888rarr infin(5)

Therefore 120591119896 is increasing as 119896 997888rarr infin Set 120591infin = lim119896997888rarrinfin 120591119896Then 120591infin le 120591119890 Now it is sufficient for us to show 120591infin = infinArguing by contradiction we assume that 120591infin = infin Thenthere exist constants 119879 gt 0 and 120576 isin (0 1) such that 119875(120591infin le119879) gt 120576 Therefore there is a constant 1198961 ge 1198960 such that119875(120591119896 le 119879) ge 120576 for all 119896 ge 1198961

Define a 1198622 function 119881 119877119899+1+ 997888rarr 119877+ by119881 (119878 1199091 1199092 119909119899) = 119878 minus 119886 minus 119886 ln 119878119886

+ 119899sum119894=1

(119909119894 minus 1 minus ln 1199091) (6)

where 119886 = min(119889119894 minus (1198780 + 1)119896119894)(119896119894 + 119886119894) (119894 = 1 2 119899) is apositive constant It is obvious that 119906 minus 1 minus ln 119906 ge 0 forall119906 ge 0Hence 119881(119878 1199091 119909119899) ge 0 By the Ito formula the followingequation can be derived

d119881 = [119878 minus 119886119878 (1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119878 minus 119886119878119899sum119894=1

1198861198941198781 + 119887119894119878119909119894 + 119886120590212minus 119899sum119894=1

119909119894 minus 1119909119894 (119889119894 + 119899sum119894=1

119896119894119909119894)119909119894 + 119899sum119894=1

119909119894 minus 11199091198941198861198941198781199091198941 + 119887119894119878

+ 119899sum119894=1

1205902119894+12 ] d119905 + (119878 minus 119886) 1205901d1198611 (119905)

+ 119899sum119894=1

(119909119894 minus 1) 120590119894+1d119861119894+1 (119905)

(7)

where

119871119881 = 119878 minus 119886119878 (1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119878 minus 119886119878119899sum119894=1

1198861198941198781 + 119887119894119878119909119894 + 119886120590212minus 119899sum119894=1

119909119894 minus 1119909119894 (119889119894 + 119899sum119894=1

119896119894119909119894)119909119894 + 119899sum119894=1

119909119894 minus 11199091198941198861198941198781199091198941 + 119887119894119878

+ 119899sum119894=1

1205902119894+12= (1198780 + 119886)(119889 + 119899sum

119894=1

119896119894119909119894) + 119899sum119894=1

1198861198861198941199091198941 + 119887119894119878 + 119886120590212

+ 119899sum119894=1

(119889119894 + 119899sum119894=1

119896119894119909119894) + 119899sum119894=1

1205902119894+12minus 119878(119889 + 119899sum

119894=1

119896119894119909119894) minus 1198861198780119878 (119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1

(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894 minus 119899sum119894=1

1198861198941198781 + 119887119894119878 (8)

Through calculations the following inequality can be ob-tained

119871119881 le (1198780 + 119886)(119889 + 119899sum119894=1

119896119894119909119894) + 119899sum119894=1

(119889119894 + 119899sum119894=1

119896119894119909119894)

+ 119886120590212 + 119899sum119894=1

1205902119894+12 + 119899sum119894=1

119886119886119894119909119894 minus 119899sum119894=1

119889119894119909119894le (1198780 + 119886) 119889 + 119899sum

119894=1

119889119894 + 119886120590212 + 119899sum119894=1

1205902119894+12+ 119899sum119894=1

[(1198780 + 119886) 119896119894 + 119896119894 + 119886119886119894 minus 119889119894] 119909119894= (1198780 + 119886) 119889 + 119899sum

119894=1

119889119894 + 119886120590212 + 119899sum119894=1

1205902119894+12+ 119899sum119894=1

(119896119894 + 119886119894) [119886 minus 119889119894 minus (1198780 + 1) 119896119894119896119894 + 119886119894 ]119909119894

(9)

Defining 119886 = min(119889119894 minus (1198780 + 1)119896119894)(119896119894 + 119886119894) (119894 = 1 2 119899)we have

119871119881 le (1198780 + 119886) 119889 + 119899sum119894=1

119889119894 + 119886120590212 + 119899sum119894=1

1205902119894+12 (10)

which yields

d119881 = 119871119881d119905 + (1198780 minus 119886) 1205901d1198611 (119905)+ 119899sum119894=1

120590119894+1 (119909119894 minus 1) d119861119894+1 (119905)

le (1198780 + 119886) 119889 + 119899sum119894=1

119889119894 + 119886120590212 + 119899sum119894=1

1205902119894+12+ (1198780 minus 119886) 1205901d1198611 (119905) + 119899sum

119894=1

120590119894+1 (119909119894 minus 1)d119861119894+1 (119905)

(11)

Integrating from 0 to 120591119896 and 119879 and taking expectation in bothsides of inequality (11) one can obtain

119864119881 (119878 (120591119896 and 119879) 119909119899 (120591119896 and 119879))le 119881 (119878 (0) 119909119899 (0))

+ ((1198780 + 119886) 119889 + 119899sum119894=1

119889119894 + 119886120590212 + 119899sum119894=1

1205902119894+12 )119879(12)

4 Complexity

Let Ω119896 = 120591119896 le 119879 for all 119896 ge 1198961 then 119875(Ω119896) ge120576 Based on the definition of the stopping time we havemi119878(119905) 1199091(119905) 119909119899(119905) le 1119896 or ma119878(119905) 1199091(119905) 119909119899(119905) ge119896 for all 120596 isin Ω119896 Hence there is a positive constant 119896 such that119881(119878(120591119896 and 119879) 119909119899(120591119896 and 119879)) ge 119896 Furthermore

119881 (119878 (0) 119909119899 (0))+ ((1198780 + 119886) 119889 + 119899sum

119894=1

119889119894 + 119886120590212 + 119899sum119894=1

1205902119894+12 )119879ge 119864119881 (119878 (120591119896 and 119879) 119909119899 (120591119896 and 119879))ge 119875 (Ω119896) 119881 (119878 (120591119896 and 119879) 119909119899 (120591119896 and 119879)) ge 120576119896

(13)

Setting 119896 997888rarr infin we have 119881(119878(0) 119909119899(0)) + ((1198780 + 119886)119889 +sum119899119894=1 119889119894+119886120590212+sum119899119894=1(1205902119894+12))119879 ge infin which is contradictorywith 119881(119878(0) 119909119899(0)) lt infin Hence 120591119890 = infin This completesthe proof

Remark 2 If we take 119879 = 119889119894119896119894 minus 1198780 (119894 = 1 2 119899) as athreshold value then the microorganisms in system (1) willsurvive as 119879 gt 1 and be extinct as 119879 le 13 The Principle of Competitive Exclusion

In this section we prove that stochastic turbidostat system (1)satisfies the principle of competitive exclusion For simplicitywe first define

⟨119909 (119905)⟩ = 1119905 int1199050119909 (119903) d119903 (14)

Lemma 3 The solution (119878(119905) 1199091(119905) 119909119899(119905)) of system (1)satisfies

lim119905997888rarrinfin

supln [119878 (119905) + sum119899119894=1 119909119894 (119905)]

ln 119905 le 1120579 119886119904 (15)

with any initial value (119878(0) 1199091(0) 119909119899(0)) isin 119877119899+1+ Here 120579is a positive constant satisfying

120582 = min 119889 1198891 119889119899 minus 119899sum119894=1

1198961198941198780

minus (120579 minus 12 or 0) (12059021 or or 1205902119899) gt 0(16)

Proof Let

119906 (119905) = 119878 (119905) + 119899sum119894=1

119909119894 (119905) (17)

and

119882(119906) = (1 + 119906)120579 (18)

where 120579 is a nonnegative constant decided later By the Itoformula we derive

d119906 = [(1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1

(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894] d119905 + 1205901119878d1198611 (119905)

+ 119899sum119894=1

120590119894+1119909119894d119861119894+1 (119905)

(19)

We can further obtain

d119882 = 120579 (1 + 119906)120579minus2(1 + 119906) [(1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1

(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894] + 120579 minus 12 (120590211198782

+ 119899sum119894=1

1205902119894+11199092119894) d119905 + 120579 (1 + 119906)120579minus1 [1205901119878d1198611 (119905)

+ 119899sum119894=1

120590119894+1119909119894d119861119894+1 (119905)]

(20)

Denote

119871119882(119906) = 120579 (1 + 119906)120579minus2 (1 + 119906)

sdot [(1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894) minus 119899sum119894=1

(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894]

+ 120579 minus 12 (120590211198782 +119899sum119894=1

1205902119894+11199092119894) = 120579 (1 + 119906)120579minus2(1

+ 119906) [1198780 (119889 + 119899sum119894=1

119896119894119909119894) minus 119878(119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1

(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894] + 120579 minus 12 (120590211198782

+ 119899sum119894=1

1205902119894+11199092119894) le 120579 (1 + 119906)120579minus2 (1 + 119906) [1198780119889

+ (1198780 119899sum119894=1

119896119894 minus min 119889 1198891 119889119899) 119906]

+ 120579 minus 12 (120590211198782 +119899sum119894=1

1205902119894+11199092119894) le 120579 (1 + 119906)120579minus2

sdot [ 119899sum119894=1

1198961198941198780 minus min 119889 1198891 119889119899

+ (120579 minus 12 or 0) (12059021 or or 1205902119899)] 1199062 + (1198780119889

Complexity 5

+ 119899sum119894=1

1198961198941198780 minus min 119889 1198891 119889119899) 119906 + (119889 + 119899sum119894=1

119896119894)

sdot 1198780 = 120579 (1 + 119906)120579minus2minus1205821199062 + (1198780119889 + 119899sum119894=1

1198961198941198780

minus min 119889 1198891 119889119899) 119906 + (119889 + 119899sum119894=1

119896119894)1198780 (21)

where

120582 = min 119889 1198891 119889119899 minus 119899sum119894=1

1198961198941198780


Let

119860 = (119889 + 119899sum119894=1

119896119894)1198780 minus min 119889 1198891 119889119899 (23)

be a constant Then

d119882(119906)le 120579 (1 + 119906)120579minus2minus1205821199062 + 119860119906 + (119889 + 119899sum

119894=1

119896119894)1198780 d119905

+ 120579 (1 + 119906)120579minus1 [1205901119878d1198611 (119905) + 119899sum119894=1

120590119894+1119909119894d119861119894+1 (119905)] (24)

For 0 lt 119902 lt 120579120582 we haved [e119902119905119882(119906)] = 119871 [e119902119905119882(119906)] d119905 + e119902119905120579 (1 + 119906)120579minus1

sdot [1205901119878d1198611 (119905) + 119899sum119894=1

120590119894+1119909119894d119861119894+1 (119905)] (25)

where

119871 [e119902119905119882(119906)] = 119902e119902119905119882(119906) + e119902119905119871119882 (119906) le 119902e119902119905 (1+ 119906)120579 + e119902119905120579 (1 + 119906)120579minus2 minus1205821199062 + 119860119906

+ (119889 + 119899sum119894=1

119896119894)1198780 = e119902119905120579 (1 + 119906)120579minus2

sdot minus (120582 minus 119902120579) 1199062 + [119860 + 2119902120579 ] 119906 + 1198780(119889 + 119899sum119894=1

119896119894)

+ 119902120579

(26)

Define a function 119891(119906) by119891 (119906) = (1 + 119906)120579minus2minus(120582 minus 119902120579) 1199062 + [119860 + 2119902120579 ] 119906

+ 1198780 (119889 + 119899sum119894=1

119896119894) + 119902120579 119906 isin 119877+(27)

and119865 fl sup119906isin119877+

119891 (119906) + 1 (28)

which gives

119871 [e119902119905119882(119906)] le 120579e119902119905119865 (29)

Integrating from 0 to 119905 and taking expectation for (25) onehas

119864 [e119902119905119882(119906)] = 119882 (119906 (0)) + 119864int1199050119871 [e119902119905119882(119906)] d119905 (30)

Then it is easy to obtain

119864 [e119902119905119882(119906)] le (1 + 119906 (0))120579 + 119864int1199050e119902119905120579119865d119905

= (1 + 119906 (0))120579 + 120579119865119902 e119902119905(31)

which leads to


sup119864 [(1 + 119906 (119905))120579] le 120579119865119902 fl 1198650 119886119904 (32)

119906(119905) is a continuous function so there is a constant 119872 gt 0such that

119864 [(1 + 119906 (119905))120579] le 119872 119905 ge 0 (33)

According to (24) for small enough 120575 gt 0 119896 = 1 2 integrating from 119896120575 to (119896 + 1)120575 and taking expectation wecan get

119864[ sup119896120575le119905le(119896+1)120575

(1 + 119906 (119905))120579]le 119864 [(1 + 119906 (119896120575))120579] + 1198681 + 1198682 le 119872 + 1198681 + 1198682

(34)

where

1198681 = 119864[ sup119896120575le119905le(119896+1)120575

1003816100381610038161003816100381610038161003816100381610038161003816int119905

119896120575120579 (1 + 119906)120579minus2

sdot minus1205821199062 + 119860119906 + 1198780(119889 + 119899sum119894=1

119896119894) d1199051003816100381610038161003816100381610038161003816100381610038161003816]

le 119864[ sup119896120575le119905le(119896+1)120575

1003816100381610038161003816100381610038161003816100381610038161003816int119905

119896120575120579 (1 + 119906)120579minus2

sdot 1198780(119889 + 119899sum119894=1

119896119894)119906 + 1198780(119889 + 119899sum119894=1

119896119894) d1199051003816100381610038161003816100381610038161003816100381610038161003816]

= 119864[ sup119896120575le119905le(119896+1)120575

1003816100381610038161003816100381610038161003816100381610038161003816int119905

119896120575120579 (1 + 119906)120579minus2

6 Complexity

sdot 1198780 (119889 + 119899sum119894=1

119896119894) (1 + 119906) d1199051003816100381610038161003816100381610038161003816100381610038161003816] le 1198780 (119889

+ 119899sum119894=1

119896119894)120579119864int(119896+1)120575119896120575

(1 + 119906)120579 d119905

le 1198621120575119864[ sup119896120575le119905le(119896+1)120575

(1 + 119906 (119905))120579] (35)

here

1198621 = 1198780(119889 + 119899sum119894=1

119896119894)120579 (36)

Applying Burkholder-Davis-Gundy inequality [41] we have

1198682 = 119864[ sup119896120575le119905le(119896+1)120575

1003816100381610038161003816100381610038161003816100381610038161003816int119905

119896120575120579 (1 + 119906)120579minus1

sdot 1205901119878d1198611 (119905) + 119899sum119894=1

120590119894+1119909119894d119861119894+1 (119905)1003816100381610038161003816100381610038161003816100381610038161003816]

le radic32119864[int(119896+1)120575119896120575

1205792 (1 + 119906)2120579minus2(120590211198782

+ 119899sum119894=1

1205902119894+11199092119894) d119905]12

le radic32119864[int(119896+1)120575119896120575

1205792 (1

+ 119906)2120579minus2 (120590211199062 +119899sum119894=1

1205902119894+11199062) d119905]12

le radic3212057912057512119864[ sup119896120575le119905le(119896+1)120575

(12059021 or or 1205902119899) (1

+ 119906)2120579]12 le radic3212057912057512 (12059021 or or 1205902119899)12

sdot 119864 [ sup119896120575le119905le(119896+1)120575

(1 + 119906)120579]

(37)

Hence we have

119864[ sup119896120575le119905le(119896+1)120575

(1 + 119906)120579]le 119864 [(1 + 119906 (119896120575))120579]

+ [1198621120575 + radic3212057912057512 (12059021 or or 1205902119899)12]times 119864[ sup119896120575le119905le(119896+1)120575

(1 + 119906)120579]

(38)

Particularly one can choose 120575 gt 0 such that

1198621120575 + radic3212057912057512 (12059021 or or 1205902119899)12 le 12 (39)

which yields

119864[ sup119896120575le119905le(119896+1)120575

(1 + 119906)120579] le 2119864 [(1 + 119906 (119896120575))120579] le 2119872 (40)

For any 120573 gt 0 the following inequality can be obtained byChebyshevrsquos inequality [41]

119875 sup119896120575le119905le(119896+1)120575

(1 + 119906 (119905))120579 ge (119896120575)1+120573

le 119864 [sup119896120575le119905le(119896+1)120575 (1 + 119906 (119905))120579](119896120575)1+120573 le 2119872

(119896120575)1+120573 (41)

Using Borel-Cantelli Lemma [41] for all 120596 isin Ω andsufficiently large 119896 we have

sup119896120575le119905le(119896+1)120575

(1 + 119906 (119905))120579 le (119896120575)1+120573 (42)

Therefore there is a 1198960(120596) for all 120596 isin Ω and 119896 ge 1198960 the aboveequation holds In addition for all 120596 isin Ω when 119896 ge 1198960 and119896120575 le 119905 le (119896 + 1)120575

ln (1 + 119906 (119905))120579ln 119905 le ln (119896120575)1+120573

ln 119905 le ln (119896120575)1+120573ln (119896120575) = 1 + 120573 (43)

which means


sup ln (1 + 119906 (119905))120579ln 119905 le 1 + 120573 119886119904 (44)

When 120573 997888rarr 0lim119905997888rarrinfin

sup ln (1 + 119906 (119905))ln 119905 le 1120579 119886119904 (45)

where 120579 is determined by (22) Therefore


supln [119878 (119905) + sum119899119894=1 119909119894 (119905)]

ln 119905 = lim119905997888rarrinfin

sup ln 119906 (119905)ln 119905

le lim119905997888rarrinfin

sup ln (1 + 119906 (119905))ln 119905 le 1120579 119886119904

(46)

This completes the proof

Lemma 4 If min119889 1198891 119889119899 gt sum119899119894=1 1198961198941198780 + ((120579 minus 1)2 or0)(12059021 or or 1205902119899) then the solution (119878(119905) 1199091(119905) 119909119899(119905)) ofsystem (1) with initial value (119878(0) 1199091(0) 119909119899(0)) isin 119877119899+1+satisfies


int1199050(119878 (119904) minus 1205821) d1198611 (119904)119905 = 0 119886119904


int1199050(1199091 (119904) minus 119909lowast1 ) d1198612 (119904)119905 = 0 119886119904


int1199050119909119894 (119904) d119861119894+1 (119904)119905 = 0 119886119904

(119894 = 2 3 119899)

(47)

where 1205821 is the break-even concentration of 1199091(119905) and 119909lowast1 is theequilibrium concentration of 1199091(119905)

Complexity 7

Proof Define

119883 (119905) = int1199050(119878 (119904) minus 1205821)d1198611 (119904)

119884 (119905) = int1199050(1199091 (119904) minus 119909lowast1 ) d1198612 (119904)

119885119894 (119905) = int1199050119909119894 (119904) d119861119894+1 (119904) 119894 = 2 3 119899

(48)

It follows from min119889 1198891 119889119899 gt sum119899119894=1 1198961198941198780 + ((120579 minus 1)2 or0)(12059021 or or 1205902119899) Lemma 3 and Burkholder-Davis-Gundyinequality [41] that

119864[ sup0le119904le119905

|119883 (119904)|120579] le 119862120579119864[int1199050(119878 (119904) minus 1205821)2 d119904]

12

le 1198621205791199051205792119864[ sup0le119904le119905

(119878 (119904) minus 1205821)120579]le 1198621205791199051205792119864[ sup

0le119904le119905(1 + 119906 (119905))120579]

le 21198721198621205791199051205792

(49)

By Doobrsquos martingale inequality [41] for arbitrary 120572 gt 0 itfollows that

119875120596 sup119896120575le119905le(119896+1)120575

|119883 (119905)|120579 gt (119896120575)1+120572+1205792

le 119864 [|119883 ((119896 + 1) 120575)|120579](119896120575)1+120572+1205792 le 2119872119862120579(119896120575)1+120572 (

(119896 + 1) 120575119896120575 )1205792

le 21+1205792119872119862120579(119896120575)1+120572

(50)

By Borel-Cantelli Lemma [41] for all 120596 isin Ω and sufficientlarge 119896 we derive

sup119896120575le119905le(119896+1)120575

|119883 (119905)|120579 le (119896120575)1+120572+1205792 (51)

Thus there exists a positive constant 1198961198830(120596) such that for all119896 ge 1198961198830 and 119896120575 le 119905 le (119896 + 1)120575ln |119883 (119905)|120579

ln 119905 le ln (119896120575)1+120572+1205792ln 119905 le ln (119896120575)1+120572+1205792

ln (119896120575) (52)

which implies

ln |119883 (119905)|120579ln 119905 le 1 + 120572 + 1205792 (53)

Taking superior limit leads to


sup ln |119883 (119905)|ln 119905 le 1 + 120572 + 1205792120579 (54)

Sending 120572 997888rarr 0 we havelim119905997888rarrinfin

sup ln |119883 (119905)|ln 119905 le 12 + 1120579 (55)

Then there exist a constant 1198791015840(120596) gt 0 and a set Ω120578 for smallenough 0 lt 120578 lt 12 minus 1120579 such that 119875(Ω120578) ge 1 minus 120578 Thus for119905 ge 1198791015840(120596) 120596 isin Ω120578 we have

ln |119883 (119905)|ln 119905 le 12 + 1120579 + 120578 (56)

which means


sup |119883 (119905)|119905 le lim119905997888rarrinfin

sup 11990512+1120579+120578119905 = 0 (57)

Together with


inf |119883 (119905)|119905 = lim119905997888rarrinfin

inf

100381610038161003816100381610038161003816int1199050 (119878 (119904) minus 1205821) d1198611 (119904)100381610038161003816100381610038161003816119905ge 0

(58)

we derive


|119883 (119905)|119905 = 0 119886119904 (59)

and


int1199050(119878 (119904) minus 1205821) d1198611 (119904)119905 = 0 119886119904 (60)

Applying the same methods we can prove that




int1199050119909119894 (119905) d119861119894+1 (119904)119905 = 0 119886119904

119894 = 2 3 119899(61)


Definition 5 Stochastic equation

d120601 (119905) = [(1198780 minus 119878 (119905)) (119889 + 119899sum119894=1

119896119894119909119894)] d119905+ 1205901120601 (119905) d1198611 (119905)

(62)

has a solution 120601(119905) weakly converging to the distribution VHere V is a probability measure of 1198771+ such that intinfin

0119911V(d119911) =

1198780 Particularly V has density (119860120590211199112119901(119911))minus1 where 119860 is anormal constant and

119901 (119911) = exp [2 (119889 + sum119899119894=1 119896119894119909119894) 119878012059021119911 ]

sdot exp [minus2 (119889 + sum119899119894=1 119896119894119909119894) 119878012059021 ] 1199112(119889+sum119899119894=1 119896119894119909119894)12059021 (63)

8 Complexity

Theorem 6 (119878(119905) 1199091(119905) 119909119899(119905)) is the solution of system(1) with any initial value (119878(0) 1199091(0) 119909119899(0)) isin 119877119899+1+ Moreover if

0 lt 1205821 lt 1205822 le sdot sdot sdot le 120582119899 lt 1198780min 119889 1198891 119889119899

gt 119899sum119894=1

1198961198941198780 + (120579 minus 12 or 0) (12059021 or or 1205902119899) 11988611989412058211 + 1198871198941205821 +

1198861198942radic119887119894 (1 + 1198871198941205821) ( 119867119889 + 119873sum119899119894=1 119896119894)12

le (119889119894 + 119873 119899sum119894=1

119896119894) + 1205902119894+12 (119894 = 2 3 119899)

and 1198861 intinfin0

119909V (d119909)1 + 1198871119909 gt 120590222 + (119889 + 119872 119899sum119894=1

119896119894)

(64)

then we havelim119905997888rarrinfin

inf ⟨1199091 (119905)⟩ gt 0 119886119904lim119905997888rarrinfin

supln119909119894 (119905)119905 le 0 119886119904 (119894 = 2 119899) (65)

where 119872 = max1199091 119909119899 119873 = min1199091 119909119899 120582119894 (119894 =1 2 119899) is the break-even concentration of 119909119894(119905) 119867 isdetermined later by (74) (1205821 119909lowast1 0 0) is the boundaryequilibrium of the corresponding deterministic model and V isdefined in Definition 5

Proof The deterministic system corresponding to (1) has anequilibrium (1205821 119909lowast1 0 0) under the condition of 1205821 lt 1198780where 1205821 and 119909lowast1 satisfy

(1198780 minus 1205821) (119889 + 1198961119909lowast1 ) = 11988611205821119909lowast11 + 11988711205821 1198891 + 1198961119909lowast1 = 119886112058211 + 11988711205821

(66)

Define 1198622 function 119881 119877119899+1+ 997888rarr 119877+ by119881 (119878 1199091 119909119899)

= 119878 minus 1205821 minus 120582 ln 1198781205821+ (1 + 11988711205821) (1199091 minus 119909lowast1 minus 119909lowast1 ln 1199091119909lowast1 )+ 119899sum119894=2

(1 + 1198871198941205821) 119909119894

(67)

Then by the Ito formula we get

d119881 = 119878 minus 1205821119878 [(1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1

1198861198941198781 + 119887119894119878119909119894] d119905 + 1205901119878d1198611 (119905) + 12 12058211198782 120590211198782d119905

+ [(1 + 11988711205821) minus (1 + 11988711205821) 119909lowast11199091 ]

sdot [minus(1198891 + 119899sum119894=1

119896119894119909119894)1199091 + 11988611198781 + 11988711198781199091] d119905

+ 12059021199091d1198612 (119905) + 12(1 + 11988711205821) 119909lowast111990921 1205902211990921d119905 + 119899sum

119894=2

(1

+ 1198871198941205821) [minus(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894 + 1198861198941198781 + 119887119894119878119909119894] d119905

+ 120590119894+1119909119894d119861119894+1 (119905) = 119871119881d119905 + (119878 minus 1205821) 1205901d1198611 (119905)

+ [(1 + 11988711205821) minus (1 + 11988711205821) 119909lowast11199091 ] 12059021199091d1198612 (119905)+ 119899sum119894=2

(1 + 1198871198941205821) 120590119894+1119909119894d119861119894+1 (119905) (68)

where

119871119881 = 119878 minus 1205821119878 [(1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1

1198861198941198781 + 119887119894119878119909119894] + 12120582112059021 + 12 (1 + 11988711205821) 119909lowast112059022+ [(1 + 11988711205821) minus (1 + 11988711205821) 119909lowast11199091 ]

sdot [minus(1198891 + 119899sum119894=1

119896119894119909119894)1199091 + 11988611198781 + 11988711198781199091]

+ 119899sum119894=2

(1 + 1198871198941205821) [minus(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894 + 1198861198941198781 + 119887119894119878119909119894]

(69)

According to (66) we have

1198780 = 1205821 + 11988611205821119909lowast1(1 + 11988711205821) (119889 + 1198961119909lowast1 ) 1198891 = 119886112058211 + 11988711205821 minus 1198961119909lowast1

(70)

It follows from some calculations that119871119881= minus(119889 + sum119899119894=1 119896119894119909119894) (119878 minus 1205821)2119878

+ 11988611205821119909lowast1 (119878 minus 1205821)sum119899119894=2 119896119894119909119894119878 (1 + 11988711205821) (119889 + 1198961119909lowast1 )+ 11988611198961119909lowast1 (119878 minus 1205821) (1199091 minus 119909lowast1 )(1 + 1198871119878) (119889 + 1198961119909lowast1 )

Complexity 9

minus 1198961 (1 + 11988711205821) (1199091 minus 119909lowast1 )2minus (1 + 11988711205821) (1199091 minus 119909lowast1 )

119899sum119894=2

119896119894119909119894+ 119899sum119894=2

119909119894 [ 11988611989412058211 + 1198871198941205821 minus (119889119894 + 119899sum119894=1

119896119894119909119894)] + 12120582112059021+ 12 (1 + 11988711205821) 119909lowast112059022

le minus(119889 + sum119899119894=1 119896119894119909119894) (119878 minus 1205821)2119878+ [ 11988611205821119909lowast1(1 + 11988711205821) (119889 + 1198961119909lowast1 ) + (1 + 11988711205821) 119909lowast1]

119899sum119894=2

119896119894119909119894

+ 11988611198961119909lowast11199091119889 + 1198961119909lowast1 + 119886111989611205821 (119909lowast1 )2119889 + 1198961119909lowast1+ 119899sum119894=2

119909119894 [minus 119899sum119894=1

119896119894 (119909119894 minus 119909lowast119894 )] + 12120582112059021+ 12 (1 + 11988711205821) 119909lowast112059022

(71)

Set

119872 = max 1199091 119909119899 119873 = min 1199091 119909119899 (72)

then

119871119881 le minus(119889 + 119873sum119899119894=1 119896119894) (119878 minus 1205821)2119878+ [ 11988611205821119909lowast1(1 + 11988711205821) (119889 + 1198961119909lowast1 ) + (1 + 11988711205821) 119909lowast1]119872 119899sum

119894=2

119896119894

+ 11988611198961119909lowast1119872119889 + 1198961119909lowast1 + 119886111989611205821 (119909lowast1 )2119889 + 1198961119909lowast1 + 119899sum119894=2

119872 119899sum119894=1

119896119894119909lowast119894 + 12sdot 120582112059021 + 12 (1 + 11988711205821) 119909lowast112059022= minus(119889 + 119873sum119899119894=1 119896119894) (119878 minus 1205821)2119878 + 119867

(73)

where

119867= [ 11988611205821119909lowast1(1 + 11988711205821) (119889 + 1198961119909lowast1 ) + (1 + 11988711205821) 119909lowast1]119872 119899sum

119894=2

119896119894

+ 11988611198961119909lowast1119872119889 + 1198961119909lowast1 + 119886111989611205821 (119909lowast1 )2119889 + 1198961119909lowast1

+ (119899 minus 1)119872 119899sum119894=1

119896119894119909lowast119894 + 12120582112059021+ 12 (1 + 11988711205821) 119909lowast112059022

(74)

Integrating (68) from 0 to 119905 and dividing by 119905 on both sidesone can obtain

119881(119905) minus 119881 (0)119905le minus119889 + 119873sum119899119894=1 119896119894119905 int119905

0

(119878 (119904) minus 1205821)2119878 (119904) d119904 + 119867+ 1205901 1119905 int119905

0(119878 (119904) minus 1205821) d1198611 (119904)

+ 1205902 (1 + 11988711205821) 1119905 int1199050(1199091 (119904) minus 119909lowast1 ) d1198612 (119905)

+ 119899sum119894=2

(1 + 1198871198941205821) 120590119894+1 int1199050119909119894 (119904) d119861119894+1 (119905)

(75)

According to Lemma 4 it follows that


sup 1119905 int1199050

(119878 (119904) minus 1205821)2119878 (119904) d119904 le 119867119889 + 119873sum119899119894=1 119896119894 (76)

Define

119881119894 (119909119894 (119905)) = ln119909119894 (119905) 119894 = 2 3 119899 (77)

Using the Ito formula we have

d119881119894 = [minus(119889119894 + 119899sum119894=1

119896119894119909119894) + 1198861198941198781 + 119887119894119878 minus 1205902119894+12 ] d119905

+ 120590119894+1d119861119894+1 (119905) le [ 119886119894 (119878 minus 1205821)(1 + 119887119894119878) (1 + 1198871198941205821)minus (119889119894 + 119873 119899sum

119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12 ] d119905

+ 120590119894+1d119861119894+1 (119905)

(78)

Then integrating (78) from 0 to 119905 on both sides and dividingall by 119905 we can get

119881119894 (119905) minus 119881119894 (0)119905le 1198861198941 + 1198871198941205821 ⟨

119878 (119905) minus 12058211 + 119887119894119878 (119905)⟩ minus (119889119894 + 119873 119899sum119894=1

119896119894)

10 Complexity

+ 11988611989412058211 + 1198871198941205821 minus1205902119894+12 + 120590119894+1119861119894+1 (119905)119905

le 1198861198942radic119887119894 (1 + 1198871198941205821) ⟨1003816100381610038161003816119878 (119905) minus 12058211003816100381610038161003816radic119878 (119905) ⟩minus (119889119894 + 119873 119899sum

119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 120590119894+1119861119894+1 (119905)119905le 1198861198942radic119887119894 (1 + 1198871198941205821) [1119905 int119905

0

(119878 (119905) minus 1205821)2119878 (119905) d119904]12

minus (119889119894 + 119873 119899sum119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 120590119894+1119861119894+1 (119905)119905 (79)

which meansln119909119894 (119905)119905

le ln119909119894 (0)119905+ 1198861198942radic119887119894 (1 + 1198871198941205821) [1119905 int119905

0

(119878 (119905) minus 1205821)2119878 (119905) d119904]12

minus (119889119894 + 119873 119899sum119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 120590119894+1119861119894+1 (119905)119905

(80)

Based on (76) lim119905997888rarrinfin (119861119894+1(119905)119905) = 0 and lim119905997888rarrinfin (ln119909119894(0)119905) = 0 we know that


supln119909119894 (119905)119905

le minus(119889119894 + 119873 119899sum119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 1198861198942radic119887119894 (1 + 1198871198941205821) ( 119867119889 + 119873sum119899119894=1 119896119894)12

(119894 = 2 119899)

(81)

If

11988611989412058211 + 1198871198941205821 +1198861198942radic119887119894 (1 + 1198871198941205821) ( 119867119889 + 119873sum119899119894=1 119896119894)

12

le (119889119894 + 119873 119899sum119894=1

119896119894) + 1205902119894+12 (82)

then we have


supln119909119894 (119905)119905 le 0 119886119904 (119894 = 2 119899) (83)

that is 119909119894(119905) (119894 = 2 119899) tends to extinction exponentiallyalmost surely

On the basis of classical comparison theoremof stochasticdifferential equation the solution 119878(119905) of system (1) and thesolution 120601(119905) of system (62) satisfy 119878(119905) le 120601(119905) According tothe Ito formula it follows thatd ln 119878 (119905)

= [(1198780 minus 119878) (119889 + sum119899119894=1 119896119894119909119894)119878 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 minus 120590212 ] d119905+ 1205901d1198611 (119905)

d ln120601 (119905)= [(1198780 minus 120601) (119889 + sum119899119894=1 )120601 minus 120590212 ] d119905 + 1205901d1198611 (119905)

(84)

Integrating the above two equations from 0 to 119905 on both sidesand dividing them by 119905 we obtainln 119878 (119905)119905= ln 119878 (0)119905

+ 1119905 int1199050[(1198780 minus 119878) (119889 + sum119899119894=1 119896119894119909119894)119878 minus 119899sum

119894=1

1198861198941199091198941 + 119887119894119878] d119905

minus 120590212 + 1119905 int11990501205901d1198611 (119905)

= (119889 + 119899sum119894=1

119896119894119909119894)1198780 ⟨1119878⟩ minus 1198861⟨ 11990911 + 1198871119878⟩

minus 119899sum119894=2

119886119894⟨ 1199091198941 + 119887119894119878⟩ minus (119889 + 119899sum119894=1

119896119894119909119894) minus 120590212+ 12059011198611 (119905)119905 + ln 119878 (0)119905

ln120601 (119905)119905= ln120601 (0)119905 + 1119905 int119905

0

(1198780 minus 120601) (119889 + sum119899119894=1 119896119894119909119894)120601 d119905 minus 120590212+ 12059011198611 (119905)119905

= (119889 + 119899sum119894=1

119896119894119909119894)1198780⟨1120601⟩ minus (119889 + 119899sum119894=1

119896119894119909119894) minus 120590212+ 12059011198611 (119905)119905 + ln120601 (0)119905

(85)

Complexity 11

Therefore

0 ge ln 119878 (119905) minus ln120601 (119905)119905= (119889 + 119899sum

119894=1

119896119894119909119894)1198780⟨1119878 minus 1120601⟩ minus 1198861⟨ 11990911 + 1198871119878⟩

minus 119899sum119894=2

119886119894⟨ 1199091198941 + 119887119894119878⟩

ge (119889 + 119873 119899sum119894=1

119896119894)1198780⟨1119878 minus 1120601⟩ minus 1198861 ⟨1199091⟩

minus 119899sum119894=2

119886119894 ⟨119909119894⟩

(86)

which implies that

⟨1119878 minus 1120601⟩

le 1(119889 + 119873sum119899119894=1 119896119894) 1198780 (1198861 ⟨1199091⟩ + 119899sum119894=2

119886119894 ⟨119909119894⟩) (87)

Next applying the Ito formula to ln1199091(119905) one deducesd ln1199091 (119905) = [minus(1198891 + 119899sum

119894=1

119896119894119909119894) + 11988611198781 + 1198871119878 minus 120590222 ] d119905+ 1205902d1198612 (119905)

ge [minus(1198891 + 119872 119899sum119894=1

119896119894) + 11988611198781 + 1198871119878 minus 120590222 ] d119905+ 1205902d1198612 (119905)

(88)

Integrating the above inequality from 0 to 119905 on both sides anddividing all by 119905 we have

ln1199091 (119905)119905ge ln1199091 (0)119905 minus 120590222 minus (119889 + 119872 119899sum

119894=1

119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩ minus 1198861⟨ 120601 minus 119878(1 + 1198871120601) (1 + 1198871119878)⟩

+ 12059021198612 (119905)119905ge ln1199091 (0)119905 minus 120590222 minus (119889 + 119872 119899sum

119894=1

119896119894)+ 1198861⟨ 1206011 + 1198871120601⟩ minus 119886111988721 ⟨1119878 minus 1120601⟩ + 12059021198612 (119905)119905

ge ln1199091 (0)119905 minus 120590222 minus (119889 + 119872 119899sum119894=1

119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩

minus 119886111988721 (119889 + 119873sum119899119894=1 119896119894) 1198780 (1198861 ⟨1199091⟩ + 119899sum119894=2

119886119894 ⟨119909119894⟩)+ 12059021198612 (119905)119905


119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩ minus 11988621 ⟨1199091⟩11988721 (119889 + 119873sum119899119894=1 119896119894) 1198780+ 12059021198612 (119905)119905

(89)

According to Lemma 3 we have

ln1199091 (119905)119905 le lim119905997888rarrinfin

sup ln 1199091 (119905)119905 le 0 (90)

which yields

11988621 ⟨1199091⟩11988721 (119889 + 119873sum119899119894=1 119896119894) 1198780 ge ln1199091 (0)119905 minus 120590222minus (119889 + 119872 119899sum

119894=1

119896119894)+ 1198861⟨ 1206011 + 1198871120601⟩ + 12059021198612 (119905)119905

(91)

that is


inf ⟨1199091 (119905)⟩ ge 11988721 (119889 + 119873sum119899119894=1 119896119894) 119878011988621 [minus120590222minus (119889 + 119872 119899sum

119894=1

119896119894) + 1198861 intinfin0

119909V (d119909)1 + 1198871119909 ] (92)

Therefore if

1198861 intinfin0

119909V (d119909)1 + 1198871119909 gt 120590222 + (119889 + 119872 119899sum119894=1

119896119894) (93)

then


inf ⟨1199091 (119905)⟩ gt 0 119886119904 (94)


Theorem 7 System (1) with (119878(0) 1199091(0) 119909119899(0)) isin 119877119899+1+admits the positive solution (119878(119905) 1199091(119905) 119909119899(119905)) Further-more if

119886119894 intinfin0

119909V (d119909)1 + 119887119894119909 lt 119889119894 + 119873 119899sum119894=1

119896119894 + 1205902119894+12 119894 = 1 2 119899 (95)

12 Complexity

then


int1199050119878 (119904) d119904119905 = 1198780 119886119904


supln 119909119894 (119905)119905 lt 0 119886119904 119894 = 1 2 119899

(96)

and 119878(119905) weakly converges to distribution V here V is defined asin Definition 5 which means the biomass of all species in thevessel will be extinct exponentially

Proof It is obvious that 119878(119905) le 120601(119905) by the classical compar-ison theorem of stochastic differential equation where 119878(119905)and 120601(119905) are the solutions of system (1) and (62) separatelyApplying the Ito formula to ln(119909119894(119905)) (119894 = 1 2 119899) wederive that

d ln 119909119894 (119905) = (minus119889119894 minus 119899sum119894=1

119896119894119909119894 + 1198861198941198781 + 119887119894119878 minus 1205902119894+12 )d119905+ 120590119894+1d119861119894+1 (119905)

le (minus119889119894 minus 119873 119899sum119894=1

119896119894 + 1198861198941206011 + 119887119894120601 minus 1205902119894+12 )d119905+ 120590119894+1d119861119894+1 (119905)

(97)

Since 119886119894119878(1 + 119887119894119878) is a monotonous increasing function and0 lt 119878(119905) le 120601(119905)ln119909119894 (119905)119905 le (minus119889119894 minus 119873 119899sum

119894=1

119896119894 minus 1205902119894+12 ) + 119886119894⟨ 1206011 + 119887119894120601⟩+ 120590119894+1119861119894+1 (119905)119905 + ln 119909119894 (0)119905

(98)

Using the conditions that lim119905997888rarrinfin (119861119894+1(119905)119905) = 0lim119905997888rarrinfin (ln119909119894(0)119905) = 0 and

119886119894 intinfin0

119909V (d119909)1 + 119887119894119909 lt 119889119894 + 119873 119899sum119894=1

119896119894 + 1205902119894+12 119894 = 1 2 119899 (99)

we know that


supln119909119894 (119905)119905 le 0 119894 = 1 2 119899 (100)

which means 119909119894(119905) (119894 = 1 2 119899) tends to zero exponen-tially almost surely

Nowwe prove that (62) is stable in distribution Set119884(119905) =120601(119905) minus 1198780 then we have

d119884 = minus(119889 + 119899sum119894=1

119896119894119909119894)119884d119905 + 1205901 (1198780 + 119884) d1198611 (119905) (101)

According to Theorem 21 in [42] we know that diffusionprocess 119884(119905) is stable in distribution when 119905 997888rarr infinTherefore 120601(119905) is also stable in distribution We further provethat 119878(119905) is stable in distribution as 119905 997888rarr infin We define

the following stochastic process 119878120576(119905) with the initial value119878120576(0) = 119878(0) byd119878120576 = [(1198780 minus 119878120576)(119889 + 119899sum

119894=1

119896119894119909119894) minus 120576119878120576] d119905+ 1205901119878120576d1198611 (119905)

(102)

Then

d (119878 minus 119878120576) = [(120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878) 119878

minus (119889 + 120576 + 119899sum119894=1

119896119894119909119894) (119878 minus 119878120576)] d119905 + 1205901 (119878minus 119878120576) d1198611 (119905)

(103)

The solution of (103) is

119878 (119905) minus 119878120576 (119905)= exp[12059011198611 (119905) minus (119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119905]

sdot int1199050exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904

(104)

According to Theorem 6 when 119905 997888rarr infin 119909119894 997888rarr 0 (119894 =2 3 119899) 1199091 997888rarr 119909lowast1 gt 0 and 119878 997888rarr 1205821 lt 1198780 Thereforefor almost all 120596 isin Ω exist119879 = 119879(120596) such that

120576 gt 119899sum119894=1

119886119894119909119894 (119905)1 + 119887119894119878 (119905) forall119905 ge 119879 (105)

In addition for all 120596 isin Ω if 119905 gt 119879 it follows that119878 (119905) minus 119878120576 (119905) = exp[12059011198611 (119905) minus (119889 + 119899sum

119894=1

119896119894119909119894 + 120576

+ 120590212 ) 119905]

sdot int1198790exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904

+ int119905119879exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

Complexity 13

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904 ge exp[12059011198611 (119905)

minus (119889 + 119872 119899sum119894=1

119896119894 + 120576 + 120590212 ) 119905]

sdot int1198790exp[(119889 + 119873 119899sum

119894=1

119896119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904(106)

and when 119905 997888rarr infin

exp[12059011198611 (119905) minus (119889 + 119872 119899sum119894=1

119896119894 + 120576 + 120590212 ) 119905] 997888rarr 01003816100381610038161003816100381610038161003816100381610038161003816int119879

0exp[(119889 + 119873 119899sum

119894=1

119896119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d1199041003816100381610038161003816100381610038161003816100381610038161003816 lt infin

(107)

which implies that


(119878 (119905) minus 119878120576 (119905)) ge lim119905997888rarrinfin

inf (119878 (119905) minus 119878120576 (119905))ge 0 119886119904 (108)

Now we consider

d (120601 (119905) minus 119878120576 (119905))= [120576119878120576 (119905) minus (119889 + 119899sum

119894=1

119896119894119909119894) (120601 (119905) minus 119878120576 (119905))] d119905+ 1205901 (120601 (119905) minus 119878120576 (119905)) d1198611 (119905)

le [120576119878120576 (119905) minus (119889 + 119873 119899sum119894=1

119896119894)(120601 (119905) minus 119878120576 (119905))] d119905+ 1205901 (120601 (119905) minus 119878120576 (119905)) d1198611 (119905)

(109)

Integrating it from 0 to 119905 one can derive

120601 (119905) minus 119878120576 (119905)le int1199050[120576119878120576 (119904) minus (119889 + 119873 119899sum

119894=1

119896119894) (120601 (119904) minus 119878120576 (119904))] d119904+ int11990501205901 (120601 (119904) minus 119878120576 (119904)) d1198611 (119904)

(110)

Taking expectation for both sides of the above inequality weobtain

119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816le 119864[int119905

0120576119878120576 (119904) minus (119889 + 119873 119899sum

119894=1

119896119894) 1003816100381610038161003816120601 (119904) minus 119878120576 (119904)1003816100381610038161003816 d119904]

le 119864[int1199050120576120601 (119904) minus (119889 + 119873 119899sum

119894=1

119896119894) 1003816100381610038161003816120601 (119904) minus 119878120576 (119904)1003816100381610038161003816 d119904] (111)

then

119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816 le 120576 sup0le119904

119864120601 (119904) (112)

which leads to

lim120576997888rarr0

inf lim119905997888rarrinfin

119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816 = 0 (113)

According to (108) (113) and 119878(119905) le 120601(119905) we can get


(120601 (119905) minus 119878 (119905)) = 0 (114)

Therefore based on the fact that 120601(119905) is stable to distributionV as 119905 997888rarr infin 119878(119905) weakly converges to distribution V Thiscompletes the proof

4 Discussion and Numerical Simulation

In this paper a turbidostat model subjected to competitionand stochastic perturbation has been investigated For system(1) we initially proved the existence and uniqueness ofglobal positive solution in Section 2 We further analyzed theprinciple of competitive exclusion for system (1) Under theconditions of Theorem 6 the species 119909119894 (119894 = 2 119899) in thevessel will be extinct and 1199091 will survive due to the fact that119909119894 (119894 = 2 119899) is inadequate competitor [2] if 0 lt 1205821 lt 1205822 lesdot sdot sdot le 120582119899 lt 1198780

Besides the above assertion we also obtain the conditionfor the extinction of system (1) which means that theprinciple of competitive exclusion becomes invalid if theconditions of Theorem 7 are satisfied In addition we havethe following remark

Remark 8 Any microorganism population 119909119894(119905) (119894 =1 2 119899) in system (1) survives and the other microorgan-ism populations in the vessel will be extinct if the followinggeneral condition holds

0 lt 120582119894 lt 120582119895 le sdot sdot sdot le 120582119896 lt 1198780119894 119895 119896 = 1 119899 and 119894 = 119895 = 119896 (115)

For a better understanding to the results obtained inTheorem 6 we offer an example as follows

Let 1198780 = 12 119889 = 012 1198891 = 032 1198892 = 031 1198861 = 21198871 = 3 1198862 = 2 1198872 = 5 1205901 = 01 1205902 = 01 1205903 = 02 1198961 = 0007and 1198962 = 001 then system (1) becomes

14 Complexity

200 400 600 800 1000 1200 1400 1600 1800 20000Time

0

01

02

03

04

05

06

07

08

Valu

es

S(t) pathx1(t) pathx2(t) path

0 100806040200

02

04x2(t) path

(a)

0

01

02

03

04

05

06

07

08

09

1

Valu

es

1 2 3 4 5 6 7 8 9 100Time


times104

(b)

Figure 1 1199091 survive and 1199092 will be extinct for system (116) with initial value 119878(0) = 05 1199091(0) = 1199092(0) = 03 (a) and (b) are the time series ofsystem with different time scale

S(t) path

01

02

03

04

05

06

07

S(t)

200 400 600 800 1000 1200 1400 1600 1800 20000Time

(a)

0

5

10

15

20

25

30

02 03 04 05 0601The distribution of S(t)

(b)

Figure 2 (a) is the time series of 119878(119905) with initial value 119878(0) = 05 (b) is the distribution interval of 119878(119905)

d119878 (119905)= [(12 minus 119878 (119905)) (012 + 00071199091 (119905) + 0011199092 (119905))minus 2119878 (119905)1 + 3119878 (119905)1199091 (119905) minus 2119878 (119905)1 + 5119878 (119905)1199092 (119905)] d119905+ 01119878 (119905) d1198611 (119905)

d1199091 (119905) = [minus (032 + 00071199091 (119905) + 0011199092 (119905)) 1199091 (119905)

+ 2119878 (119905)1 + 3119878 (119905)1199091 (119905)] d119905 + 011199091 (119905) d1198612 (119905) d1199092 (119905) = [minus (031 + 00071199091 (119905) + 0011199092 (119905)) 1199092 (119905)

+ 2119878 (119905)1 + 5119878 (119905)1199092 (119905)] d119905 + 021199092 (119905) d1198613 (119905) (116)

We derive 119909lowast1 = 01913 119909lowast2 = 0 1205821 = 03102 and 1205822 =07128 Obviously 1205821 lt 1205822 lt 1198780 The concentration of

Complexity 15

01

02

03

04

05

06

07

08

09

1

S(t)

S(t) path

1 2 3 4 5 6 7 8 9 100Time times10

4

(a)


0

5

10

15

20

25

30

(b)


x1(t) path

005

01

015

02

025

03

035

04

x1(t)

200 400 600 800 1000 1200 1400 1600 1800 20000Time

(a)

01 015 02 025 03 035 04 045 05005The distribution of x1(t)

0

5

10

15

20

25

30

35

40

45

50

(b)


nutrient at equilibrium is 119878lowast = 1205821 = 03102 Hence thespecies besides 1199091(119905) will be extinct eventually as Theorem 6shows (see Figure 1) From the biological viewpoint thenatural resource for population will be fluctuated because ofall kinds of reasons such as excessive development or naturaldisasters This phenomenon also occurs in microorganismscultivation If the temperature or lightness is various (thedensity of white noise 120590119894 represents this phenomenon insystem (1)) the dynamical behaviors such as digestion abilityfor microorganisms will be various at different points That

is the stochastic factors will cause the variations of dynam-ical behaviors and this phenomenon is interacted betweennutrient 119878(119905) andmicroorganisms119909119894 (119905) in the turbidostatThestochastic factors always cause the fluctuation of 119878(119905) and119909119894 (119905)(see Figures 1ndash5)

Although the concentration of nutrients varies signif-icantly in the system it always fluctuates around 1205821 =03102 (the black line in Figures 1(b) 2(a) and 3(a)) andhas a stationary distribution in the end (see Figures 2(b) and3(b)) The concentration of 1199091(119905) fluctuates around 119909lowast1 =

16 Complexity

1 2 3 4 5 6 7 8 9 100Time

005

01

015

02

025

03

035

04

045

05

x1(t)

x1(t) path

times104

(a)


0

5

10

15

20

25

30

35

40

45

50

(b)


01913 (the black line in Figure 1(b)) and there exists astationary distribution (see Figures 4(b) and 5(b)) Thisfluctuation phenomenon basically comes from the randomfactors which leads to variation of concentration for 119878(119905) and119909119894(119905)The species 1199092(119905)will be extinct since the concentrationof nutrient 119878lowast = 1205821 = 03102 is less than its break-evenconcentration 1205822 = 07128 (see Figure 1)

To further confirm the results obtained inTheorem 7 wegive the following example

Set 1198780 = 12 119889 = 08 1198891 = 09 1198892 = 09 1198861 = 1 1198871 = 041198862 = 1 1198872 = 06 1205901 = 02 1205902 = 08 1205903 = 08 1198961 = 02 and1198962 = 02 then system (1) arrives at

d119878 (119905) = [(12 minus 119878 (119905)) (08 + 021199091 (119905) + 021199092 (119905))minus 119878 (119905)1 + 04119878 (119905)1199091 (119905) minus 119878 (119905)1 + 06119878 (t)1199092 (119905)] d119905+ 02119878 (119905) d1198611 (119905)

d1199091 (119905) = [minus (09 + 021199091 (119905) + 021199092 (119905)) 1199091 (119905)+ 119878 (119905)1 + 04119878 (119905)1199091 (119905)] d119905 + 081199091 (119905) d1198612 (119905)

d1199092 (119905) = [minus (09 + 021199091 (119905) + 021199092 (119905)) 1199092 (119905)+ 119878 (119905)1 + 06119878 (119905)1199092 (119905)] d119905 + 081199092 (119905) d1198613 (119905)

(117)

It follows from simple calculations to obtain (1119886119894)[119889119894 +119873(1198961 + 1198962) + 1205902119894+12] asymp 122 gt 1198780 which satisfies the

conditions of Theorem 7 Hence 1199091(119905) and 1199092(119905) tend tozero exponentially which implies 119909lowast1 = 0 119909lowast2 = 0 (seeFigure 6)

In fact if the density of white noise 120590119894+1 increased(such as extremely high temperature) so that (1119886119894)[119889119894 +119873(1198961 + 1198962) + 1205902119894+12] gt 1198780 that is the environ-ment is not suitable for microbial survival both adequateand inadequate competitors will be extinct in the end(see Figure 6)

To sum up the competition from interspecies andstochastic factors from environment may affect the dynam-ical behaviors of species Inevitably some species becomeadequate competitors and survive after a long time and theother species in the system become inadequate competitorsand die out in the end However adequate and inadequatecompetitors may be extinct under strong enough stochasticdisturbance

Data Availability

No data were used to support this study

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

Weare very grateful to Prof Sigurdur FHafstein ofUniversityof Iceland who offered invaluable assistance This work

Complexity 17

0 20018016014012010080604020Time

0

02

04

06

08

1

12

14

16

18

2Va

lues


0 109876543210

01

02

03 x1(t) pathx2(t) path

(a)

0 10000900080007000600050004000300020001000Time

0

05

1

15

2

25

Valu

es


x1(t) pathx2(t) path

0

02

04

0 87654321

(b)

Figure 6 1199091 and 1199092 die out for system (117) with initial value 119878(0) = 05 1199091(0) = 1199092(0) = 03 (a) and (b) are the time series of system withdifferent time scale

was supported by the National Natural Science Founda-tion of China (grant numbers 11561022 and 11701163) andthe China Postdoctoral Science Foundation (grant number2014M562008)

References

[1] A Novick and L Szilard ldquoDescription of the chemostatrdquoScience vol 112 no 2920 pp 715-716 1950

[2] H L Smith and P Waltman The Theory of the ChemostatCambridge University Press 1995

[3] M Ioannides and I Larramendy-Valverde ldquoOn geometryand scale of a stochastic chemostatrdquo Communications inStatisticsmdashTheory and Methods vol 42 no 16 pp 2902ndash29112013

[4] P De Leenheer and H Smith ldquoFeedback control for chemostatmodelsrdquo Journal of Mathematical Biology vol 46 no 1 pp 48ndash70 2003

[5] P Collet S Martinez S Meleard and S Martin ldquoStochasticmodels for a chemostat and long-time behaviorrdquo Advances inApplied Probability vol 45 no 3 pp 822ndash836 2013

[6] X Meng L Wang and T Zhang ldquoGlobal dynamics analysis ofa nonlinear impulsive stochastic chemostat system in a pollutedenvironmentrdquo Journal of AppliedAnalysis andComputation vol6 no 3 pp 865ndash875 2016

[7] C Xu S Yuan and T Zhang ldquoAverage break-even concen-tration in a simple chemostat model with telegraph noiserdquoNonlinear Analysis Hybrid Systems vol 29 pp 373ndash382 2018

[8] N Panikov Microbial Growth Kinetics Chapman amp Hall NewYork NY USA 1995

[9] J Flegr ldquoTwo distinct types of natural selection in turbidostat-like and chemostat-like ecosystemsrdquo Journal of TheoreticalBiology vol 188 no 1 pp 121ndash126 1997

[10] B Li ldquoCompetition in a turbidostat for an inhibitory nutrientrdquoJournal of Biological Dynamics vol 2 no 2 pp 208ndash220 2008

[11] Z Li andL Chen ldquoPeriodic solution of a turbidostatmodel withimpulsive state feedback controlrdquo Nonlinear Dynamics vol 58no 3 pp 525ndash538 2009

[12] A Cammarota and M Miccio ldquoCompetition of two microbialspecies in a turbidostatrdquoComputer AidedChemical Engineeringvol 28 no C pp 331ndash336 2010

[13] T Zhang T Zhang and X Meng ldquoStability analysis of achemostatmodel withmaintenance energyrdquoAppliedMathemat-ics Letters vol 68 pp 1ndash7 2017

[14] N Champagnat P-E Jabin and S Meleard ldquoAdaptation ina stochastic multi-resources chemostat modelrdquo Journal deMathematiques Pures et Appliquees vol 101 no 6 pp 755ndash7882014

[15] X Lv X Meng and X Wang ldquoExtinction and stationarydistribution of an impulsive stochastic chemostat model withnonlinear perturbationrdquoChaos Solitons amp Fractals vol 110 pp273ndash279 2018

[16] G J Butler and G S Wolkowicz ldquoPredator-mediated competi-tion in the chemostatrdquo Journal of Mathematical Biology vol 24no 2 pp 167ndash191 1986

[17] RMay Stability andComplexity inModel Ecosystems PrincetonUniversity Press 1973

[18] J Grasman M de Gee andO van Herwaarden ldquoBreakdown ofa chemostat exposed to stochastic noiserdquo Journal of EngineeringMathematics vol 53 no 3-4 pp 291ndash300 2005

[19] F Campillo M Joannides and I Larramendy-ValverdeldquoStochastic modeling of the chemostatrdquo Ecological Modellingvol 222 no 15 pp 2676ndash2689 2011

[20] D Zhao and S Yuan ldquoCritical result on the break-evenconcentration in a single-species stochastic chemostat modelrdquoJournal of Mathematical Analysis and Applications vol 434 no2 pp 1336ndash1345 2016

[21] Q Zhang and D Jiang ldquoCompetitive exclusion in a stochasticchemostat model with Holling type II functional responserdquo

18 Complexity

Journal of Mathematical Chemistry vol 54 no 3 pp 777ndash7912016

[22] C Xu and S Yuan ldquoAn analogue of break-even concentrationin a simple stochastic chemostat modelrdquo Applied MathematicsLetters vol 48 pp 62ndash68 2015

[23] H M Davey C L Davey A M Woodward A N EdmondsA W Lee and D B Kell ldquoOscillatory stochastic and chaoticgrowth rate fluctuations in permittistatically controlled yeastculturesrdquo BioSystems vol 39 no 1 pp 43ndash61 1996

[24] G Liu X Wang X Meng and S Gao ldquoExtinction andpersistence in mean of a novel delay impulsive stochasticinfected predator-prey system with jumpsrdquo Complexity vol2017 Article ID 1950970 15 pages 2017

[25] X ZMeng S N Zhao T Feng and T H Zhang ldquoDynamics ofa novel nonlinear stochastic SIS epidemic model with doubleepidemic hypothesisrdquo Journal of Mathematical Analysis andApplications vol 433 no 1 pp 227ndash242 2016

[26] X Yu S Yuan and T Zhang ldquoThe effects of toxin-producingphytoplankton and environmental fluctuations on the plank-tonic bloomsrdquoNonlinearDynamics vol 91 no 3 pp 1653ndash16682018

[27] X Yu S Yuan and T Zhang ldquoPersistence and ergodicityof a stochastic single species model with Allee effect underregime switchingrdquo Communications in Nonlinear Science andNumerical Simulation vol 59 pp 359ndash374 2018

[28] Y Zhao S Yuan and T Zhang ldquoStochastic periodic solution ofa non-autonomous toxic-producing phytoplankton allelopathymodel with environmental fluctuationrdquo Communications inNonlinear Science and Numerical Simulation vol 44 pp 266ndash276 2017

[29] M Ghoul and S Mitri ldquoThe ecology and evolution of microbialcompetitionrdquo Trends in Microbiology vol 24 no 10 pp 833ndash845 2016

[30] J P Kaye and S C Hart ldquoCompetition for nitrogen betweenplants and soil microorganismsrdquo Trends in Ecology amp Evolutionvol 12 no 4 pp 139ndash143 1997

[31] G S Wolkowicz H Xia and S Ruan ldquoCompetition in thechemostat a distributed delay model and its global asymptoticbehaviorrdquo SIAM Journal on Applied Mathematics vol 57 no 5pp 1281ndash1310 1997

[32] M Liu K Wang and Q Wu ldquoSurvival analysis of stochasticcompetitive models in a polluted environment and stochasticcompetitive exclusion principlerdquo Bulletin of Mathematical Biol-ogy vol 73 no 9 pp 1969ndash2012 2011

[33] C Xu S Yuan and T Zhang ldquoSensitivity analysis and feedbackcontrol of noise-induced extinction for competition chemostatmodel with mutualismrdquo Physica A Statistical Mechanics and itsApplications vol 505 pp 891ndash902 2018

[34] Y Zhao S Yuan and JMa ldquoSurvival and stationary distributionanalysis of a stochastic competitive model of three species in apolluted environmentrdquoBulletin ofMathematical Biology vol 77no 7 pp 1285ndash1326 2015

[35] Y Sanling H Maoan and M Zhien ldquoCompetition in thechemostat convergence of a model with delayed response ingrowthrdquo Chaos Solitons amp Fractals vol 17 no 4 pp 659ndash6672003

[36] S B Hsu ldquoLimiting behavior for competing speciesrdquo SIAMJournal on Applied Mathematics vol 34 no 4 pp 760ndash7631978

[37] C Xu and S Yuan ldquoCompetition in the chemostat a stochasticmulti-speciesmodel and its asymptotic behaviorrdquoMathematicalBiosciences vol 280 pp 1ndash9 2016

[38] DVoulgarelis AVelayudhan andF Smith ldquoStochastic analysisof a full system of two competing populations in a chemostatrdquoChemical Engineering Science vol 175 pp 424ndash444 2018

[39] Y Zhao S Yuan and Q Zhang ldquoThe effect of Levy noise onthe survival of a stochastic competitive model in an impulsivepolluted environmentrdquo Applied Mathematical Modelling Sim-ulation and Computation for Engineering and EnvironmentalSystems vol 40 no 17-18 pp 7583ndash7600 2016

[40] Y Zhao S Yuan and T Zhang ldquoThe stationary distributionand ergodicity of a stochastic phytoplankton allelopathy modelunder regime switchingrdquo Communications in Nonlinear Scienceand Numerical Simulation vol 37 pp 131ndash142 2016

[41] X Mao Stochastic Differential Equations and ApplicationsHorwood Chichester UK 2nd edition 1997

[42] G K Basak and R N Bhattacharya ldquoStability in distributionfor a class of singular diffusionsrdquo Annals of Probability vol 20no 1 pp 312ndash321 1992

Hindawiwwwhindawicom Volume 2018

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Volume 2018

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Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

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Dierential EquationsInternational Journal of

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Submit your manuscripts atwwwhindawicom

Research ArticleDynamical Analysis of a StochasticMultispecies Turbidostat Model

Yu Mu12 Zuxiong Li 13 Huili Xiang 1 and Hailing Wang 1

1Department of Mathematics Hubei University for Nationalities Enshi Hubei 445000 China2Department of Mathematics City University of Hong Kong Hong Kong China3School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan Hubei 430074 China

Correspondence should be addressed to Zuxiong Li lizx0427126com

Received 31 July 2018 Accepted 4 December 2018 Published 10 January 2019

Academic Editor Marcelo Messias

Copyright copy 2019 Yu Mu et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A stochastic turbidostat system in which the dilution rate is subject to white noise is investigated in this paper First of all sufficientconditions of the competitive exclusion among microorganisms are obtained by employing the techniques of stochastic analysisFurthermore the results demonstrate that the competition among microorganisms and stochastic disturbance will affect thedynamical behaviors of microorganisms Finally the theoretical results obtained in this contribution are illustrated by numericalsimulations

1 Introduction

The chemostat and turbidostat two types of devices forcontinuous cultivation of microorganism have been utilizedto analyze population dynamics Novick et al [1] first pro-posed the mechanism of chemostat in 1950 and then anincreasing number of researchers have devoted themselvesto investigate chemostat systems [2ndash7] However there existsome drawbacks in chemostat model with a constant dilutionrate such as thewaste of substrate and higher viscosity causedby the mass transfer efficiency Models with the dilution raterelated to the state ofmicroorganism which can be called tur-bidostat (see [8]) can overcome the above drawbacks and behelpful to design optimal strategies to improve the substrateutilization Flegr [9] investigated turbidostat system with twospecies from numerical simulation and De Leenheer et al[4] analyzed the system theoretically Li [10] systematicallyinvestigated the turbidostat model and established sufficientconditions of coexistence of two species Subsequently manyresearchers investigated turbidostat and chemostat systemsMany valuable and interesting results were obtained [11ndash16]

According to May [17] the parameters of the systemsuch as birth rate death rate and the input concentration ofnutrient are inevitably disturbed by environmental factors

In recent years more and more stochastic ecological mod-els were chosen to describe the dynamics of populationsGrasmanet et al [18] established a stochastic chemostatmodel with three trophic levels Using singular perturbationmethods they obtained the expected breakdown time andanalyzed the influence of stochastic factors on dynamicbehaviors of the system in detail Considering the stochasticfactors in chemostat Campillo et al [19] proposed stochasticsystems with different population scales and they inves-tigated these systems and derived the field of validity fordifferent population scales Collet et al [5] pointed out thatthe population size in the device was determined not only bythe concentration of the nutrient but also by the stochasticbirth and death rates By analyzing the long time behaviorsof microbes they derived the conditions for the globalexistence of the solutions of the system and the existenceof quasi-stationary distribution Meng et al [6] constructedan impulsive stochastic chemostat model and investigatedthe extinction and permanence of the microorganisms Theyalso pointed out that small perturbation from white noisecould cause the extinction of microorganisms Xu et al [7]investigated a chemostat model containing telegraph noisewith the help of Markov chain and derived the break-evenconcentrationwhich determines the persistence or extinction

HindawiComplexityVolume 2019 Article ID 4681205 18 pageshttpsdoiorg10115520194681205

2 Complexity




d119878 (119905) = [(1198780 minus 119878 (119905)) (119889 + 119899sum119894=1

119896119894119909119894 (119905))

minus 119899sum119894=1

119886119894119878 (119905)1 + 119887119894119878 (119905)119909119894 (119905)] d119905 + 1205901119878 (119905) d1198611 (119905)

d119909119894 (119905) = [minus(119889119894 + 119899sum119894=1

119896119894119909119894 (119905)) 119909119894 (119905)

+ 119886119894119878 (119905)1 + 119887119894119878 (119905)119909119894 (119905)] d119905 + 120590119894+1119909119894 (119905) d119861119894+1 (119905) 119894 = 1 2 119899

(1)









d119906 (119905) = [(1198780 minus e119906(119905)) (119889 + sum119899119894=1 119896119894eV119894(119905))e119906(119905)

minus 119899sum119894=1

119886119894eV119894(119905)1 + 119887119894e119906(119905) minus120590212 ] d119905 + 1205901d1198611 (119905)

dV119894 (119905) = [minus(119889119894 + 119899sum119894=1

119896119894eV119894(119905)) + 119886119894e119906(119905)1 + 119887119894e119906(119905) minus1205902119894+12 ] d119905

+ 120590119894+1d119861119894+1 (119905) 119894 = 1 2 119899

(3)



(4)

Complexity 3



as 119896 997888rarr infin(5)



+ 119899sum119894=1

(119909119894 minus 1 minus ln 1199091) (6)


d119881 = [119878 minus 119886119878 (1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119878 minus 119886119878119899sum119894=1

1198861198941198781 + 119887119894119878119909119894 + 119886120590212minus 119899sum119894=1

119909119894 minus 1119909119894 (119889119894 + 119899sum119894=1

119896119894119909119894)119909119894 + 119899sum119894=1

119909119894 minus 11199091198941198861198941198781199091198941 + 119887119894119878

+ 119899sum119894=1

1205902119894+12 ] d119905 + (119878 minus 119886) 1205901d1198611 (119905)

+ 119899sum119894=1

(119909119894 minus 1) 120590119894+1d119861119894+1 (119905)

(7)

where

119871119881 = 119878 minus 119886119878 (1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119878 minus 119886119878119899sum119894=1

1198861198941198781 + 119887119894119878119909119894 + 119886120590212minus 119899sum119894=1

119909119894 minus 1119909119894 (119889119894 + 119899sum119894=1

119896119894119909119894)119909119894 + 119899sum119894=1

119909119894 minus 11199091198941198861198941198781199091198941 + 119887119894119878

+ 119899sum119894=1

1205902119894+12= (1198780 + 119886)(119889 + 119899sum

119894=1

119896119894119909119894) + 119899sum119894=1

1198861198861198941199091198941 + 119887119894119878 + 119886120590212

+ 119899sum119894=1

(119889119894 + 119899sum119894=1

119896119894119909119894) + 119899sum119894=1

1205902119894+12minus 119878(119889 + 119899sum

119894=1

119896119894119909119894) minus 1198861198780119878 (119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1

(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894 minus 119899sum119894=1

1198861198941198781 + 119887119894119878 (8)


119871119881 le (1198780 + 119886)(119889 + 119899sum119894=1

119896119894119909119894) + 119899sum119894=1

(119889119894 + 119899sum119894=1

119896119894119909119894)

+ 119886120590212 + 119899sum119894=1

1205902119894+12 + 119899sum119894=1

119886119886119894119909119894 minus 119899sum119894=1

119889119894119909119894le (1198780 + 119886) 119889 + 119899sum

119894=1

119889119894 + 119886120590212 + 119899sum119894=1

1205902119894+12+ 119899sum119894=1

[(1198780 + 119886) 119896119894 + 119896119894 + 119886119886119894 minus 119889119894] 119909119894= (1198780 + 119886) 119889 + 119899sum

119894=1

119889119894 + 119886120590212 + 119899sum119894=1

1205902119894+12+ 119899sum119894=1

(119896119894 + 119886119894) [119886 minus 119889119894 minus (1198780 + 1) 119896119894119896119894 + 119886119894 ]119909119894

(9)


119871119881 le (1198780 + 119886) 119889 + 119899sum119894=1

119889119894 + 119886120590212 + 119899sum119894=1

1205902119894+12 (10)

which yields

d119881 = 119871119881d119905 + (1198780 minus 119886) 1205901d1198611 (119905)+ 119899sum119894=1

120590119894+1 (119909119894 minus 1) d119861119894+1 (119905)

le (1198780 + 119886) 119889 + 119899sum119894=1

119889119894 + 119886120590212 + 119899sum119894=1

1205902119894+12+ (1198780 minus 119886) 1205901d1198611 (119905) + 119899sum

119894=1

120590119894+1 (119909119894 minus 1)d119861119894+1 (119905)

(11)


119864119881 (119878 (120591119896 and 119879) 119909119899 (120591119896 and 119879))le 119881 (119878 (0) 119909119899 (0))

+ ((1198780 + 119886) 119889 + 119899sum119894=1

119889119894 + 119886120590212 + 119899sum119894=1

1205902119894+12 )119879(12)

4 Complexity


119881 (119878 (0) 119909119899 (0))+ ((1198780 + 119886) 119889 + 119899sum

119894=1

119889119894 + 119886120590212 + 119899sum119894=1


(13)




⟨119909 (119905)⟩ = 1119905 int1199050119909 (119903) d119903 (14)



supln [119878 (119905) + sum119899119894=1 119909119894 (119905)]

ln 119905 le 1120579 119886119904 (15)


120582 = min 119889 1198891 119889119899 minus 119899sum119894=1

1198961198941198780


Proof Let

119906 (119905) = 119878 (119905) + 119899sum119894=1

119909119894 (119905) (17)

and

119882(119906) = (1 + 119906)120579 (18)


d119906 = [(1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1

(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894] d119905 + 1205901119878d1198611 (119905)

+ 119899sum119894=1

120590119894+1119909119894d119861119894+1 (119905)

(19)


d119882 = 120579 (1 + 119906)120579minus2(1 + 119906) [(1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1

(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894] + 120579 minus 12 (120590211198782

+ 119899sum119894=1

1205902119894+11199092119894) d119905 + 120579 (1 + 119906)120579minus1 [1205901119878d1198611 (119905)

+ 119899sum119894=1

120590119894+1119909119894d119861119894+1 (119905)]

(20)

Denote

119871119882(119906) = 120579 (1 + 119906)120579minus2 (1 + 119906)

sdot [(1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894) minus 119899sum119894=1

(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894]

+ 120579 minus 12 (120590211198782 +119899sum119894=1

1205902119894+11199092119894) = 120579 (1 + 119906)120579minus2(1

+ 119906) [1198780 (119889 + 119899sum119894=1

119896119894119909119894) minus 119878(119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1

(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894] + 120579 minus 12 (120590211198782

+ 119899sum119894=1

1205902119894+11199092119894) le 120579 (1 + 119906)120579minus2 (1 + 119906) [1198780119889

+ (1198780 119899sum119894=1

119896119894 minus min 119889 1198891 119889119899) 119906]

+ 120579 minus 12 (120590211198782 +119899sum119894=1

1205902119894+11199092119894) le 120579 (1 + 119906)120579minus2

sdot [ 119899sum119894=1

1198961198941198780 minus min 119889 1198891 119889119899

+ (120579 minus 12 or 0) (12059021 or or 1205902119899)] 1199062 + (1198780119889

Complexity 5

+ 119899sum119894=1

1198961198941198780 minus min 119889 1198891 119889119899) 119906 + (119889 + 119899sum119894=1

119896119894)


1198961198941198780

minus min 119889 1198891 119889119899) 119906 + (119889 + 119899sum119894=1

119896119894)1198780 (21)

where

120582 = min 119889 1198891 119889119899 minus 119899sum119894=1

1198961198941198780


Let

119860 = (119889 + 119899sum119894=1

119896119894)1198780 minus min 119889 1198891 119889119899 (23)

be a constant Then


119894=1

119896119894)1198780 d119905

+ 120579 (1 + 119906)120579minus1 [1205901119878d1198611 (119905) + 119899sum119894=1

120590119894+1119909119894d119861119894+1 (119905)] (24)


sdot [1205901119878d1198611 (119905) + 119899sum119894=1

120590119894+1119909119894d119861119894+1 (119905)] (25)

where


+ (119889 + 119899sum119894=1

119896119894)1198780 = e119902119905120579 (1 + 119906)120579minus2


119896119894)

+ 119902120579

(26)


+ 1198780 (119889 + 119899sum119894=1

119896119894) + 119902120579 119906 isin 119877+(27)

and119865 fl sup119906isin119877+

119891 (119906) + 1 (28)

which gives

119871 [e119902119905119882(119906)] le 120579e119902119905119865 (29)


119864 [e119902119905119882(119906)] = 119882 (119906 (0)) + 119864int1199050119871 [e119902119905119882(119906)] d119905 (30)


119864 [e119902119905119882(119906)] le (1 + 119906 (0))120579 + 119864int1199050e119902119905120579119865d119905

= (1 + 119906 (0))120579 + 120579119865119902 e119902119905(31)

which leads to


sup119864 [(1 + 119906 (119905))120579] le 120579119865119902 fl 1198650 119886119904 (32)


119864 [(1 + 119906 (119905))120579] le 119872 119905 ge 0 (33)


119864[ sup119896120575le119905le(119896+1)120575

(1 + 119906 (119905))120579]le 119864 [(1 + 119906 (119896120575))120579] + 1198681 + 1198682 le 119872 + 1198681 + 1198682

(34)

where

1198681 = 119864[ sup119896120575le119905le(119896+1)120575

1003816100381610038161003816100381610038161003816100381610038161003816int119905

119896120575120579 (1 + 119906)120579minus2

sdot minus1205821199062 + 119860119906 + 1198780(119889 + 119899sum119894=1

119896119894) d1199051003816100381610038161003816100381610038161003816100381610038161003816]

le 119864[ sup119896120575le119905le(119896+1)120575

1003816100381610038161003816100381610038161003816100381610038161003816int119905

119896120575120579 (1 + 119906)120579minus2

sdot 1198780(119889 + 119899sum119894=1

119896119894)119906 + 1198780(119889 + 119899sum119894=1

119896119894) d1199051003816100381610038161003816100381610038161003816100381610038161003816]

= 119864[ sup119896120575le119905le(119896+1)120575

1003816100381610038161003816100381610038161003816100381610038161003816int119905

119896120575120579 (1 + 119906)120579minus2

6 Complexity

sdot 1198780 (119889 + 119899sum119894=1

119896119894) (1 + 119906) d1199051003816100381610038161003816100381610038161003816100381610038161003816] le 1198780 (119889

+ 119899sum119894=1

119896119894)120579119864int(119896+1)120575119896120575

(1 + 119906)120579 d119905

le 1198621120575119864[ sup119896120575le119905le(119896+1)120575

(1 + 119906 (119905))120579] (35)

here

1198621 = 1198780(119889 + 119899sum119894=1

119896119894)120579 (36)


1198682 = 119864[ sup119896120575le119905le(119896+1)120575

1003816100381610038161003816100381610038161003816100381610038161003816int119905

119896120575120579 (1 + 119906)120579minus1

sdot 1205901119878d1198611 (119905) + 119899sum119894=1

120590119894+1119909119894d119861119894+1 (119905)1003816100381610038161003816100381610038161003816100381610038161003816]

le radic32119864[int(119896+1)120575119896120575

1205792 (1 + 119906)2120579minus2(120590211198782

+ 119899sum119894=1

1205902119894+11199092119894) d119905]12

le radic32119864[int(119896+1)120575119896120575

1205792 (1

+ 119906)2120579minus2 (120590211199062 +119899sum119894=1

1205902119894+11199062) d119905]12

le radic3212057912057512119864[ sup119896120575le119905le(119896+1)120575

(12059021 or or 1205902119899) (1

+ 119906)2120579]12 le radic3212057912057512 (12059021 or or 1205902119899)12

sdot 119864 [ sup119896120575le119905le(119896+1)120575

(1 + 119906)120579]

(37)

Hence we have

119864[ sup119896120575le119905le(119896+1)120575

(1 + 119906)120579]le 119864 [(1 + 119906 (119896120575))120579]


(1 + 119906)120579]

(38)


1198621120575 + radic3212057912057512 (12059021 or or 1205902119899)12 le 12 (39)

which yields

119864[ sup119896120575le119905le(119896+1)120575

(1 + 119906)120579] le 2119864 [(1 + 119906 (119896120575))120579] le 2119872 (40)


119875 sup119896120575le119905le(119896+1)120575

(1 + 119906 (119905))120579 ge (119896120575)1+120573

le 119864 [sup119896120575le119905le(119896+1)120575 (1 + 119906 (119905))120579](119896120575)1+120573 le 2119872

(119896120575)1+120573 (41)


sup119896120575le119905le(119896+1)120575

(1 + 119906 (119905))120579 le (119896120575)1+120573 (42)


ln (1 + 119906 (119905))120579ln 119905 le ln (119896120575)1+120573

ln 119905 le ln (119896120575)1+120573ln (119896120575) = 1 + 120573 (43)

which means


sup ln (1 + 119906 (119905))120579ln 119905 le 1 + 120573 119886119904 (44)


sup ln (1 + 119906 (119905))ln 119905 le 1120579 119886119904 (45)



supln [119878 (119905) + sum119899119894=1 119909119894 (119905)]

ln 119905 = lim119905997888rarrinfin

sup ln 119906 (119905)ln 119905


sup ln (1 + 119906 (119905))ln 119905 le 1120579 119886119904

(46)




int1199050(119878 (119904) minus 1205821) d1198611 (119904)119905 = 0 119886119904




int1199050119909119894 (119904) d119861119894+1 (119904)119905 = 0 119886119904

(119894 = 2 3 119899)

(47)


Complexity 7

Proof Define

119883 (119905) = int1199050(119878 (119904) minus 1205821)d1198611 (119904)


119885119894 (119905) = int1199050119909119894 (119904) d119861119894+1 (119904) 119894 = 2 3 119899

(48)


119864[ sup0le119904le119905

|119883 (119904)|120579] le 119862120579119864[int1199050(119878 (119904) minus 1205821)2 d119904]

12

le 1198621205791199051205792119864[ sup0le119904le119905

(119878 (119904) minus 1205821)120579]le 1198621205791199051205792119864[ sup

0le119904le119905(1 + 119906 (119905))120579]

le 21198721198621205791199051205792

(49)


119875120596 sup119896120575le119905le(119896+1)120575

|119883 (119905)|120579 gt (119896120575)1+120572+1205792

le 119864 [|119883 ((119896 + 1) 120575)|120579](119896120575)1+120572+1205792 le 2119872119862120579(119896120575)1+120572 (

(119896 + 1) 120575119896120575 )1205792

le 21+1205792119872119862120579(119896120575)1+120572

(50)


sup119896120575le119905le(119896+1)120575

|119883 (119905)|120579 le (119896120575)1+120572+1205792 (51)


ln 119905 le ln (119896120575)1+120572+1205792ln 119905 le ln (119896120575)1+120572+1205792

ln (119896120575) (52)

which implies

ln |119883 (119905)|120579ln 119905 le 1 + 120572 + 1205792 (53)



sup ln |119883 (119905)|ln 119905 le 1 + 120572 + 1205792120579 (54)


sup ln |119883 (119905)|ln 119905 le 12 + 1120579 (55)


ln |119883 (119905)|ln 119905 le 12 + 1120579 + 120578 (56)

which means



sup 11990512+1120579+120578119905 = 0 (57)

Together with



inf

100381610038161003816100381610038161003816int1199050 (119878 (119904) minus 1205821) d1198611 (119904)100381610038161003816100381610038161003816119905ge 0

(58)

we derive


|119883 (119905)|119905 = 0 119886119904 (59)

and


int1199050(119878 (119904) minus 1205821) d1198611 (119904)119905 = 0 119886119904 (60)





int1199050119909119894 (119905) d119861119894+1 (119904)119905 = 0 119886119904

119894 = 2 3 119899(61)



d120601 (119905) = [(1198780 minus 119878 (119905)) (119889 + 119899sum119894=1

119896119894119909119894)] d119905+ 1205901120601 (119905) d1198611 (119905)

(62)


0119911V(d119911) =


119901 (119911) = exp [2 (119889 + sum119899119894=1 119896119894119909119894) 119878012059021119911 ]


8 Complexity



gt 119899sum119894=1

1198961198941198780 + (120579 minus 12 or 0) (12059021 or or 1205902119899) 11988611989412058211 + 1198871198941205821 +

1198861198942radic119887119894 (1 + 1198871198941205821) ( 119867119889 + 119873sum119899119894=1 119896119894)12

le (119889119894 + 119873 119899sum119894=1

119896119894) + 1205902119894+12 (119894 = 2 3 119899)


119909V (d119909)1 + 1198871119909 gt 120590222 + (119889 + 119872 119899sum119894=1

119896119894)

(64)



supln119909119894 (119905)119905 le 0 119886119904 (119894 = 2 119899) (65)




(66)



(1 + 1198871198941205821) 119909119894

(67)


d119881 = 119878 minus 1205821119878 [(1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1

1198861198941198781 + 119887119894119878119909119894] d119905 + 1205901119878d1198611 (119905) + 12 12058211198782 120590211198782d119905

+ [(1 + 11988711205821) minus (1 + 11988711205821) 119909lowast11199091 ]

sdot [minus(1198891 + 119899sum119894=1

119896119894119909119894)1199091 + 11988611198781 + 11988711198781199091] d119905

+ 12059021199091d1198612 (119905) + 12(1 + 11988711205821) 119909lowast111990921 1205902211990921d119905 + 119899sum

119894=2

(1

+ 1198871198941205821) [minus(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894 + 1198861198941198781 + 119887119894119878119909119894] d119905

+ 120590119894+1119909119894d119861119894+1 (119905) = 119871119881d119905 + (119878 minus 1205821) 1205901d1198611 (119905)


(1 + 1198871198941205821) 120590119894+1119909119894d119861119894+1 (119905) (68)

where

119871119881 = 119878 minus 1205821119878 [(1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1


sdot [minus(1198891 + 119899sum119894=1

119896119894119909119894)1199091 + 11988611198781 + 11988711198781199091]

+ 119899sum119894=2

(1 + 1198871198941205821) [minus(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894 + 1198861198941198781 + 119887119894119878119909119894]

(69)



(70)



Complexity 9


119899sum119894=2

119896119894119909119894+ 119899sum119894=2

119909119894 [ 11988611989412058211 + 1198871198941205821 minus (119889119894 + 119899sum119894=1

119896119894119909119894)] + 12120582112059021+ 12 (1 + 11988711205821) 119909lowast112059022


119899sum119894=2

119896119894119909119894


119909119894 [minus 119899sum119894=1


(71)

Set

119872 = max 1199091 119909119899 119873 = min 1199091 119909119899 (72)

then


119894=2

119896119894


119872 119899sum119894=1


(73)

where


119894=2

119896119894


+ (119899 minus 1)119872 119899sum119894=1

119896119894119909lowast119894 + 12120582112059021+ 12 (1 + 11988711205821) 119909lowast112059022

(74)



0

(119878 (119904) minus 1205821)2119878 (119904) d119904 + 119867+ 1205901 1119905 int119905

0(119878 (119904) minus 1205821) d1198611 (119904)


+ 119899sum119894=2

(1 + 1198871198941205821) 120590119894+1 int1199050119909119894 (119904) d119861119894+1 (119905)

(75)



sup 1119905 int1199050

(119878 (119904) minus 1205821)2119878 (119904) d119904 le 119867119889 + 119873sum119899119894=1 119896119894 (76)

Define

119881119894 (119909119894 (119905)) = ln119909119894 (119905) 119894 = 2 3 119899 (77)


d119881119894 = [minus(119889119894 + 119899sum119894=1

119896119894119909119894) + 1198861198941198781 + 119887119894119878 minus 1205902119894+12 ] d119905


119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12 ] d119905

+ 120590119894+1d119861119894+1 (119905)

(78)


119881119894 (119905) minus 119881119894 (0)119905le 1198861198941 + 1198871198941205821 ⟨

119878 (119905) minus 12058211 + 119887119894119878 (119905)⟩ minus (119889119894 + 119873 119899sum119894=1

119896119894)

10 Complexity

+ 11988611989412058211 + 1198871198941205821 minus1205902119894+12 + 120590119894+1119861119894+1 (119905)119905


119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 120590119894+1119861119894+1 (119905)119905le 1198861198942radic119887119894 (1 + 1198871198941205821) [1119905 int119905

0

(119878 (119905) minus 1205821)2119878 (119905) d119904]12

minus (119889119894 + 119873 119899sum119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 120590119894+1119861119894+1 (119905)119905 (79)

which meansln119909119894 (119905)119905

le ln119909119894 (0)119905+ 1198861198942radic119887119894 (1 + 1198871198941205821) [1119905 int119905

0

(119878 (119905) minus 1205821)2119878 (119905) d119904]12

minus (119889119894 + 119873 119899sum119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 120590119894+1119861119894+1 (119905)119905

(80)



supln119909119894 (119905)119905

le minus(119889119894 + 119873 119899sum119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 1198861198942radic119887119894 (1 + 1198871198941205821) ( 119867119889 + 119873sum119899119894=1 119896119894)12

(119894 = 2 119899)

(81)

If

11988611989412058211 + 1198871198941205821 +1198861198942radic119887119894 (1 + 1198871198941205821) ( 119867119889 + 119873sum119899119894=1 119896119894)

12

le (119889119894 + 119873 119899sum119894=1

119896119894) + 1205902119894+12 (82)

then we have


supln119909119894 (119905)119905 le 0 119886119904 (119894 = 2 119899) (83)




1198861198941199091198941 + 119887119894119878 minus 120590212 ] d119905+ 1205901d1198611 (119905)


(84)



119894=1

1198861198941199091198941 + 119887119894119878] d119905

minus 120590212 + 1119905 int11990501205901d1198611 (119905)

= (119889 + 119899sum119894=1

119896119894119909119894)1198780 ⟨1119878⟩ minus 1198861⟨ 11990911 + 1198871119878⟩

minus 119899sum119894=2

119886119894⟨ 1199091198941 + 119887119894119878⟩ minus (119889 + 119899sum119894=1

119896119894119909119894) minus 120590212+ 12059011198611 (119905)119905 + ln 119878 (0)119905

ln120601 (119905)119905= ln120601 (0)119905 + 1119905 int119905

0

(1198780 minus 120601) (119889 + sum119899119894=1 119896119894119909119894)120601 d119905 minus 120590212+ 12059011198611 (119905)119905

= (119889 + 119899sum119894=1

119896119894119909119894)1198780⟨1120601⟩ minus (119889 + 119899sum119894=1

119896119894119909119894) minus 120590212+ 12059011198611 (119905)119905 + ln120601 (0)119905

(85)

Complexity 11

Therefore

0 ge ln 119878 (119905) minus ln120601 (119905)119905= (119889 + 119899sum

119894=1

119896119894119909119894)1198780⟨1119878 minus 1120601⟩ minus 1198861⟨ 11990911 + 1198871119878⟩

minus 119899sum119894=2

119886119894⟨ 1199091198941 + 119887119894119878⟩

ge (119889 + 119873 119899sum119894=1

119896119894)1198780⟨1119878 minus 1120601⟩ minus 1198861 ⟨1199091⟩

minus 119899sum119894=2

119886119894 ⟨119909119894⟩

(86)

which implies that

⟨1119878 minus 1120601⟩

le 1(119889 + 119873sum119899119894=1 119896119894) 1198780 (1198861 ⟨1199091⟩ + 119899sum119894=2

119886119894 ⟨119909119894⟩) (87)


119894=1

119896119894119909119894) + 11988611198781 + 1198871119878 minus 120590222 ] d119905+ 1205902d1198612 (119905)

ge [minus(1198891 + 119872 119899sum119894=1

119896119894) + 11988611198781 + 1198871119878 minus 120590222 ] d119905+ 1205902d1198612 (119905)

(88)



119894=1

119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩ minus 1198861⟨ 120601 minus 119878(1 + 1198871120601) (1 + 1198871119878)⟩


119894=1

119896119894)+ 1198861⟨ 1206011 + 1198871120601⟩ minus 119886111988721 ⟨1119878 minus 1120601⟩ + 12059021198612 (119905)119905


119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩

minus 119886111988721 (119889 + 119873sum119899119894=1 119896119894) 1198780 (1198861 ⟨1199091⟩ + 119899sum119894=2

119886119894 ⟨119909119894⟩)+ 12059021198612 (119905)119905


119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩ minus 11988621 ⟨1199091⟩11988721 (119889 + 119873sum119899119894=1 119896119894) 1198780+ 12059021198612 (119905)119905

(89)



sup ln 1199091 (119905)119905 le 0 (90)

which yields


119894=1

119896119894)+ 1198861⟨ 1206011 + 1198871120601⟩ + 12059021198612 (119905)119905

(91)

that is



119894=1

119896119894) + 1198861 intinfin0

119909V (d119909)1 + 1198871119909 ] (92)

Therefore if

1198861 intinfin0

119909V (d119909)1 + 1198871119909 gt 120590222 + (119889 + 119872 119899sum119894=1

119896119894) (93)

then


inf ⟨1199091 (119905)⟩ gt 0 119886119904 (94)



119886119894 intinfin0

119909V (d119909)1 + 119887119894119909 lt 119889119894 + 119873 119899sum119894=1

119896119894 + 1205902119894+12 119894 = 1 2 119899 (95)

12 Complexity

then


int1199050119878 (119904) d119904119905 = 1198780 119886119904


supln 119909119894 (119905)119905 lt 0 119886119904 119894 = 1 2 119899

(96)




119896119894119909119894 + 1198861198941198781 + 119887119894119878 minus 1205902119894+12 )d119905+ 120590119894+1d119861119894+1 (119905)


119896119894 + 1198861198941206011 + 119887119894120601 minus 1205902119894+12 )d119905+ 120590119894+1d119861119894+1 (119905)

(97)


119894=1

119896119894 minus 1205902119894+12 ) + 119886119894⟨ 1206011 + 119887119894120601⟩+ 120590119894+1119861119894+1 (119905)119905 + ln 119909119894 (0)119905

(98)


119886119894 intinfin0

119909V (d119909)1 + 119887119894119909 lt 119889119894 + 119873 119899sum119894=1

119896119894 + 1205902119894+12 119894 = 1 2 119899 (99)

we know that


supln119909119894 (119905)119905 le 0 119894 = 1 2 119899 (100)



d119884 = minus(119889 + 119899sum119894=1

119896119894119909119894)119884d119905 + 1205901 (1198780 + 119884) d1198611 (119905) (101)



119894=1

119896119894119909119894) minus 120576119878120576] d119905+ 1205901119878120576d1198611 (119905)

(102)

Then

d (119878 minus 119878120576) = [(120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878) 119878

minus (119889 + 120576 + 119899sum119894=1

119896119894119909119894) (119878 minus 119878120576)] d119905 + 1205901 (119878minus 119878120576) d1198611 (119905)

(103)



119894=1

119896119894119909119894 + 120576 + 120590212 ) 119905]

sdot int1199050exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904

(104)


120576 gt 119899sum119894=1

119886119894119909119894 (119905)1 + 119887119894119878 (119905) forall119905 ge 119879 (105)


119894=1

119896119894119909119894 + 120576

+ 120590212 ) 119905]

sdot int1198790exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904

+ int119905119879exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

Complexity 13

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904 ge exp[12059011198611 (119905)

minus (119889 + 119872 119899sum119894=1

119896119894 + 120576 + 120590212 ) 119905]

sdot int1198790exp[(119889 + 119873 119899sum

119894=1

119896119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904(106)


exp[12059011198611 (119905) minus (119889 + 119872 119899sum119894=1

119896119894 + 120576 + 120590212 ) 119905] 997888rarr 01003816100381610038161003816100381610038161003816100381610038161003816int119879

0exp[(119889 + 119873 119899sum

119894=1

119896119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d1199041003816100381610038161003816100381610038161003816100381610038161003816 lt infin

(107)

which implies that



inf (119878 (119905) minus 119878120576 (119905))ge 0 119886119904 (108)

Now we consider

d (120601 (119905) minus 119878120576 (119905))= [120576119878120576 (119905) minus (119889 + 119899sum

119894=1

119896119894119909119894) (120601 (119905) minus 119878120576 (119905))] d119905+ 1205901 (120601 (119905) minus 119878120576 (119905)) d1198611 (119905)

le [120576119878120576 (119905) minus (119889 + 119873 119899sum119894=1

119896119894)(120601 (119905) minus 119878120576 (119905))] d119905+ 1205901 (120601 (119905) minus 119878120576 (119905)) d1198611 (119905)

(109)



119894=1


(110)


119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816le 119864[int119905

0120576119878120576 (119904) minus (119889 + 119873 119899sum

119894=1

119896119894) 1003816100381610038161003816120601 (119904) minus 119878120576 (119904)1003816100381610038161003816 d119904]

le 119864[int1199050120576120601 (119904) minus (119889 + 119873 119899sum

119894=1

119896119894) 1003816100381610038161003816120601 (119904) minus 119878120576 (119904)1003816100381610038161003816 d119904] (111)

then

119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816 le 120576 sup0le119904

119864120601 (119904) (112)

which leads to

lim120576997888rarr0


119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816 = 0 (113)



(120601 (119905) minus 119878 (119905)) = 0 (114)









14 Complexity

200 400 600 800 1000 1200 1400 1600 1800 20000Time

0

01

02

03

04

05

06

07

08

Valu

es


0 100806040200

02

04x2(t) path

(a)

0

01

02

03

04

05

06

07

08

09

1

Valu

es

1 2 3 4 5 6 7 8 9 100Time


times104

(b)


S(t) path

01

02

03

04

05

06

07

S(t)

200 400 600 800 1000 1200 1400 1600 1800 20000Time

(a)

0

5

10

15

20

25

30


(b)



d1199091 (119905) = [minus (032 + 00071199091 (119905) + 0011199092 (119905)) 1199091 (119905)

+ 2119878 (119905)1 + 3119878 (119905)1199091 (119905)] d119905 + 011199091 (119905) d1198612 (119905) d1199092 (119905) = [minus (031 + 00071199091 (119905) + 0011199092 (119905)) 1199092 (119905)

+ 2119878 (119905)1 + 5119878 (119905)1199092 (119905)] d119905 + 021199092 (119905) d1198613 (119905) (116)


Complexity 15

01

02

03

04

05

06

07

08

09

1

S(t)

S(t) path

1 2 3 4 5 6 7 8 9 100Time times10

4

(a)


0

5

10

15

20

25

30

(b)


x1(t) path

005

01

015

02

025

03

035

04

x1(t)

200 400 600 800 1000 1200 1400 1600 1800 20000Time

(a)


0

5

10

15

20

25

30

35

40

45

50

(b)





16 Complexity

1 2 3 4 5 6 7 8 9 100Time

005

01

015

02

025

03

035

04

045

05

x1(t)

x1(t) path

times104

(a)


0

5

10

15

20

25

30

35

40

45

50

(b)






d1199091 (119905) = [minus (09 + 021199091 (119905) + 021199092 (119905)) 1199091 (119905)+ 119878 (119905)1 + 04119878 (119905)1199091 (119905)] d119905 + 081199091 (119905) d1198612 (119905)

d1199092 (119905) = [minus (09 + 021199091 (119905) + 021199092 (119905)) 1199092 (119905)+ 119878 (119905)1 + 06119878 (119905)1199092 (119905)] d119905 + 081199092 (119905) d1198613 (119905)

(117)





Data Availability




Acknowledgments


Complexity 17

0 20018016014012010080604020Time

0

02

04

06

08

1

12

14

16

18

2Va

lues


0 109876543210

01

02


(a)

0 10000900080007000600050004000300020001000Time

0

05

1

15

2

25

Valu

es



0

02

04

0 87654321

(b)



References






















18 Complexity






























Journal of












Journal of







Volume 2018






Volume 2018








2 Complexity




d119878 (119905) = [(1198780 minus 119878 (119905)) (119889 + 119899sum119894=1

119896119894119909119894 (119905))

minus 119899sum119894=1

119886119894119878 (119905)1 + 119887119894119878 (119905)119909119894 (119905)] d119905 + 1205901119878 (119905) d1198611 (119905)

d119909119894 (119905) = [minus(119889119894 + 119899sum119894=1

119896119894119909119894 (119905)) 119909119894 (119905)

+ 119886119894119878 (119905)1 + 119887119894119878 (119905)119909119894 (119905)] d119905 + 120590119894+1119909119894 (119905) d119861119894+1 (119905) 119894 = 1 2 119899

(1)









d119906 (119905) = [(1198780 minus e119906(119905)) (119889 + sum119899119894=1 119896119894eV119894(119905))e119906(119905)

minus 119899sum119894=1

119886119894eV119894(119905)1 + 119887119894e119906(119905) minus120590212 ] d119905 + 1205901d1198611 (119905)

dV119894 (119905) = [minus(119889119894 + 119899sum119894=1

119896119894eV119894(119905)) + 119886119894e119906(119905)1 + 119887119894e119906(119905) minus1205902119894+12 ] d119905

+ 120590119894+1d119861119894+1 (119905) 119894 = 1 2 119899

(3)



(4)

Complexity 3



as 119896 997888rarr infin(5)



+ 119899sum119894=1

(119909119894 minus 1 minus ln 1199091) (6)


d119881 = [119878 minus 119886119878 (1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119878 minus 119886119878119899sum119894=1

1198861198941198781 + 119887119894119878119909119894 + 119886120590212minus 119899sum119894=1

119909119894 minus 1119909119894 (119889119894 + 119899sum119894=1

119896119894119909119894)119909119894 + 119899sum119894=1

119909119894 minus 11199091198941198861198941198781199091198941 + 119887119894119878

+ 119899sum119894=1

1205902119894+12 ] d119905 + (119878 minus 119886) 1205901d1198611 (119905)

+ 119899sum119894=1

(119909119894 minus 1) 120590119894+1d119861119894+1 (119905)

(7)

where

119871119881 = 119878 minus 119886119878 (1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119878 minus 119886119878119899sum119894=1

1198861198941198781 + 119887119894119878119909119894 + 119886120590212minus 119899sum119894=1

119909119894 minus 1119909119894 (119889119894 + 119899sum119894=1

119896119894119909119894)119909119894 + 119899sum119894=1

119909119894 minus 11199091198941198861198941198781199091198941 + 119887119894119878

+ 119899sum119894=1

1205902119894+12= (1198780 + 119886)(119889 + 119899sum

119894=1

119896119894119909119894) + 119899sum119894=1

1198861198861198941199091198941 + 119887119894119878 + 119886120590212

+ 119899sum119894=1

(119889119894 + 119899sum119894=1

119896119894119909119894) + 119899sum119894=1

1205902119894+12minus 119878(119889 + 119899sum

119894=1

119896119894119909119894) minus 1198861198780119878 (119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1

(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894 minus 119899sum119894=1

1198861198941198781 + 119887119894119878 (8)


119871119881 le (1198780 + 119886)(119889 + 119899sum119894=1

119896119894119909119894) + 119899sum119894=1

(119889119894 + 119899sum119894=1

119896119894119909119894)

+ 119886120590212 + 119899sum119894=1

1205902119894+12 + 119899sum119894=1

119886119886119894119909119894 minus 119899sum119894=1

119889119894119909119894le (1198780 + 119886) 119889 + 119899sum

119894=1

119889119894 + 119886120590212 + 119899sum119894=1

1205902119894+12+ 119899sum119894=1

[(1198780 + 119886) 119896119894 + 119896119894 + 119886119886119894 minus 119889119894] 119909119894= (1198780 + 119886) 119889 + 119899sum

119894=1

119889119894 + 119886120590212 + 119899sum119894=1

1205902119894+12+ 119899sum119894=1

(119896119894 + 119886119894) [119886 minus 119889119894 minus (1198780 + 1) 119896119894119896119894 + 119886119894 ]119909119894

(9)


119871119881 le (1198780 + 119886) 119889 + 119899sum119894=1

119889119894 + 119886120590212 + 119899sum119894=1

1205902119894+12 (10)

which yields

d119881 = 119871119881d119905 + (1198780 minus 119886) 1205901d1198611 (119905)+ 119899sum119894=1

120590119894+1 (119909119894 minus 1) d119861119894+1 (119905)

le (1198780 + 119886) 119889 + 119899sum119894=1

119889119894 + 119886120590212 + 119899sum119894=1

1205902119894+12+ (1198780 minus 119886) 1205901d1198611 (119905) + 119899sum

119894=1

120590119894+1 (119909119894 minus 1)d119861119894+1 (119905)

(11)


119864119881 (119878 (120591119896 and 119879) 119909119899 (120591119896 and 119879))le 119881 (119878 (0) 119909119899 (0))

+ ((1198780 + 119886) 119889 + 119899sum119894=1

119889119894 + 119886120590212 + 119899sum119894=1

1205902119894+12 )119879(12)

4 Complexity


119881 (119878 (0) 119909119899 (0))+ ((1198780 + 119886) 119889 + 119899sum

119894=1

119889119894 + 119886120590212 + 119899sum119894=1


(13)




⟨119909 (119905)⟩ = 1119905 int1199050119909 (119903) d119903 (14)



supln [119878 (119905) + sum119899119894=1 119909119894 (119905)]

ln 119905 le 1120579 119886119904 (15)


120582 = min 119889 1198891 119889119899 minus 119899sum119894=1

1198961198941198780


Proof Let

119906 (119905) = 119878 (119905) + 119899sum119894=1

119909119894 (119905) (17)

and

119882(119906) = (1 + 119906)120579 (18)


d119906 = [(1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1

(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894] d119905 + 1205901119878d1198611 (119905)

+ 119899sum119894=1

120590119894+1119909119894d119861119894+1 (119905)

(19)


d119882 = 120579 (1 + 119906)120579minus2(1 + 119906) [(1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1

(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894] + 120579 minus 12 (120590211198782

+ 119899sum119894=1

1205902119894+11199092119894) d119905 + 120579 (1 + 119906)120579minus1 [1205901119878d1198611 (119905)

+ 119899sum119894=1

120590119894+1119909119894d119861119894+1 (119905)]

(20)

Denote

119871119882(119906) = 120579 (1 + 119906)120579minus2 (1 + 119906)

sdot [(1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894) minus 119899sum119894=1

(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894]

+ 120579 minus 12 (120590211198782 +119899sum119894=1

1205902119894+11199092119894) = 120579 (1 + 119906)120579minus2(1

+ 119906) [1198780 (119889 + 119899sum119894=1

119896119894119909119894) minus 119878(119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1

(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894] + 120579 minus 12 (120590211198782

+ 119899sum119894=1

1205902119894+11199092119894) le 120579 (1 + 119906)120579minus2 (1 + 119906) [1198780119889

+ (1198780 119899sum119894=1

119896119894 minus min 119889 1198891 119889119899) 119906]

+ 120579 minus 12 (120590211198782 +119899sum119894=1

1205902119894+11199092119894) le 120579 (1 + 119906)120579minus2

sdot [ 119899sum119894=1

1198961198941198780 minus min 119889 1198891 119889119899

+ (120579 minus 12 or 0) (12059021 or or 1205902119899)] 1199062 + (1198780119889

Complexity 5

+ 119899sum119894=1

1198961198941198780 minus min 119889 1198891 119889119899) 119906 + (119889 + 119899sum119894=1

119896119894)


1198961198941198780

minus min 119889 1198891 119889119899) 119906 + (119889 + 119899sum119894=1

119896119894)1198780 (21)

where

120582 = min 119889 1198891 119889119899 minus 119899sum119894=1

1198961198941198780


Let

119860 = (119889 + 119899sum119894=1

119896119894)1198780 minus min 119889 1198891 119889119899 (23)

be a constant Then


119894=1

119896119894)1198780 d119905

+ 120579 (1 + 119906)120579minus1 [1205901119878d1198611 (119905) + 119899sum119894=1

120590119894+1119909119894d119861119894+1 (119905)] (24)


sdot [1205901119878d1198611 (119905) + 119899sum119894=1

120590119894+1119909119894d119861119894+1 (119905)] (25)

where


+ (119889 + 119899sum119894=1

119896119894)1198780 = e119902119905120579 (1 + 119906)120579minus2


119896119894)

+ 119902120579

(26)


+ 1198780 (119889 + 119899sum119894=1

119896119894) + 119902120579 119906 isin 119877+(27)

and119865 fl sup119906isin119877+

119891 (119906) + 1 (28)

which gives

119871 [e119902119905119882(119906)] le 120579e119902119905119865 (29)


119864 [e119902119905119882(119906)] = 119882 (119906 (0)) + 119864int1199050119871 [e119902119905119882(119906)] d119905 (30)


119864 [e119902119905119882(119906)] le (1 + 119906 (0))120579 + 119864int1199050e119902119905120579119865d119905

= (1 + 119906 (0))120579 + 120579119865119902 e119902119905(31)

which leads to


sup119864 [(1 + 119906 (119905))120579] le 120579119865119902 fl 1198650 119886119904 (32)


119864 [(1 + 119906 (119905))120579] le 119872 119905 ge 0 (33)


119864[ sup119896120575le119905le(119896+1)120575

(1 + 119906 (119905))120579]le 119864 [(1 + 119906 (119896120575))120579] + 1198681 + 1198682 le 119872 + 1198681 + 1198682

(34)

where

1198681 = 119864[ sup119896120575le119905le(119896+1)120575

1003816100381610038161003816100381610038161003816100381610038161003816int119905

119896120575120579 (1 + 119906)120579minus2

sdot minus1205821199062 + 119860119906 + 1198780(119889 + 119899sum119894=1

119896119894) d1199051003816100381610038161003816100381610038161003816100381610038161003816]

le 119864[ sup119896120575le119905le(119896+1)120575

1003816100381610038161003816100381610038161003816100381610038161003816int119905

119896120575120579 (1 + 119906)120579minus2

sdot 1198780(119889 + 119899sum119894=1

119896119894)119906 + 1198780(119889 + 119899sum119894=1

119896119894) d1199051003816100381610038161003816100381610038161003816100381610038161003816]

= 119864[ sup119896120575le119905le(119896+1)120575

1003816100381610038161003816100381610038161003816100381610038161003816int119905

119896120575120579 (1 + 119906)120579minus2

6 Complexity

sdot 1198780 (119889 + 119899sum119894=1

119896119894) (1 + 119906) d1199051003816100381610038161003816100381610038161003816100381610038161003816] le 1198780 (119889

+ 119899sum119894=1

119896119894)120579119864int(119896+1)120575119896120575

(1 + 119906)120579 d119905

le 1198621120575119864[ sup119896120575le119905le(119896+1)120575

(1 + 119906 (119905))120579] (35)

here

1198621 = 1198780(119889 + 119899sum119894=1

119896119894)120579 (36)


1198682 = 119864[ sup119896120575le119905le(119896+1)120575

1003816100381610038161003816100381610038161003816100381610038161003816int119905

119896120575120579 (1 + 119906)120579minus1

sdot 1205901119878d1198611 (119905) + 119899sum119894=1

120590119894+1119909119894d119861119894+1 (119905)1003816100381610038161003816100381610038161003816100381610038161003816]

le radic32119864[int(119896+1)120575119896120575

1205792 (1 + 119906)2120579minus2(120590211198782

+ 119899sum119894=1

1205902119894+11199092119894) d119905]12

le radic32119864[int(119896+1)120575119896120575

1205792 (1

+ 119906)2120579minus2 (120590211199062 +119899sum119894=1

1205902119894+11199062) d119905]12

le radic3212057912057512119864[ sup119896120575le119905le(119896+1)120575

(12059021 or or 1205902119899) (1

+ 119906)2120579]12 le radic3212057912057512 (12059021 or or 1205902119899)12

sdot 119864 [ sup119896120575le119905le(119896+1)120575

(1 + 119906)120579]

(37)

Hence we have

119864[ sup119896120575le119905le(119896+1)120575

(1 + 119906)120579]le 119864 [(1 + 119906 (119896120575))120579]


(1 + 119906)120579]

(38)


1198621120575 + radic3212057912057512 (12059021 or or 1205902119899)12 le 12 (39)

which yields

119864[ sup119896120575le119905le(119896+1)120575

(1 + 119906)120579] le 2119864 [(1 + 119906 (119896120575))120579] le 2119872 (40)


119875 sup119896120575le119905le(119896+1)120575

(1 + 119906 (119905))120579 ge (119896120575)1+120573

le 119864 [sup119896120575le119905le(119896+1)120575 (1 + 119906 (119905))120579](119896120575)1+120573 le 2119872

(119896120575)1+120573 (41)


sup119896120575le119905le(119896+1)120575

(1 + 119906 (119905))120579 le (119896120575)1+120573 (42)


ln (1 + 119906 (119905))120579ln 119905 le ln (119896120575)1+120573

ln 119905 le ln (119896120575)1+120573ln (119896120575) = 1 + 120573 (43)

which means


sup ln (1 + 119906 (119905))120579ln 119905 le 1 + 120573 119886119904 (44)


sup ln (1 + 119906 (119905))ln 119905 le 1120579 119886119904 (45)



supln [119878 (119905) + sum119899119894=1 119909119894 (119905)]

ln 119905 = lim119905997888rarrinfin

sup ln 119906 (119905)ln 119905


sup ln (1 + 119906 (119905))ln 119905 le 1120579 119886119904

(46)




int1199050(119878 (119904) minus 1205821) d1198611 (119904)119905 = 0 119886119904




int1199050119909119894 (119904) d119861119894+1 (119904)119905 = 0 119886119904

(119894 = 2 3 119899)

(47)


Complexity 7

Proof Define

119883 (119905) = int1199050(119878 (119904) minus 1205821)d1198611 (119904)


119885119894 (119905) = int1199050119909119894 (119904) d119861119894+1 (119904) 119894 = 2 3 119899

(48)


119864[ sup0le119904le119905

|119883 (119904)|120579] le 119862120579119864[int1199050(119878 (119904) minus 1205821)2 d119904]

12

le 1198621205791199051205792119864[ sup0le119904le119905

(119878 (119904) minus 1205821)120579]le 1198621205791199051205792119864[ sup

0le119904le119905(1 + 119906 (119905))120579]

le 21198721198621205791199051205792

(49)


119875120596 sup119896120575le119905le(119896+1)120575

|119883 (119905)|120579 gt (119896120575)1+120572+1205792

le 119864 [|119883 ((119896 + 1) 120575)|120579](119896120575)1+120572+1205792 le 2119872119862120579(119896120575)1+120572 (

(119896 + 1) 120575119896120575 )1205792

le 21+1205792119872119862120579(119896120575)1+120572

(50)


sup119896120575le119905le(119896+1)120575

|119883 (119905)|120579 le (119896120575)1+120572+1205792 (51)


ln 119905 le ln (119896120575)1+120572+1205792ln 119905 le ln (119896120575)1+120572+1205792

ln (119896120575) (52)

which implies

ln |119883 (119905)|120579ln 119905 le 1 + 120572 + 1205792 (53)



sup ln |119883 (119905)|ln 119905 le 1 + 120572 + 1205792120579 (54)


sup ln |119883 (119905)|ln 119905 le 12 + 1120579 (55)


ln |119883 (119905)|ln 119905 le 12 + 1120579 + 120578 (56)

which means



sup 11990512+1120579+120578119905 = 0 (57)

Together with



inf

100381610038161003816100381610038161003816int1199050 (119878 (119904) minus 1205821) d1198611 (119904)100381610038161003816100381610038161003816119905ge 0

(58)

we derive


|119883 (119905)|119905 = 0 119886119904 (59)

and


int1199050(119878 (119904) minus 1205821) d1198611 (119904)119905 = 0 119886119904 (60)





int1199050119909119894 (119905) d119861119894+1 (119904)119905 = 0 119886119904

119894 = 2 3 119899(61)



d120601 (119905) = [(1198780 minus 119878 (119905)) (119889 + 119899sum119894=1

119896119894119909119894)] d119905+ 1205901120601 (119905) d1198611 (119905)

(62)


0119911V(d119911) =


119901 (119911) = exp [2 (119889 + sum119899119894=1 119896119894119909119894) 119878012059021119911 ]


8 Complexity



gt 119899sum119894=1

1198961198941198780 + (120579 minus 12 or 0) (12059021 or or 1205902119899) 11988611989412058211 + 1198871198941205821 +

1198861198942radic119887119894 (1 + 1198871198941205821) ( 119867119889 + 119873sum119899119894=1 119896119894)12

le (119889119894 + 119873 119899sum119894=1

119896119894) + 1205902119894+12 (119894 = 2 3 119899)


119909V (d119909)1 + 1198871119909 gt 120590222 + (119889 + 119872 119899sum119894=1

119896119894)

(64)



supln119909119894 (119905)119905 le 0 119886119904 (119894 = 2 119899) (65)




(66)



(1 + 1198871198941205821) 119909119894

(67)


d119881 = 119878 minus 1205821119878 [(1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1

1198861198941198781 + 119887119894119878119909119894] d119905 + 1205901119878d1198611 (119905) + 12 12058211198782 120590211198782d119905

+ [(1 + 11988711205821) minus (1 + 11988711205821) 119909lowast11199091 ]

sdot [minus(1198891 + 119899sum119894=1

119896119894119909119894)1199091 + 11988611198781 + 11988711198781199091] d119905

+ 12059021199091d1198612 (119905) + 12(1 + 11988711205821) 119909lowast111990921 1205902211990921d119905 + 119899sum

119894=2

(1

+ 1198871198941205821) [minus(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894 + 1198861198941198781 + 119887119894119878119909119894] d119905

+ 120590119894+1119909119894d119861119894+1 (119905) = 119871119881d119905 + (119878 minus 1205821) 1205901d1198611 (119905)


(1 + 1198871198941205821) 120590119894+1119909119894d119861119894+1 (119905) (68)

where

119871119881 = 119878 minus 1205821119878 [(1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1


sdot [minus(1198891 + 119899sum119894=1

119896119894119909119894)1199091 + 11988611198781 + 11988711198781199091]

+ 119899sum119894=2

(1 + 1198871198941205821) [minus(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894 + 1198861198941198781 + 119887119894119878119909119894]

(69)



(70)



Complexity 9


119899sum119894=2

119896119894119909119894+ 119899sum119894=2

119909119894 [ 11988611989412058211 + 1198871198941205821 minus (119889119894 + 119899sum119894=1

119896119894119909119894)] + 12120582112059021+ 12 (1 + 11988711205821) 119909lowast112059022


119899sum119894=2

119896119894119909119894


119909119894 [minus 119899sum119894=1


(71)

Set

119872 = max 1199091 119909119899 119873 = min 1199091 119909119899 (72)

then


119894=2

119896119894


119872 119899sum119894=1


(73)

where


119894=2

119896119894


+ (119899 minus 1)119872 119899sum119894=1

119896119894119909lowast119894 + 12120582112059021+ 12 (1 + 11988711205821) 119909lowast112059022

(74)



0

(119878 (119904) minus 1205821)2119878 (119904) d119904 + 119867+ 1205901 1119905 int119905

0(119878 (119904) minus 1205821) d1198611 (119904)


+ 119899sum119894=2

(1 + 1198871198941205821) 120590119894+1 int1199050119909119894 (119904) d119861119894+1 (119905)

(75)



sup 1119905 int1199050

(119878 (119904) minus 1205821)2119878 (119904) d119904 le 119867119889 + 119873sum119899119894=1 119896119894 (76)

Define

119881119894 (119909119894 (119905)) = ln119909119894 (119905) 119894 = 2 3 119899 (77)


d119881119894 = [minus(119889119894 + 119899sum119894=1

119896119894119909119894) + 1198861198941198781 + 119887119894119878 minus 1205902119894+12 ] d119905


119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12 ] d119905

+ 120590119894+1d119861119894+1 (119905)

(78)


119881119894 (119905) minus 119881119894 (0)119905le 1198861198941 + 1198871198941205821 ⟨

119878 (119905) minus 12058211 + 119887119894119878 (119905)⟩ minus (119889119894 + 119873 119899sum119894=1

119896119894)

10 Complexity

+ 11988611989412058211 + 1198871198941205821 minus1205902119894+12 + 120590119894+1119861119894+1 (119905)119905


119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 120590119894+1119861119894+1 (119905)119905le 1198861198942radic119887119894 (1 + 1198871198941205821) [1119905 int119905

0

(119878 (119905) minus 1205821)2119878 (119905) d119904]12

minus (119889119894 + 119873 119899sum119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 120590119894+1119861119894+1 (119905)119905 (79)

which meansln119909119894 (119905)119905

le ln119909119894 (0)119905+ 1198861198942radic119887119894 (1 + 1198871198941205821) [1119905 int119905

0

(119878 (119905) minus 1205821)2119878 (119905) d119904]12

minus (119889119894 + 119873 119899sum119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 120590119894+1119861119894+1 (119905)119905

(80)



supln119909119894 (119905)119905

le minus(119889119894 + 119873 119899sum119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 1198861198942radic119887119894 (1 + 1198871198941205821) ( 119867119889 + 119873sum119899119894=1 119896119894)12

(119894 = 2 119899)

(81)

If

11988611989412058211 + 1198871198941205821 +1198861198942radic119887119894 (1 + 1198871198941205821) ( 119867119889 + 119873sum119899119894=1 119896119894)

12

le (119889119894 + 119873 119899sum119894=1

119896119894) + 1205902119894+12 (82)

then we have


supln119909119894 (119905)119905 le 0 119886119904 (119894 = 2 119899) (83)




1198861198941199091198941 + 119887119894119878 minus 120590212 ] d119905+ 1205901d1198611 (119905)


(84)



119894=1

1198861198941199091198941 + 119887119894119878] d119905

minus 120590212 + 1119905 int11990501205901d1198611 (119905)

= (119889 + 119899sum119894=1

119896119894119909119894)1198780 ⟨1119878⟩ minus 1198861⟨ 11990911 + 1198871119878⟩

minus 119899sum119894=2

119886119894⟨ 1199091198941 + 119887119894119878⟩ minus (119889 + 119899sum119894=1

119896119894119909119894) minus 120590212+ 12059011198611 (119905)119905 + ln 119878 (0)119905

ln120601 (119905)119905= ln120601 (0)119905 + 1119905 int119905

0

(1198780 minus 120601) (119889 + sum119899119894=1 119896119894119909119894)120601 d119905 minus 120590212+ 12059011198611 (119905)119905

= (119889 + 119899sum119894=1

119896119894119909119894)1198780⟨1120601⟩ minus (119889 + 119899sum119894=1

119896119894119909119894) minus 120590212+ 12059011198611 (119905)119905 + ln120601 (0)119905

(85)

Complexity 11

Therefore

0 ge ln 119878 (119905) minus ln120601 (119905)119905= (119889 + 119899sum

119894=1

119896119894119909119894)1198780⟨1119878 minus 1120601⟩ minus 1198861⟨ 11990911 + 1198871119878⟩

minus 119899sum119894=2

119886119894⟨ 1199091198941 + 119887119894119878⟩

ge (119889 + 119873 119899sum119894=1

119896119894)1198780⟨1119878 minus 1120601⟩ minus 1198861 ⟨1199091⟩

minus 119899sum119894=2

119886119894 ⟨119909119894⟩

(86)

which implies that

⟨1119878 minus 1120601⟩

le 1(119889 + 119873sum119899119894=1 119896119894) 1198780 (1198861 ⟨1199091⟩ + 119899sum119894=2

119886119894 ⟨119909119894⟩) (87)


119894=1

119896119894119909119894) + 11988611198781 + 1198871119878 minus 120590222 ] d119905+ 1205902d1198612 (119905)

ge [minus(1198891 + 119872 119899sum119894=1

119896119894) + 11988611198781 + 1198871119878 minus 120590222 ] d119905+ 1205902d1198612 (119905)

(88)



119894=1

119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩ minus 1198861⟨ 120601 minus 119878(1 + 1198871120601) (1 + 1198871119878)⟩


119894=1

119896119894)+ 1198861⟨ 1206011 + 1198871120601⟩ minus 119886111988721 ⟨1119878 minus 1120601⟩ + 12059021198612 (119905)119905


119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩

minus 119886111988721 (119889 + 119873sum119899119894=1 119896119894) 1198780 (1198861 ⟨1199091⟩ + 119899sum119894=2

119886119894 ⟨119909119894⟩)+ 12059021198612 (119905)119905


119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩ minus 11988621 ⟨1199091⟩11988721 (119889 + 119873sum119899119894=1 119896119894) 1198780+ 12059021198612 (119905)119905

(89)



sup ln 1199091 (119905)119905 le 0 (90)

which yields


119894=1

119896119894)+ 1198861⟨ 1206011 + 1198871120601⟩ + 12059021198612 (119905)119905

(91)

that is



119894=1

119896119894) + 1198861 intinfin0

119909V (d119909)1 + 1198871119909 ] (92)

Therefore if

1198861 intinfin0

119909V (d119909)1 + 1198871119909 gt 120590222 + (119889 + 119872 119899sum119894=1

119896119894) (93)

then


inf ⟨1199091 (119905)⟩ gt 0 119886119904 (94)



119886119894 intinfin0

119909V (d119909)1 + 119887119894119909 lt 119889119894 + 119873 119899sum119894=1

119896119894 + 1205902119894+12 119894 = 1 2 119899 (95)

12 Complexity

then


int1199050119878 (119904) d119904119905 = 1198780 119886119904


supln 119909119894 (119905)119905 lt 0 119886119904 119894 = 1 2 119899

(96)




119896119894119909119894 + 1198861198941198781 + 119887119894119878 minus 1205902119894+12 )d119905+ 120590119894+1d119861119894+1 (119905)


119896119894 + 1198861198941206011 + 119887119894120601 minus 1205902119894+12 )d119905+ 120590119894+1d119861119894+1 (119905)

(97)


119894=1

119896119894 minus 1205902119894+12 ) + 119886119894⟨ 1206011 + 119887119894120601⟩+ 120590119894+1119861119894+1 (119905)119905 + ln 119909119894 (0)119905

(98)


119886119894 intinfin0

119909V (d119909)1 + 119887119894119909 lt 119889119894 + 119873 119899sum119894=1

119896119894 + 1205902119894+12 119894 = 1 2 119899 (99)

we know that


supln119909119894 (119905)119905 le 0 119894 = 1 2 119899 (100)



d119884 = minus(119889 + 119899sum119894=1

119896119894119909119894)119884d119905 + 1205901 (1198780 + 119884) d1198611 (119905) (101)



119894=1

119896119894119909119894) minus 120576119878120576] d119905+ 1205901119878120576d1198611 (119905)

(102)

Then

d (119878 minus 119878120576) = [(120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878) 119878

minus (119889 + 120576 + 119899sum119894=1

119896119894119909119894) (119878 minus 119878120576)] d119905 + 1205901 (119878minus 119878120576) d1198611 (119905)

(103)



119894=1

119896119894119909119894 + 120576 + 120590212 ) 119905]

sdot int1199050exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904

(104)


120576 gt 119899sum119894=1

119886119894119909119894 (119905)1 + 119887119894119878 (119905) forall119905 ge 119879 (105)


119894=1

119896119894119909119894 + 120576

+ 120590212 ) 119905]

sdot int1198790exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904

+ int119905119879exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

Complexity 13

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904 ge exp[12059011198611 (119905)

minus (119889 + 119872 119899sum119894=1

119896119894 + 120576 + 120590212 ) 119905]

sdot int1198790exp[(119889 + 119873 119899sum

119894=1

119896119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904(106)


exp[12059011198611 (119905) minus (119889 + 119872 119899sum119894=1

119896119894 + 120576 + 120590212 ) 119905] 997888rarr 01003816100381610038161003816100381610038161003816100381610038161003816int119879

0exp[(119889 + 119873 119899sum

119894=1

119896119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d1199041003816100381610038161003816100381610038161003816100381610038161003816 lt infin

(107)

which implies that



inf (119878 (119905) minus 119878120576 (119905))ge 0 119886119904 (108)

Now we consider

d (120601 (119905) minus 119878120576 (119905))= [120576119878120576 (119905) minus (119889 + 119899sum

119894=1

119896119894119909119894) (120601 (119905) minus 119878120576 (119905))] d119905+ 1205901 (120601 (119905) minus 119878120576 (119905)) d1198611 (119905)

le [120576119878120576 (119905) minus (119889 + 119873 119899sum119894=1

119896119894)(120601 (119905) minus 119878120576 (119905))] d119905+ 1205901 (120601 (119905) minus 119878120576 (119905)) d1198611 (119905)

(109)



119894=1


(110)


119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816le 119864[int119905

0120576119878120576 (119904) minus (119889 + 119873 119899sum

119894=1

119896119894) 1003816100381610038161003816120601 (119904) minus 119878120576 (119904)1003816100381610038161003816 d119904]

le 119864[int1199050120576120601 (119904) minus (119889 + 119873 119899sum

119894=1

119896119894) 1003816100381610038161003816120601 (119904) minus 119878120576 (119904)1003816100381610038161003816 d119904] (111)

then

119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816 le 120576 sup0le119904

119864120601 (119904) (112)

which leads to

lim120576997888rarr0


119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816 = 0 (113)



(120601 (119905) minus 119878 (119905)) = 0 (114)









14 Complexity

200 400 600 800 1000 1200 1400 1600 1800 20000Time

0

01

02

03

04

05

06

07

08

Valu

es


0 100806040200

02

04x2(t) path

(a)

0

01

02

03

04

05

06

07

08

09

1

Valu

es

1 2 3 4 5 6 7 8 9 100Time


times104

(b)


S(t) path

01

02

03

04

05

06

07

S(t)

200 400 600 800 1000 1200 1400 1600 1800 20000Time

(a)

0

5

10

15

20

25

30


(b)



d1199091 (119905) = [minus (032 + 00071199091 (119905) + 0011199092 (119905)) 1199091 (119905)

+ 2119878 (119905)1 + 3119878 (119905)1199091 (119905)] d119905 + 011199091 (119905) d1198612 (119905) d1199092 (119905) = [minus (031 + 00071199091 (119905) + 0011199092 (119905)) 1199092 (119905)

+ 2119878 (119905)1 + 5119878 (119905)1199092 (119905)] d119905 + 021199092 (119905) d1198613 (119905) (116)


Complexity 15

01

02

03

04

05

06

07

08

09

1

S(t)

S(t) path

1 2 3 4 5 6 7 8 9 100Time times10

4

(a)


0

5

10

15

20

25

30

(b)


x1(t) path

005

01

015

02

025

03

035

04

x1(t)

200 400 600 800 1000 1200 1400 1600 1800 20000Time

(a)


0

5

10

15

20

25

30

35

40

45

50

(b)





16 Complexity

1 2 3 4 5 6 7 8 9 100Time

005

01

015

02

025

03

035

04

045

05

x1(t)

x1(t) path

times104

(a)


0

5

10

15

20

25

30

35

40

45

50

(b)






d1199091 (119905) = [minus (09 + 021199091 (119905) + 021199092 (119905)) 1199091 (119905)+ 119878 (119905)1 + 04119878 (119905)1199091 (119905)] d119905 + 081199091 (119905) d1198612 (119905)

d1199092 (119905) = [minus (09 + 021199091 (119905) + 021199092 (119905)) 1199092 (119905)+ 119878 (119905)1 + 06119878 (119905)1199092 (119905)] d119905 + 081199092 (119905) d1198613 (119905)

(117)





Data Availability




Acknowledgments


Complexity 17

0 20018016014012010080604020Time

0

02

04

06

08

1

12

14

16

18

2Va

lues


0 109876543210

01

02


(a)

0 10000900080007000600050004000300020001000Time

0

05

1

15

2

25

Valu

es



0

02

04

0 87654321

(b)



References






















18 Complexity






























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 3



as 119896 997888rarr infin(5)



+ 119899sum119894=1

(119909119894 minus 1 minus ln 1199091) (6)


d119881 = [119878 minus 119886119878 (1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119878 minus 119886119878119899sum119894=1

1198861198941198781 + 119887119894119878119909119894 + 119886120590212minus 119899sum119894=1

119909119894 minus 1119909119894 (119889119894 + 119899sum119894=1

119896119894119909119894)119909119894 + 119899sum119894=1

119909119894 minus 11199091198941198861198941198781199091198941 + 119887119894119878

+ 119899sum119894=1

1205902119894+12 ] d119905 + (119878 minus 119886) 1205901d1198611 (119905)

+ 119899sum119894=1

(119909119894 minus 1) 120590119894+1d119861119894+1 (119905)

(7)

where

119871119881 = 119878 minus 119886119878 (1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119878 minus 119886119878119899sum119894=1

1198861198941198781 + 119887119894119878119909119894 + 119886120590212minus 119899sum119894=1

119909119894 minus 1119909119894 (119889119894 + 119899sum119894=1

119896119894119909119894)119909119894 + 119899sum119894=1

119909119894 minus 11199091198941198861198941198781199091198941 + 119887119894119878

+ 119899sum119894=1

1205902119894+12= (1198780 + 119886)(119889 + 119899sum

119894=1

119896119894119909119894) + 119899sum119894=1

1198861198861198941199091198941 + 119887119894119878 + 119886120590212

+ 119899sum119894=1

(119889119894 + 119899sum119894=1

119896119894119909119894) + 119899sum119894=1

1205902119894+12minus 119878(119889 + 119899sum

119894=1

119896119894119909119894) minus 1198861198780119878 (119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1

(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894 minus 119899sum119894=1

1198861198941198781 + 119887119894119878 (8)


119871119881 le (1198780 + 119886)(119889 + 119899sum119894=1

119896119894119909119894) + 119899sum119894=1

(119889119894 + 119899sum119894=1

119896119894119909119894)

+ 119886120590212 + 119899sum119894=1

1205902119894+12 + 119899sum119894=1

119886119886119894119909119894 minus 119899sum119894=1

119889119894119909119894le (1198780 + 119886) 119889 + 119899sum

119894=1

119889119894 + 119886120590212 + 119899sum119894=1

1205902119894+12+ 119899sum119894=1

[(1198780 + 119886) 119896119894 + 119896119894 + 119886119886119894 minus 119889119894] 119909119894= (1198780 + 119886) 119889 + 119899sum

119894=1

119889119894 + 119886120590212 + 119899sum119894=1

1205902119894+12+ 119899sum119894=1

(119896119894 + 119886119894) [119886 minus 119889119894 minus (1198780 + 1) 119896119894119896119894 + 119886119894 ]119909119894

(9)


119871119881 le (1198780 + 119886) 119889 + 119899sum119894=1

119889119894 + 119886120590212 + 119899sum119894=1

1205902119894+12 (10)

which yields

d119881 = 119871119881d119905 + (1198780 minus 119886) 1205901d1198611 (119905)+ 119899sum119894=1

120590119894+1 (119909119894 minus 1) d119861119894+1 (119905)

le (1198780 + 119886) 119889 + 119899sum119894=1

119889119894 + 119886120590212 + 119899sum119894=1

1205902119894+12+ (1198780 minus 119886) 1205901d1198611 (119905) + 119899sum

119894=1

120590119894+1 (119909119894 minus 1)d119861119894+1 (119905)

(11)


119864119881 (119878 (120591119896 and 119879) 119909119899 (120591119896 and 119879))le 119881 (119878 (0) 119909119899 (0))

+ ((1198780 + 119886) 119889 + 119899sum119894=1

119889119894 + 119886120590212 + 119899sum119894=1

1205902119894+12 )119879(12)

4 Complexity


119881 (119878 (0) 119909119899 (0))+ ((1198780 + 119886) 119889 + 119899sum

119894=1

119889119894 + 119886120590212 + 119899sum119894=1


(13)




⟨119909 (119905)⟩ = 1119905 int1199050119909 (119903) d119903 (14)



supln [119878 (119905) + sum119899119894=1 119909119894 (119905)]

ln 119905 le 1120579 119886119904 (15)


120582 = min 119889 1198891 119889119899 minus 119899sum119894=1

1198961198941198780


Proof Let

119906 (119905) = 119878 (119905) + 119899sum119894=1

119909119894 (119905) (17)

and

119882(119906) = (1 + 119906)120579 (18)


d119906 = [(1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1

(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894] d119905 + 1205901119878d1198611 (119905)

+ 119899sum119894=1

120590119894+1119909119894d119861119894+1 (119905)

(19)


d119882 = 120579 (1 + 119906)120579minus2(1 + 119906) [(1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1

(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894] + 120579 minus 12 (120590211198782

+ 119899sum119894=1

1205902119894+11199092119894) d119905 + 120579 (1 + 119906)120579minus1 [1205901119878d1198611 (119905)

+ 119899sum119894=1

120590119894+1119909119894d119861119894+1 (119905)]

(20)

Denote

119871119882(119906) = 120579 (1 + 119906)120579minus2 (1 + 119906)

sdot [(1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894) minus 119899sum119894=1

(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894]

+ 120579 minus 12 (120590211198782 +119899sum119894=1

1205902119894+11199092119894) = 120579 (1 + 119906)120579minus2(1

+ 119906) [1198780 (119889 + 119899sum119894=1

119896119894119909119894) minus 119878(119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1

(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894] + 120579 minus 12 (120590211198782

+ 119899sum119894=1

1205902119894+11199092119894) le 120579 (1 + 119906)120579minus2 (1 + 119906) [1198780119889

+ (1198780 119899sum119894=1

119896119894 minus min 119889 1198891 119889119899) 119906]

+ 120579 minus 12 (120590211198782 +119899sum119894=1

1205902119894+11199092119894) le 120579 (1 + 119906)120579minus2

sdot [ 119899sum119894=1

1198961198941198780 minus min 119889 1198891 119889119899

+ (120579 minus 12 or 0) (12059021 or or 1205902119899)] 1199062 + (1198780119889

Complexity 5

+ 119899sum119894=1

1198961198941198780 minus min 119889 1198891 119889119899) 119906 + (119889 + 119899sum119894=1

119896119894)


1198961198941198780

minus min 119889 1198891 119889119899) 119906 + (119889 + 119899sum119894=1

119896119894)1198780 (21)

where

120582 = min 119889 1198891 119889119899 minus 119899sum119894=1

1198961198941198780


Let

119860 = (119889 + 119899sum119894=1

119896119894)1198780 minus min 119889 1198891 119889119899 (23)

be a constant Then


119894=1

119896119894)1198780 d119905

+ 120579 (1 + 119906)120579minus1 [1205901119878d1198611 (119905) + 119899sum119894=1

120590119894+1119909119894d119861119894+1 (119905)] (24)


sdot [1205901119878d1198611 (119905) + 119899sum119894=1

120590119894+1119909119894d119861119894+1 (119905)] (25)

where


+ (119889 + 119899sum119894=1

119896119894)1198780 = e119902119905120579 (1 + 119906)120579minus2


119896119894)

+ 119902120579

(26)


+ 1198780 (119889 + 119899sum119894=1

119896119894) + 119902120579 119906 isin 119877+(27)

and119865 fl sup119906isin119877+

119891 (119906) + 1 (28)

which gives

119871 [e119902119905119882(119906)] le 120579e119902119905119865 (29)


119864 [e119902119905119882(119906)] = 119882 (119906 (0)) + 119864int1199050119871 [e119902119905119882(119906)] d119905 (30)


119864 [e119902119905119882(119906)] le (1 + 119906 (0))120579 + 119864int1199050e119902119905120579119865d119905

= (1 + 119906 (0))120579 + 120579119865119902 e119902119905(31)

which leads to


sup119864 [(1 + 119906 (119905))120579] le 120579119865119902 fl 1198650 119886119904 (32)


119864 [(1 + 119906 (119905))120579] le 119872 119905 ge 0 (33)


119864[ sup119896120575le119905le(119896+1)120575

(1 + 119906 (119905))120579]le 119864 [(1 + 119906 (119896120575))120579] + 1198681 + 1198682 le 119872 + 1198681 + 1198682

(34)

where

1198681 = 119864[ sup119896120575le119905le(119896+1)120575

1003816100381610038161003816100381610038161003816100381610038161003816int119905

119896120575120579 (1 + 119906)120579minus2

sdot minus1205821199062 + 119860119906 + 1198780(119889 + 119899sum119894=1

119896119894) d1199051003816100381610038161003816100381610038161003816100381610038161003816]

le 119864[ sup119896120575le119905le(119896+1)120575

1003816100381610038161003816100381610038161003816100381610038161003816int119905

119896120575120579 (1 + 119906)120579minus2

sdot 1198780(119889 + 119899sum119894=1

119896119894)119906 + 1198780(119889 + 119899sum119894=1

119896119894) d1199051003816100381610038161003816100381610038161003816100381610038161003816]

= 119864[ sup119896120575le119905le(119896+1)120575

1003816100381610038161003816100381610038161003816100381610038161003816int119905

119896120575120579 (1 + 119906)120579minus2

6 Complexity

sdot 1198780 (119889 + 119899sum119894=1

119896119894) (1 + 119906) d1199051003816100381610038161003816100381610038161003816100381610038161003816] le 1198780 (119889

+ 119899sum119894=1

119896119894)120579119864int(119896+1)120575119896120575

(1 + 119906)120579 d119905

le 1198621120575119864[ sup119896120575le119905le(119896+1)120575

(1 + 119906 (119905))120579] (35)

here

1198621 = 1198780(119889 + 119899sum119894=1

119896119894)120579 (36)


1198682 = 119864[ sup119896120575le119905le(119896+1)120575

1003816100381610038161003816100381610038161003816100381610038161003816int119905

119896120575120579 (1 + 119906)120579minus1

sdot 1205901119878d1198611 (119905) + 119899sum119894=1

120590119894+1119909119894d119861119894+1 (119905)1003816100381610038161003816100381610038161003816100381610038161003816]

le radic32119864[int(119896+1)120575119896120575

1205792 (1 + 119906)2120579minus2(120590211198782

+ 119899sum119894=1

1205902119894+11199092119894) d119905]12

le radic32119864[int(119896+1)120575119896120575

1205792 (1

+ 119906)2120579minus2 (120590211199062 +119899sum119894=1

1205902119894+11199062) d119905]12

le radic3212057912057512119864[ sup119896120575le119905le(119896+1)120575

(12059021 or or 1205902119899) (1

+ 119906)2120579]12 le radic3212057912057512 (12059021 or or 1205902119899)12

sdot 119864 [ sup119896120575le119905le(119896+1)120575

(1 + 119906)120579]

(37)

Hence we have

119864[ sup119896120575le119905le(119896+1)120575

(1 + 119906)120579]le 119864 [(1 + 119906 (119896120575))120579]


(1 + 119906)120579]

(38)


1198621120575 + radic3212057912057512 (12059021 or or 1205902119899)12 le 12 (39)

which yields

119864[ sup119896120575le119905le(119896+1)120575

(1 + 119906)120579] le 2119864 [(1 + 119906 (119896120575))120579] le 2119872 (40)


119875 sup119896120575le119905le(119896+1)120575

(1 + 119906 (119905))120579 ge (119896120575)1+120573

le 119864 [sup119896120575le119905le(119896+1)120575 (1 + 119906 (119905))120579](119896120575)1+120573 le 2119872

(119896120575)1+120573 (41)


sup119896120575le119905le(119896+1)120575

(1 + 119906 (119905))120579 le (119896120575)1+120573 (42)


ln (1 + 119906 (119905))120579ln 119905 le ln (119896120575)1+120573

ln 119905 le ln (119896120575)1+120573ln (119896120575) = 1 + 120573 (43)

which means


sup ln (1 + 119906 (119905))120579ln 119905 le 1 + 120573 119886119904 (44)


sup ln (1 + 119906 (119905))ln 119905 le 1120579 119886119904 (45)



supln [119878 (119905) + sum119899119894=1 119909119894 (119905)]

ln 119905 = lim119905997888rarrinfin

sup ln 119906 (119905)ln 119905


sup ln (1 + 119906 (119905))ln 119905 le 1120579 119886119904

(46)




int1199050(119878 (119904) minus 1205821) d1198611 (119904)119905 = 0 119886119904




int1199050119909119894 (119904) d119861119894+1 (119904)119905 = 0 119886119904

(119894 = 2 3 119899)

(47)


Complexity 7

Proof Define

119883 (119905) = int1199050(119878 (119904) minus 1205821)d1198611 (119904)


119885119894 (119905) = int1199050119909119894 (119904) d119861119894+1 (119904) 119894 = 2 3 119899

(48)


119864[ sup0le119904le119905

|119883 (119904)|120579] le 119862120579119864[int1199050(119878 (119904) minus 1205821)2 d119904]

12

le 1198621205791199051205792119864[ sup0le119904le119905

(119878 (119904) minus 1205821)120579]le 1198621205791199051205792119864[ sup

0le119904le119905(1 + 119906 (119905))120579]

le 21198721198621205791199051205792

(49)


119875120596 sup119896120575le119905le(119896+1)120575

|119883 (119905)|120579 gt (119896120575)1+120572+1205792

le 119864 [|119883 ((119896 + 1) 120575)|120579](119896120575)1+120572+1205792 le 2119872119862120579(119896120575)1+120572 (

(119896 + 1) 120575119896120575 )1205792

le 21+1205792119872119862120579(119896120575)1+120572

(50)


sup119896120575le119905le(119896+1)120575

|119883 (119905)|120579 le (119896120575)1+120572+1205792 (51)


ln 119905 le ln (119896120575)1+120572+1205792ln 119905 le ln (119896120575)1+120572+1205792

ln (119896120575) (52)

which implies

ln |119883 (119905)|120579ln 119905 le 1 + 120572 + 1205792 (53)



sup ln |119883 (119905)|ln 119905 le 1 + 120572 + 1205792120579 (54)


sup ln |119883 (119905)|ln 119905 le 12 + 1120579 (55)


ln |119883 (119905)|ln 119905 le 12 + 1120579 + 120578 (56)

which means



sup 11990512+1120579+120578119905 = 0 (57)

Together with



inf

100381610038161003816100381610038161003816int1199050 (119878 (119904) minus 1205821) d1198611 (119904)100381610038161003816100381610038161003816119905ge 0

(58)

we derive


|119883 (119905)|119905 = 0 119886119904 (59)

and


int1199050(119878 (119904) minus 1205821) d1198611 (119904)119905 = 0 119886119904 (60)





int1199050119909119894 (119905) d119861119894+1 (119904)119905 = 0 119886119904

119894 = 2 3 119899(61)



d120601 (119905) = [(1198780 minus 119878 (119905)) (119889 + 119899sum119894=1

119896119894119909119894)] d119905+ 1205901120601 (119905) d1198611 (119905)

(62)


0119911V(d119911) =


119901 (119911) = exp [2 (119889 + sum119899119894=1 119896119894119909119894) 119878012059021119911 ]


8 Complexity



gt 119899sum119894=1

1198961198941198780 + (120579 minus 12 or 0) (12059021 or or 1205902119899) 11988611989412058211 + 1198871198941205821 +

1198861198942radic119887119894 (1 + 1198871198941205821) ( 119867119889 + 119873sum119899119894=1 119896119894)12

le (119889119894 + 119873 119899sum119894=1

119896119894) + 1205902119894+12 (119894 = 2 3 119899)


119909V (d119909)1 + 1198871119909 gt 120590222 + (119889 + 119872 119899sum119894=1

119896119894)

(64)



supln119909119894 (119905)119905 le 0 119886119904 (119894 = 2 119899) (65)




(66)



(1 + 1198871198941205821) 119909119894

(67)


d119881 = 119878 minus 1205821119878 [(1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1

1198861198941198781 + 119887119894119878119909119894] d119905 + 1205901119878d1198611 (119905) + 12 12058211198782 120590211198782d119905

+ [(1 + 11988711205821) minus (1 + 11988711205821) 119909lowast11199091 ]

sdot [minus(1198891 + 119899sum119894=1

119896119894119909119894)1199091 + 11988611198781 + 11988711198781199091] d119905

+ 12059021199091d1198612 (119905) + 12(1 + 11988711205821) 119909lowast111990921 1205902211990921d119905 + 119899sum

119894=2

(1

+ 1198871198941205821) [minus(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894 + 1198861198941198781 + 119887119894119878119909119894] d119905

+ 120590119894+1119909119894d119861119894+1 (119905) = 119871119881d119905 + (119878 minus 1205821) 1205901d1198611 (119905)


(1 + 1198871198941205821) 120590119894+1119909119894d119861119894+1 (119905) (68)

where

119871119881 = 119878 minus 1205821119878 [(1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1


sdot [minus(1198891 + 119899sum119894=1

119896119894119909119894)1199091 + 11988611198781 + 11988711198781199091]

+ 119899sum119894=2

(1 + 1198871198941205821) [minus(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894 + 1198861198941198781 + 119887119894119878119909119894]

(69)



(70)



Complexity 9


119899sum119894=2

119896119894119909119894+ 119899sum119894=2

119909119894 [ 11988611989412058211 + 1198871198941205821 minus (119889119894 + 119899sum119894=1

119896119894119909119894)] + 12120582112059021+ 12 (1 + 11988711205821) 119909lowast112059022


119899sum119894=2

119896119894119909119894


119909119894 [minus 119899sum119894=1


(71)

Set

119872 = max 1199091 119909119899 119873 = min 1199091 119909119899 (72)

then


119894=2

119896119894


119872 119899sum119894=1


(73)

where


119894=2

119896119894


+ (119899 minus 1)119872 119899sum119894=1

119896119894119909lowast119894 + 12120582112059021+ 12 (1 + 11988711205821) 119909lowast112059022

(74)



0

(119878 (119904) minus 1205821)2119878 (119904) d119904 + 119867+ 1205901 1119905 int119905

0(119878 (119904) minus 1205821) d1198611 (119904)


+ 119899sum119894=2

(1 + 1198871198941205821) 120590119894+1 int1199050119909119894 (119904) d119861119894+1 (119905)

(75)



sup 1119905 int1199050

(119878 (119904) minus 1205821)2119878 (119904) d119904 le 119867119889 + 119873sum119899119894=1 119896119894 (76)

Define

119881119894 (119909119894 (119905)) = ln119909119894 (119905) 119894 = 2 3 119899 (77)


d119881119894 = [minus(119889119894 + 119899sum119894=1

119896119894119909119894) + 1198861198941198781 + 119887119894119878 minus 1205902119894+12 ] d119905


119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12 ] d119905

+ 120590119894+1d119861119894+1 (119905)

(78)


119881119894 (119905) minus 119881119894 (0)119905le 1198861198941 + 1198871198941205821 ⟨

119878 (119905) minus 12058211 + 119887119894119878 (119905)⟩ minus (119889119894 + 119873 119899sum119894=1

119896119894)

10 Complexity

+ 11988611989412058211 + 1198871198941205821 minus1205902119894+12 + 120590119894+1119861119894+1 (119905)119905


119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 120590119894+1119861119894+1 (119905)119905le 1198861198942radic119887119894 (1 + 1198871198941205821) [1119905 int119905

0

(119878 (119905) minus 1205821)2119878 (119905) d119904]12

minus (119889119894 + 119873 119899sum119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 120590119894+1119861119894+1 (119905)119905 (79)

which meansln119909119894 (119905)119905

le ln119909119894 (0)119905+ 1198861198942radic119887119894 (1 + 1198871198941205821) [1119905 int119905

0

(119878 (119905) minus 1205821)2119878 (119905) d119904]12

minus (119889119894 + 119873 119899sum119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 120590119894+1119861119894+1 (119905)119905

(80)



supln119909119894 (119905)119905

le minus(119889119894 + 119873 119899sum119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 1198861198942radic119887119894 (1 + 1198871198941205821) ( 119867119889 + 119873sum119899119894=1 119896119894)12

(119894 = 2 119899)

(81)

If

11988611989412058211 + 1198871198941205821 +1198861198942radic119887119894 (1 + 1198871198941205821) ( 119867119889 + 119873sum119899119894=1 119896119894)

12

le (119889119894 + 119873 119899sum119894=1

119896119894) + 1205902119894+12 (82)

then we have


supln119909119894 (119905)119905 le 0 119886119904 (119894 = 2 119899) (83)




1198861198941199091198941 + 119887119894119878 minus 120590212 ] d119905+ 1205901d1198611 (119905)


(84)



119894=1

1198861198941199091198941 + 119887119894119878] d119905

minus 120590212 + 1119905 int11990501205901d1198611 (119905)

= (119889 + 119899sum119894=1

119896119894119909119894)1198780 ⟨1119878⟩ minus 1198861⟨ 11990911 + 1198871119878⟩

minus 119899sum119894=2

119886119894⟨ 1199091198941 + 119887119894119878⟩ minus (119889 + 119899sum119894=1

119896119894119909119894) minus 120590212+ 12059011198611 (119905)119905 + ln 119878 (0)119905

ln120601 (119905)119905= ln120601 (0)119905 + 1119905 int119905

0

(1198780 minus 120601) (119889 + sum119899119894=1 119896119894119909119894)120601 d119905 minus 120590212+ 12059011198611 (119905)119905

= (119889 + 119899sum119894=1

119896119894119909119894)1198780⟨1120601⟩ minus (119889 + 119899sum119894=1

119896119894119909119894) minus 120590212+ 12059011198611 (119905)119905 + ln120601 (0)119905

(85)

Complexity 11

Therefore

0 ge ln 119878 (119905) minus ln120601 (119905)119905= (119889 + 119899sum

119894=1

119896119894119909119894)1198780⟨1119878 minus 1120601⟩ minus 1198861⟨ 11990911 + 1198871119878⟩

minus 119899sum119894=2

119886119894⟨ 1199091198941 + 119887119894119878⟩

ge (119889 + 119873 119899sum119894=1

119896119894)1198780⟨1119878 minus 1120601⟩ minus 1198861 ⟨1199091⟩

minus 119899sum119894=2

119886119894 ⟨119909119894⟩

(86)

which implies that

⟨1119878 minus 1120601⟩

le 1(119889 + 119873sum119899119894=1 119896119894) 1198780 (1198861 ⟨1199091⟩ + 119899sum119894=2

119886119894 ⟨119909119894⟩) (87)


119894=1

119896119894119909119894) + 11988611198781 + 1198871119878 minus 120590222 ] d119905+ 1205902d1198612 (119905)

ge [minus(1198891 + 119872 119899sum119894=1

119896119894) + 11988611198781 + 1198871119878 minus 120590222 ] d119905+ 1205902d1198612 (119905)

(88)



119894=1

119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩ minus 1198861⟨ 120601 minus 119878(1 + 1198871120601) (1 + 1198871119878)⟩


119894=1

119896119894)+ 1198861⟨ 1206011 + 1198871120601⟩ minus 119886111988721 ⟨1119878 minus 1120601⟩ + 12059021198612 (119905)119905


119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩

minus 119886111988721 (119889 + 119873sum119899119894=1 119896119894) 1198780 (1198861 ⟨1199091⟩ + 119899sum119894=2

119886119894 ⟨119909119894⟩)+ 12059021198612 (119905)119905


119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩ minus 11988621 ⟨1199091⟩11988721 (119889 + 119873sum119899119894=1 119896119894) 1198780+ 12059021198612 (119905)119905

(89)



sup ln 1199091 (119905)119905 le 0 (90)

which yields


119894=1

119896119894)+ 1198861⟨ 1206011 + 1198871120601⟩ + 12059021198612 (119905)119905

(91)

that is



119894=1

119896119894) + 1198861 intinfin0

119909V (d119909)1 + 1198871119909 ] (92)

Therefore if

1198861 intinfin0

119909V (d119909)1 + 1198871119909 gt 120590222 + (119889 + 119872 119899sum119894=1

119896119894) (93)

then


inf ⟨1199091 (119905)⟩ gt 0 119886119904 (94)



119886119894 intinfin0

119909V (d119909)1 + 119887119894119909 lt 119889119894 + 119873 119899sum119894=1

119896119894 + 1205902119894+12 119894 = 1 2 119899 (95)

12 Complexity

then


int1199050119878 (119904) d119904119905 = 1198780 119886119904


supln 119909119894 (119905)119905 lt 0 119886119904 119894 = 1 2 119899

(96)




119896119894119909119894 + 1198861198941198781 + 119887119894119878 minus 1205902119894+12 )d119905+ 120590119894+1d119861119894+1 (119905)


119896119894 + 1198861198941206011 + 119887119894120601 minus 1205902119894+12 )d119905+ 120590119894+1d119861119894+1 (119905)

(97)


119894=1

119896119894 minus 1205902119894+12 ) + 119886119894⟨ 1206011 + 119887119894120601⟩+ 120590119894+1119861119894+1 (119905)119905 + ln 119909119894 (0)119905

(98)


119886119894 intinfin0

119909V (d119909)1 + 119887119894119909 lt 119889119894 + 119873 119899sum119894=1

119896119894 + 1205902119894+12 119894 = 1 2 119899 (99)

we know that


supln119909119894 (119905)119905 le 0 119894 = 1 2 119899 (100)



d119884 = minus(119889 + 119899sum119894=1

119896119894119909119894)119884d119905 + 1205901 (1198780 + 119884) d1198611 (119905) (101)



119894=1

119896119894119909119894) minus 120576119878120576] d119905+ 1205901119878120576d1198611 (119905)

(102)

Then

d (119878 minus 119878120576) = [(120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878) 119878

minus (119889 + 120576 + 119899sum119894=1

119896119894119909119894) (119878 minus 119878120576)] d119905 + 1205901 (119878minus 119878120576) d1198611 (119905)

(103)



119894=1

119896119894119909119894 + 120576 + 120590212 ) 119905]

sdot int1199050exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904

(104)


120576 gt 119899sum119894=1

119886119894119909119894 (119905)1 + 119887119894119878 (119905) forall119905 ge 119879 (105)


119894=1

119896119894119909119894 + 120576

+ 120590212 ) 119905]

sdot int1198790exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904

+ int119905119879exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

Complexity 13

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904 ge exp[12059011198611 (119905)

minus (119889 + 119872 119899sum119894=1

119896119894 + 120576 + 120590212 ) 119905]

sdot int1198790exp[(119889 + 119873 119899sum

119894=1

119896119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904(106)


exp[12059011198611 (119905) minus (119889 + 119872 119899sum119894=1

119896119894 + 120576 + 120590212 ) 119905] 997888rarr 01003816100381610038161003816100381610038161003816100381610038161003816int119879

0exp[(119889 + 119873 119899sum

119894=1

119896119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d1199041003816100381610038161003816100381610038161003816100381610038161003816 lt infin

(107)

which implies that



inf (119878 (119905) minus 119878120576 (119905))ge 0 119886119904 (108)

Now we consider

d (120601 (119905) minus 119878120576 (119905))= [120576119878120576 (119905) minus (119889 + 119899sum

119894=1

119896119894119909119894) (120601 (119905) minus 119878120576 (119905))] d119905+ 1205901 (120601 (119905) minus 119878120576 (119905)) d1198611 (119905)

le [120576119878120576 (119905) minus (119889 + 119873 119899sum119894=1

119896119894)(120601 (119905) minus 119878120576 (119905))] d119905+ 1205901 (120601 (119905) minus 119878120576 (119905)) d1198611 (119905)

(109)



119894=1


(110)


119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816le 119864[int119905

0120576119878120576 (119904) minus (119889 + 119873 119899sum

119894=1

119896119894) 1003816100381610038161003816120601 (119904) minus 119878120576 (119904)1003816100381610038161003816 d119904]

le 119864[int1199050120576120601 (119904) minus (119889 + 119873 119899sum

119894=1

119896119894) 1003816100381610038161003816120601 (119904) minus 119878120576 (119904)1003816100381610038161003816 d119904] (111)

then

119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816 le 120576 sup0le119904

119864120601 (119904) (112)

which leads to

lim120576997888rarr0


119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816 = 0 (113)



(120601 (119905) minus 119878 (119905)) = 0 (114)









14 Complexity

200 400 600 800 1000 1200 1400 1600 1800 20000Time

0

01

02

03

04

05

06

07

08

Valu

es


0 100806040200

02

04x2(t) path

(a)

0

01

02

03

04

05

06

07

08

09

1

Valu

es

1 2 3 4 5 6 7 8 9 100Time


times104

(b)


S(t) path

01

02

03

04

05

06

07

S(t)

200 400 600 800 1000 1200 1400 1600 1800 20000Time

(a)

0

5

10

15

20

25

30


(b)



d1199091 (119905) = [minus (032 + 00071199091 (119905) + 0011199092 (119905)) 1199091 (119905)

+ 2119878 (119905)1 + 3119878 (119905)1199091 (119905)] d119905 + 011199091 (119905) d1198612 (119905) d1199092 (119905) = [minus (031 + 00071199091 (119905) + 0011199092 (119905)) 1199092 (119905)

+ 2119878 (119905)1 + 5119878 (119905)1199092 (119905)] d119905 + 021199092 (119905) d1198613 (119905) (116)


Complexity 15

01

02

03

04

05

06

07

08

09

1

S(t)

S(t) path

1 2 3 4 5 6 7 8 9 100Time times10

4

(a)


0

5

10

15

20

25

30

(b)


x1(t) path

005

01

015

02

025

03

035

04

x1(t)

200 400 600 800 1000 1200 1400 1600 1800 20000Time

(a)


0

5

10

15

20

25

30

35

40

45

50

(b)





16 Complexity

1 2 3 4 5 6 7 8 9 100Time

005

01

015

02

025

03

035

04

045

05

x1(t)

x1(t) path

times104

(a)


0

5

10

15

20

25

30

35

40

45

50

(b)






d1199091 (119905) = [minus (09 + 021199091 (119905) + 021199092 (119905)) 1199091 (119905)+ 119878 (119905)1 + 04119878 (119905)1199091 (119905)] d119905 + 081199091 (119905) d1198612 (119905)

d1199092 (119905) = [minus (09 + 021199091 (119905) + 021199092 (119905)) 1199092 (119905)+ 119878 (119905)1 + 06119878 (119905)1199092 (119905)] d119905 + 081199092 (119905) d1198613 (119905)

(117)





Data Availability




Acknowledgments


Complexity 17

0 20018016014012010080604020Time

0

02

04

06

08

1

12

14

16

18

2Va

lues


0 109876543210

01

02


(a)

0 10000900080007000600050004000300020001000Time

0

05

1

15

2

25

Valu

es



0

02

04

0 87654321

(b)



References






















18 Complexity






























Journal of












Journal of







Volume 2018






Volume 2018








4 Complexity


119881 (119878 (0) 119909119899 (0))+ ((1198780 + 119886) 119889 + 119899sum

119894=1

119889119894 + 119886120590212 + 119899sum119894=1


(13)




⟨119909 (119905)⟩ = 1119905 int1199050119909 (119903) d119903 (14)



supln [119878 (119905) + sum119899119894=1 119909119894 (119905)]

ln 119905 le 1120579 119886119904 (15)


120582 = min 119889 1198891 119889119899 minus 119899sum119894=1

1198961198941198780


Proof Let

119906 (119905) = 119878 (119905) + 119899sum119894=1

119909119894 (119905) (17)

and

119882(119906) = (1 + 119906)120579 (18)


d119906 = [(1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1

(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894] d119905 + 1205901119878d1198611 (119905)

+ 119899sum119894=1

120590119894+1119909119894d119861119894+1 (119905)

(19)


d119882 = 120579 (1 + 119906)120579minus2(1 + 119906) [(1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1

(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894] + 120579 minus 12 (120590211198782

+ 119899sum119894=1

1205902119894+11199092119894) d119905 + 120579 (1 + 119906)120579minus1 [1205901119878d1198611 (119905)

+ 119899sum119894=1

120590119894+1119909119894d119861119894+1 (119905)]

(20)

Denote

119871119882(119906) = 120579 (1 + 119906)120579minus2 (1 + 119906)

sdot [(1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894) minus 119899sum119894=1

(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894]

+ 120579 minus 12 (120590211198782 +119899sum119894=1

1205902119894+11199092119894) = 120579 (1 + 119906)120579minus2(1

+ 119906) [1198780 (119889 + 119899sum119894=1

119896119894119909119894) minus 119878(119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1

(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894] + 120579 minus 12 (120590211198782

+ 119899sum119894=1

1205902119894+11199092119894) le 120579 (1 + 119906)120579minus2 (1 + 119906) [1198780119889

+ (1198780 119899sum119894=1

119896119894 minus min 119889 1198891 119889119899) 119906]

+ 120579 minus 12 (120590211198782 +119899sum119894=1

1205902119894+11199092119894) le 120579 (1 + 119906)120579minus2

sdot [ 119899sum119894=1

1198961198941198780 minus min 119889 1198891 119889119899

+ (120579 minus 12 or 0) (12059021 or or 1205902119899)] 1199062 + (1198780119889

Complexity 5

+ 119899sum119894=1

1198961198941198780 minus min 119889 1198891 119889119899) 119906 + (119889 + 119899sum119894=1

119896119894)


1198961198941198780

minus min 119889 1198891 119889119899) 119906 + (119889 + 119899sum119894=1

119896119894)1198780 (21)

where

120582 = min 119889 1198891 119889119899 minus 119899sum119894=1

1198961198941198780


Let

119860 = (119889 + 119899sum119894=1

119896119894)1198780 minus min 119889 1198891 119889119899 (23)

be a constant Then


119894=1

119896119894)1198780 d119905

+ 120579 (1 + 119906)120579minus1 [1205901119878d1198611 (119905) + 119899sum119894=1

120590119894+1119909119894d119861119894+1 (119905)] (24)


sdot [1205901119878d1198611 (119905) + 119899sum119894=1

120590119894+1119909119894d119861119894+1 (119905)] (25)

where


+ (119889 + 119899sum119894=1

119896119894)1198780 = e119902119905120579 (1 + 119906)120579minus2


119896119894)

+ 119902120579

(26)


+ 1198780 (119889 + 119899sum119894=1

119896119894) + 119902120579 119906 isin 119877+(27)

and119865 fl sup119906isin119877+

119891 (119906) + 1 (28)

which gives

119871 [e119902119905119882(119906)] le 120579e119902119905119865 (29)


119864 [e119902119905119882(119906)] = 119882 (119906 (0)) + 119864int1199050119871 [e119902119905119882(119906)] d119905 (30)


119864 [e119902119905119882(119906)] le (1 + 119906 (0))120579 + 119864int1199050e119902119905120579119865d119905

= (1 + 119906 (0))120579 + 120579119865119902 e119902119905(31)

which leads to


sup119864 [(1 + 119906 (119905))120579] le 120579119865119902 fl 1198650 119886119904 (32)


119864 [(1 + 119906 (119905))120579] le 119872 119905 ge 0 (33)


119864[ sup119896120575le119905le(119896+1)120575

(1 + 119906 (119905))120579]le 119864 [(1 + 119906 (119896120575))120579] + 1198681 + 1198682 le 119872 + 1198681 + 1198682

(34)

where

1198681 = 119864[ sup119896120575le119905le(119896+1)120575

1003816100381610038161003816100381610038161003816100381610038161003816int119905

119896120575120579 (1 + 119906)120579minus2

sdot minus1205821199062 + 119860119906 + 1198780(119889 + 119899sum119894=1

119896119894) d1199051003816100381610038161003816100381610038161003816100381610038161003816]

le 119864[ sup119896120575le119905le(119896+1)120575

1003816100381610038161003816100381610038161003816100381610038161003816int119905

119896120575120579 (1 + 119906)120579minus2

sdot 1198780(119889 + 119899sum119894=1

119896119894)119906 + 1198780(119889 + 119899sum119894=1

119896119894) d1199051003816100381610038161003816100381610038161003816100381610038161003816]

= 119864[ sup119896120575le119905le(119896+1)120575

1003816100381610038161003816100381610038161003816100381610038161003816int119905

119896120575120579 (1 + 119906)120579minus2

6 Complexity

sdot 1198780 (119889 + 119899sum119894=1

119896119894) (1 + 119906) d1199051003816100381610038161003816100381610038161003816100381610038161003816] le 1198780 (119889

+ 119899sum119894=1

119896119894)120579119864int(119896+1)120575119896120575

(1 + 119906)120579 d119905

le 1198621120575119864[ sup119896120575le119905le(119896+1)120575

(1 + 119906 (119905))120579] (35)

here

1198621 = 1198780(119889 + 119899sum119894=1

119896119894)120579 (36)


1198682 = 119864[ sup119896120575le119905le(119896+1)120575

1003816100381610038161003816100381610038161003816100381610038161003816int119905

119896120575120579 (1 + 119906)120579minus1

sdot 1205901119878d1198611 (119905) + 119899sum119894=1

120590119894+1119909119894d119861119894+1 (119905)1003816100381610038161003816100381610038161003816100381610038161003816]

le radic32119864[int(119896+1)120575119896120575

1205792 (1 + 119906)2120579minus2(120590211198782

+ 119899sum119894=1

1205902119894+11199092119894) d119905]12

le radic32119864[int(119896+1)120575119896120575

1205792 (1

+ 119906)2120579minus2 (120590211199062 +119899sum119894=1

1205902119894+11199062) d119905]12

le radic3212057912057512119864[ sup119896120575le119905le(119896+1)120575

(12059021 or or 1205902119899) (1

+ 119906)2120579]12 le radic3212057912057512 (12059021 or or 1205902119899)12

sdot 119864 [ sup119896120575le119905le(119896+1)120575

(1 + 119906)120579]

(37)

Hence we have

119864[ sup119896120575le119905le(119896+1)120575

(1 + 119906)120579]le 119864 [(1 + 119906 (119896120575))120579]


(1 + 119906)120579]

(38)


1198621120575 + radic3212057912057512 (12059021 or or 1205902119899)12 le 12 (39)

which yields

119864[ sup119896120575le119905le(119896+1)120575

(1 + 119906)120579] le 2119864 [(1 + 119906 (119896120575))120579] le 2119872 (40)


119875 sup119896120575le119905le(119896+1)120575

(1 + 119906 (119905))120579 ge (119896120575)1+120573

le 119864 [sup119896120575le119905le(119896+1)120575 (1 + 119906 (119905))120579](119896120575)1+120573 le 2119872

(119896120575)1+120573 (41)


sup119896120575le119905le(119896+1)120575

(1 + 119906 (119905))120579 le (119896120575)1+120573 (42)


ln (1 + 119906 (119905))120579ln 119905 le ln (119896120575)1+120573

ln 119905 le ln (119896120575)1+120573ln (119896120575) = 1 + 120573 (43)

which means


sup ln (1 + 119906 (119905))120579ln 119905 le 1 + 120573 119886119904 (44)


sup ln (1 + 119906 (119905))ln 119905 le 1120579 119886119904 (45)



supln [119878 (119905) + sum119899119894=1 119909119894 (119905)]

ln 119905 = lim119905997888rarrinfin

sup ln 119906 (119905)ln 119905


sup ln (1 + 119906 (119905))ln 119905 le 1120579 119886119904

(46)




int1199050(119878 (119904) minus 1205821) d1198611 (119904)119905 = 0 119886119904




int1199050119909119894 (119904) d119861119894+1 (119904)119905 = 0 119886119904

(119894 = 2 3 119899)

(47)


Complexity 7

Proof Define

119883 (119905) = int1199050(119878 (119904) minus 1205821)d1198611 (119904)


119885119894 (119905) = int1199050119909119894 (119904) d119861119894+1 (119904) 119894 = 2 3 119899

(48)


119864[ sup0le119904le119905

|119883 (119904)|120579] le 119862120579119864[int1199050(119878 (119904) minus 1205821)2 d119904]

12

le 1198621205791199051205792119864[ sup0le119904le119905

(119878 (119904) minus 1205821)120579]le 1198621205791199051205792119864[ sup

0le119904le119905(1 + 119906 (119905))120579]

le 21198721198621205791199051205792

(49)


119875120596 sup119896120575le119905le(119896+1)120575

|119883 (119905)|120579 gt (119896120575)1+120572+1205792

le 119864 [|119883 ((119896 + 1) 120575)|120579](119896120575)1+120572+1205792 le 2119872119862120579(119896120575)1+120572 (

(119896 + 1) 120575119896120575 )1205792

le 21+1205792119872119862120579(119896120575)1+120572

(50)


sup119896120575le119905le(119896+1)120575

|119883 (119905)|120579 le (119896120575)1+120572+1205792 (51)


ln 119905 le ln (119896120575)1+120572+1205792ln 119905 le ln (119896120575)1+120572+1205792

ln (119896120575) (52)

which implies

ln |119883 (119905)|120579ln 119905 le 1 + 120572 + 1205792 (53)



sup ln |119883 (119905)|ln 119905 le 1 + 120572 + 1205792120579 (54)


sup ln |119883 (119905)|ln 119905 le 12 + 1120579 (55)


ln |119883 (119905)|ln 119905 le 12 + 1120579 + 120578 (56)

which means



sup 11990512+1120579+120578119905 = 0 (57)

Together with



inf

100381610038161003816100381610038161003816int1199050 (119878 (119904) minus 1205821) d1198611 (119904)100381610038161003816100381610038161003816119905ge 0

(58)

we derive


|119883 (119905)|119905 = 0 119886119904 (59)

and


int1199050(119878 (119904) minus 1205821) d1198611 (119904)119905 = 0 119886119904 (60)





int1199050119909119894 (119905) d119861119894+1 (119904)119905 = 0 119886119904

119894 = 2 3 119899(61)



d120601 (119905) = [(1198780 minus 119878 (119905)) (119889 + 119899sum119894=1

119896119894119909119894)] d119905+ 1205901120601 (119905) d1198611 (119905)

(62)


0119911V(d119911) =


119901 (119911) = exp [2 (119889 + sum119899119894=1 119896119894119909119894) 119878012059021119911 ]


8 Complexity



gt 119899sum119894=1

1198961198941198780 + (120579 minus 12 or 0) (12059021 or or 1205902119899) 11988611989412058211 + 1198871198941205821 +

1198861198942radic119887119894 (1 + 1198871198941205821) ( 119867119889 + 119873sum119899119894=1 119896119894)12

le (119889119894 + 119873 119899sum119894=1

119896119894) + 1205902119894+12 (119894 = 2 3 119899)


119909V (d119909)1 + 1198871119909 gt 120590222 + (119889 + 119872 119899sum119894=1

119896119894)

(64)



supln119909119894 (119905)119905 le 0 119886119904 (119894 = 2 119899) (65)




(66)



(1 + 1198871198941205821) 119909119894

(67)


d119881 = 119878 minus 1205821119878 [(1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1

1198861198941198781 + 119887119894119878119909119894] d119905 + 1205901119878d1198611 (119905) + 12 12058211198782 120590211198782d119905

+ [(1 + 11988711205821) minus (1 + 11988711205821) 119909lowast11199091 ]

sdot [minus(1198891 + 119899sum119894=1

119896119894119909119894)1199091 + 11988611198781 + 11988711198781199091] d119905

+ 12059021199091d1198612 (119905) + 12(1 + 11988711205821) 119909lowast111990921 1205902211990921d119905 + 119899sum

119894=2

(1

+ 1198871198941205821) [minus(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894 + 1198861198941198781 + 119887119894119878119909119894] d119905

+ 120590119894+1119909119894d119861119894+1 (119905) = 119871119881d119905 + (119878 minus 1205821) 1205901d1198611 (119905)


(1 + 1198871198941205821) 120590119894+1119909119894d119861119894+1 (119905) (68)

where

119871119881 = 119878 minus 1205821119878 [(1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1


sdot [minus(1198891 + 119899sum119894=1

119896119894119909119894)1199091 + 11988611198781 + 11988711198781199091]

+ 119899sum119894=2

(1 + 1198871198941205821) [minus(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894 + 1198861198941198781 + 119887119894119878119909119894]

(69)



(70)



Complexity 9


119899sum119894=2

119896119894119909119894+ 119899sum119894=2

119909119894 [ 11988611989412058211 + 1198871198941205821 minus (119889119894 + 119899sum119894=1

119896119894119909119894)] + 12120582112059021+ 12 (1 + 11988711205821) 119909lowast112059022


119899sum119894=2

119896119894119909119894


119909119894 [minus 119899sum119894=1


(71)

Set

119872 = max 1199091 119909119899 119873 = min 1199091 119909119899 (72)

then


119894=2

119896119894


119872 119899sum119894=1


(73)

where


119894=2

119896119894


+ (119899 minus 1)119872 119899sum119894=1

119896119894119909lowast119894 + 12120582112059021+ 12 (1 + 11988711205821) 119909lowast112059022

(74)



0

(119878 (119904) minus 1205821)2119878 (119904) d119904 + 119867+ 1205901 1119905 int119905

0(119878 (119904) minus 1205821) d1198611 (119904)


+ 119899sum119894=2

(1 + 1198871198941205821) 120590119894+1 int1199050119909119894 (119904) d119861119894+1 (119905)

(75)



sup 1119905 int1199050

(119878 (119904) minus 1205821)2119878 (119904) d119904 le 119867119889 + 119873sum119899119894=1 119896119894 (76)

Define

119881119894 (119909119894 (119905)) = ln119909119894 (119905) 119894 = 2 3 119899 (77)


d119881119894 = [minus(119889119894 + 119899sum119894=1

119896119894119909119894) + 1198861198941198781 + 119887119894119878 minus 1205902119894+12 ] d119905


119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12 ] d119905

+ 120590119894+1d119861119894+1 (119905)

(78)


119881119894 (119905) minus 119881119894 (0)119905le 1198861198941 + 1198871198941205821 ⟨

119878 (119905) minus 12058211 + 119887119894119878 (119905)⟩ minus (119889119894 + 119873 119899sum119894=1

119896119894)

10 Complexity

+ 11988611989412058211 + 1198871198941205821 minus1205902119894+12 + 120590119894+1119861119894+1 (119905)119905


119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 120590119894+1119861119894+1 (119905)119905le 1198861198942radic119887119894 (1 + 1198871198941205821) [1119905 int119905

0

(119878 (119905) minus 1205821)2119878 (119905) d119904]12

minus (119889119894 + 119873 119899sum119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 120590119894+1119861119894+1 (119905)119905 (79)

which meansln119909119894 (119905)119905

le ln119909119894 (0)119905+ 1198861198942radic119887119894 (1 + 1198871198941205821) [1119905 int119905

0

(119878 (119905) minus 1205821)2119878 (119905) d119904]12

minus (119889119894 + 119873 119899sum119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 120590119894+1119861119894+1 (119905)119905

(80)



supln119909119894 (119905)119905

le minus(119889119894 + 119873 119899sum119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 1198861198942radic119887119894 (1 + 1198871198941205821) ( 119867119889 + 119873sum119899119894=1 119896119894)12

(119894 = 2 119899)

(81)

If

11988611989412058211 + 1198871198941205821 +1198861198942radic119887119894 (1 + 1198871198941205821) ( 119867119889 + 119873sum119899119894=1 119896119894)

12

le (119889119894 + 119873 119899sum119894=1

119896119894) + 1205902119894+12 (82)

then we have


supln119909119894 (119905)119905 le 0 119886119904 (119894 = 2 119899) (83)




1198861198941199091198941 + 119887119894119878 minus 120590212 ] d119905+ 1205901d1198611 (119905)


(84)



119894=1

1198861198941199091198941 + 119887119894119878] d119905

minus 120590212 + 1119905 int11990501205901d1198611 (119905)

= (119889 + 119899sum119894=1

119896119894119909119894)1198780 ⟨1119878⟩ minus 1198861⟨ 11990911 + 1198871119878⟩

minus 119899sum119894=2

119886119894⟨ 1199091198941 + 119887119894119878⟩ minus (119889 + 119899sum119894=1

119896119894119909119894) minus 120590212+ 12059011198611 (119905)119905 + ln 119878 (0)119905

ln120601 (119905)119905= ln120601 (0)119905 + 1119905 int119905

0

(1198780 minus 120601) (119889 + sum119899119894=1 119896119894119909119894)120601 d119905 minus 120590212+ 12059011198611 (119905)119905

= (119889 + 119899sum119894=1

119896119894119909119894)1198780⟨1120601⟩ minus (119889 + 119899sum119894=1

119896119894119909119894) minus 120590212+ 12059011198611 (119905)119905 + ln120601 (0)119905

(85)

Complexity 11

Therefore

0 ge ln 119878 (119905) minus ln120601 (119905)119905= (119889 + 119899sum

119894=1

119896119894119909119894)1198780⟨1119878 minus 1120601⟩ minus 1198861⟨ 11990911 + 1198871119878⟩

minus 119899sum119894=2

119886119894⟨ 1199091198941 + 119887119894119878⟩

ge (119889 + 119873 119899sum119894=1

119896119894)1198780⟨1119878 minus 1120601⟩ minus 1198861 ⟨1199091⟩

minus 119899sum119894=2

119886119894 ⟨119909119894⟩

(86)

which implies that

⟨1119878 minus 1120601⟩

le 1(119889 + 119873sum119899119894=1 119896119894) 1198780 (1198861 ⟨1199091⟩ + 119899sum119894=2

119886119894 ⟨119909119894⟩) (87)


119894=1

119896119894119909119894) + 11988611198781 + 1198871119878 minus 120590222 ] d119905+ 1205902d1198612 (119905)

ge [minus(1198891 + 119872 119899sum119894=1

119896119894) + 11988611198781 + 1198871119878 minus 120590222 ] d119905+ 1205902d1198612 (119905)

(88)



119894=1

119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩ minus 1198861⟨ 120601 minus 119878(1 + 1198871120601) (1 + 1198871119878)⟩


119894=1

119896119894)+ 1198861⟨ 1206011 + 1198871120601⟩ minus 119886111988721 ⟨1119878 minus 1120601⟩ + 12059021198612 (119905)119905


119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩

minus 119886111988721 (119889 + 119873sum119899119894=1 119896119894) 1198780 (1198861 ⟨1199091⟩ + 119899sum119894=2

119886119894 ⟨119909119894⟩)+ 12059021198612 (119905)119905


119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩ minus 11988621 ⟨1199091⟩11988721 (119889 + 119873sum119899119894=1 119896119894) 1198780+ 12059021198612 (119905)119905

(89)



sup ln 1199091 (119905)119905 le 0 (90)

which yields


119894=1

119896119894)+ 1198861⟨ 1206011 + 1198871120601⟩ + 12059021198612 (119905)119905

(91)

that is



119894=1

119896119894) + 1198861 intinfin0

119909V (d119909)1 + 1198871119909 ] (92)

Therefore if

1198861 intinfin0

119909V (d119909)1 + 1198871119909 gt 120590222 + (119889 + 119872 119899sum119894=1

119896119894) (93)

then


inf ⟨1199091 (119905)⟩ gt 0 119886119904 (94)



119886119894 intinfin0

119909V (d119909)1 + 119887119894119909 lt 119889119894 + 119873 119899sum119894=1

119896119894 + 1205902119894+12 119894 = 1 2 119899 (95)

12 Complexity

then


int1199050119878 (119904) d119904119905 = 1198780 119886119904


supln 119909119894 (119905)119905 lt 0 119886119904 119894 = 1 2 119899

(96)




119896119894119909119894 + 1198861198941198781 + 119887119894119878 minus 1205902119894+12 )d119905+ 120590119894+1d119861119894+1 (119905)


119896119894 + 1198861198941206011 + 119887119894120601 minus 1205902119894+12 )d119905+ 120590119894+1d119861119894+1 (119905)

(97)


119894=1

119896119894 minus 1205902119894+12 ) + 119886119894⟨ 1206011 + 119887119894120601⟩+ 120590119894+1119861119894+1 (119905)119905 + ln 119909119894 (0)119905

(98)


119886119894 intinfin0

119909V (d119909)1 + 119887119894119909 lt 119889119894 + 119873 119899sum119894=1

119896119894 + 1205902119894+12 119894 = 1 2 119899 (99)

we know that


supln119909119894 (119905)119905 le 0 119894 = 1 2 119899 (100)



d119884 = minus(119889 + 119899sum119894=1

119896119894119909119894)119884d119905 + 1205901 (1198780 + 119884) d1198611 (119905) (101)



119894=1

119896119894119909119894) minus 120576119878120576] d119905+ 1205901119878120576d1198611 (119905)

(102)

Then

d (119878 minus 119878120576) = [(120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878) 119878

minus (119889 + 120576 + 119899sum119894=1

119896119894119909119894) (119878 minus 119878120576)] d119905 + 1205901 (119878minus 119878120576) d1198611 (119905)

(103)



119894=1

119896119894119909119894 + 120576 + 120590212 ) 119905]

sdot int1199050exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904

(104)


120576 gt 119899sum119894=1

119886119894119909119894 (119905)1 + 119887119894119878 (119905) forall119905 ge 119879 (105)


119894=1

119896119894119909119894 + 120576

+ 120590212 ) 119905]

sdot int1198790exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904

+ int119905119879exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

Complexity 13

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904 ge exp[12059011198611 (119905)

minus (119889 + 119872 119899sum119894=1

119896119894 + 120576 + 120590212 ) 119905]

sdot int1198790exp[(119889 + 119873 119899sum

119894=1

119896119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904(106)


exp[12059011198611 (119905) minus (119889 + 119872 119899sum119894=1

119896119894 + 120576 + 120590212 ) 119905] 997888rarr 01003816100381610038161003816100381610038161003816100381610038161003816int119879

0exp[(119889 + 119873 119899sum

119894=1

119896119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d1199041003816100381610038161003816100381610038161003816100381610038161003816 lt infin

(107)

which implies that



inf (119878 (119905) minus 119878120576 (119905))ge 0 119886119904 (108)

Now we consider

d (120601 (119905) minus 119878120576 (119905))= [120576119878120576 (119905) minus (119889 + 119899sum

119894=1

119896119894119909119894) (120601 (119905) minus 119878120576 (119905))] d119905+ 1205901 (120601 (119905) minus 119878120576 (119905)) d1198611 (119905)

le [120576119878120576 (119905) minus (119889 + 119873 119899sum119894=1

119896119894)(120601 (119905) minus 119878120576 (119905))] d119905+ 1205901 (120601 (119905) minus 119878120576 (119905)) d1198611 (119905)

(109)



119894=1


(110)


119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816le 119864[int119905

0120576119878120576 (119904) minus (119889 + 119873 119899sum

119894=1

119896119894) 1003816100381610038161003816120601 (119904) minus 119878120576 (119904)1003816100381610038161003816 d119904]

le 119864[int1199050120576120601 (119904) minus (119889 + 119873 119899sum

119894=1

119896119894) 1003816100381610038161003816120601 (119904) minus 119878120576 (119904)1003816100381610038161003816 d119904] (111)

then

119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816 le 120576 sup0le119904

119864120601 (119904) (112)

which leads to

lim120576997888rarr0


119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816 = 0 (113)



(120601 (119905) minus 119878 (119905)) = 0 (114)









14 Complexity

200 400 600 800 1000 1200 1400 1600 1800 20000Time

0

01

02

03

04

05

06

07

08

Valu

es


0 100806040200

02

04x2(t) path

(a)

0

01

02

03

04

05

06

07

08

09

1

Valu

es

1 2 3 4 5 6 7 8 9 100Time


times104

(b)


S(t) path

01

02

03

04

05

06

07

S(t)

200 400 600 800 1000 1200 1400 1600 1800 20000Time

(a)

0

5

10

15

20

25

30


(b)



d1199091 (119905) = [minus (032 + 00071199091 (119905) + 0011199092 (119905)) 1199091 (119905)

+ 2119878 (119905)1 + 3119878 (119905)1199091 (119905)] d119905 + 011199091 (119905) d1198612 (119905) d1199092 (119905) = [minus (031 + 00071199091 (119905) + 0011199092 (119905)) 1199092 (119905)

+ 2119878 (119905)1 + 5119878 (119905)1199092 (119905)] d119905 + 021199092 (119905) d1198613 (119905) (116)


Complexity 15

01

02

03

04

05

06

07

08

09

1

S(t)

S(t) path

1 2 3 4 5 6 7 8 9 100Time times10

4

(a)


0

5

10

15

20

25

30

(b)


x1(t) path

005

01

015

02

025

03

035

04

x1(t)

200 400 600 800 1000 1200 1400 1600 1800 20000Time

(a)


0

5

10

15

20

25

30

35

40

45

50

(b)





16 Complexity

1 2 3 4 5 6 7 8 9 100Time

005

01

015

02

025

03

035

04

045

05

x1(t)

x1(t) path

times104

(a)


0

5

10

15

20

25

30

35

40

45

50

(b)






d1199091 (119905) = [minus (09 + 021199091 (119905) + 021199092 (119905)) 1199091 (119905)+ 119878 (119905)1 + 04119878 (119905)1199091 (119905)] d119905 + 081199091 (119905) d1198612 (119905)

d1199092 (119905) = [minus (09 + 021199091 (119905) + 021199092 (119905)) 1199092 (119905)+ 119878 (119905)1 + 06119878 (119905)1199092 (119905)] d119905 + 081199092 (119905) d1198613 (119905)

(117)





Data Availability




Acknowledgments


Complexity 17

0 20018016014012010080604020Time

0

02

04

06

08

1

12

14

16

18

2Va

lues


0 109876543210

01

02


(a)

0 10000900080007000600050004000300020001000Time

0

05

1

15

2

25

Valu

es



0

02

04

0 87654321

(b)



References






















18 Complexity






























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 5

+ 119899sum119894=1

1198961198941198780 minus min 119889 1198891 119889119899) 119906 + (119889 + 119899sum119894=1

119896119894)


1198961198941198780

minus min 119889 1198891 119889119899) 119906 + (119889 + 119899sum119894=1

119896119894)1198780 (21)

where

120582 = min 119889 1198891 119889119899 minus 119899sum119894=1

1198961198941198780


Let

119860 = (119889 + 119899sum119894=1

119896119894)1198780 minus min 119889 1198891 119889119899 (23)

be a constant Then


119894=1

119896119894)1198780 d119905

+ 120579 (1 + 119906)120579minus1 [1205901119878d1198611 (119905) + 119899sum119894=1

120590119894+1119909119894d119861119894+1 (119905)] (24)


sdot [1205901119878d1198611 (119905) + 119899sum119894=1

120590119894+1119909119894d119861119894+1 (119905)] (25)

where


+ (119889 + 119899sum119894=1

119896119894)1198780 = e119902119905120579 (1 + 119906)120579minus2


119896119894)

+ 119902120579

(26)


+ 1198780 (119889 + 119899sum119894=1

119896119894) + 119902120579 119906 isin 119877+(27)

and119865 fl sup119906isin119877+

119891 (119906) + 1 (28)

which gives

119871 [e119902119905119882(119906)] le 120579e119902119905119865 (29)


119864 [e119902119905119882(119906)] = 119882 (119906 (0)) + 119864int1199050119871 [e119902119905119882(119906)] d119905 (30)


119864 [e119902119905119882(119906)] le (1 + 119906 (0))120579 + 119864int1199050e119902119905120579119865d119905

= (1 + 119906 (0))120579 + 120579119865119902 e119902119905(31)

which leads to


sup119864 [(1 + 119906 (119905))120579] le 120579119865119902 fl 1198650 119886119904 (32)


119864 [(1 + 119906 (119905))120579] le 119872 119905 ge 0 (33)


119864[ sup119896120575le119905le(119896+1)120575

(1 + 119906 (119905))120579]le 119864 [(1 + 119906 (119896120575))120579] + 1198681 + 1198682 le 119872 + 1198681 + 1198682

(34)

where

1198681 = 119864[ sup119896120575le119905le(119896+1)120575

1003816100381610038161003816100381610038161003816100381610038161003816int119905

119896120575120579 (1 + 119906)120579minus2

sdot minus1205821199062 + 119860119906 + 1198780(119889 + 119899sum119894=1

119896119894) d1199051003816100381610038161003816100381610038161003816100381610038161003816]

le 119864[ sup119896120575le119905le(119896+1)120575

1003816100381610038161003816100381610038161003816100381610038161003816int119905

119896120575120579 (1 + 119906)120579minus2

sdot 1198780(119889 + 119899sum119894=1

119896119894)119906 + 1198780(119889 + 119899sum119894=1

119896119894) d1199051003816100381610038161003816100381610038161003816100381610038161003816]

= 119864[ sup119896120575le119905le(119896+1)120575

1003816100381610038161003816100381610038161003816100381610038161003816int119905

119896120575120579 (1 + 119906)120579minus2

6 Complexity

sdot 1198780 (119889 + 119899sum119894=1

119896119894) (1 + 119906) d1199051003816100381610038161003816100381610038161003816100381610038161003816] le 1198780 (119889

+ 119899sum119894=1

119896119894)120579119864int(119896+1)120575119896120575

(1 + 119906)120579 d119905

le 1198621120575119864[ sup119896120575le119905le(119896+1)120575

(1 + 119906 (119905))120579] (35)

here

1198621 = 1198780(119889 + 119899sum119894=1

119896119894)120579 (36)


1198682 = 119864[ sup119896120575le119905le(119896+1)120575

1003816100381610038161003816100381610038161003816100381610038161003816int119905

119896120575120579 (1 + 119906)120579minus1

sdot 1205901119878d1198611 (119905) + 119899sum119894=1

120590119894+1119909119894d119861119894+1 (119905)1003816100381610038161003816100381610038161003816100381610038161003816]

le radic32119864[int(119896+1)120575119896120575

1205792 (1 + 119906)2120579minus2(120590211198782

+ 119899sum119894=1

1205902119894+11199092119894) d119905]12

le radic32119864[int(119896+1)120575119896120575

1205792 (1

+ 119906)2120579minus2 (120590211199062 +119899sum119894=1

1205902119894+11199062) d119905]12

le radic3212057912057512119864[ sup119896120575le119905le(119896+1)120575

(12059021 or or 1205902119899) (1

+ 119906)2120579]12 le radic3212057912057512 (12059021 or or 1205902119899)12

sdot 119864 [ sup119896120575le119905le(119896+1)120575

(1 + 119906)120579]

(37)

Hence we have

119864[ sup119896120575le119905le(119896+1)120575

(1 + 119906)120579]le 119864 [(1 + 119906 (119896120575))120579]


(1 + 119906)120579]

(38)


1198621120575 + radic3212057912057512 (12059021 or or 1205902119899)12 le 12 (39)

which yields

119864[ sup119896120575le119905le(119896+1)120575

(1 + 119906)120579] le 2119864 [(1 + 119906 (119896120575))120579] le 2119872 (40)


119875 sup119896120575le119905le(119896+1)120575

(1 + 119906 (119905))120579 ge (119896120575)1+120573

le 119864 [sup119896120575le119905le(119896+1)120575 (1 + 119906 (119905))120579](119896120575)1+120573 le 2119872

(119896120575)1+120573 (41)


sup119896120575le119905le(119896+1)120575

(1 + 119906 (119905))120579 le (119896120575)1+120573 (42)


ln (1 + 119906 (119905))120579ln 119905 le ln (119896120575)1+120573

ln 119905 le ln (119896120575)1+120573ln (119896120575) = 1 + 120573 (43)

which means


sup ln (1 + 119906 (119905))120579ln 119905 le 1 + 120573 119886119904 (44)


sup ln (1 + 119906 (119905))ln 119905 le 1120579 119886119904 (45)



supln [119878 (119905) + sum119899119894=1 119909119894 (119905)]

ln 119905 = lim119905997888rarrinfin

sup ln 119906 (119905)ln 119905


sup ln (1 + 119906 (119905))ln 119905 le 1120579 119886119904

(46)




int1199050(119878 (119904) minus 1205821) d1198611 (119904)119905 = 0 119886119904




int1199050119909119894 (119904) d119861119894+1 (119904)119905 = 0 119886119904

(119894 = 2 3 119899)

(47)


Complexity 7

Proof Define

119883 (119905) = int1199050(119878 (119904) minus 1205821)d1198611 (119904)


119885119894 (119905) = int1199050119909119894 (119904) d119861119894+1 (119904) 119894 = 2 3 119899

(48)


119864[ sup0le119904le119905

|119883 (119904)|120579] le 119862120579119864[int1199050(119878 (119904) minus 1205821)2 d119904]

12

le 1198621205791199051205792119864[ sup0le119904le119905

(119878 (119904) minus 1205821)120579]le 1198621205791199051205792119864[ sup

0le119904le119905(1 + 119906 (119905))120579]

le 21198721198621205791199051205792

(49)


119875120596 sup119896120575le119905le(119896+1)120575

|119883 (119905)|120579 gt (119896120575)1+120572+1205792

le 119864 [|119883 ((119896 + 1) 120575)|120579](119896120575)1+120572+1205792 le 2119872119862120579(119896120575)1+120572 (

(119896 + 1) 120575119896120575 )1205792

le 21+1205792119872119862120579(119896120575)1+120572

(50)


sup119896120575le119905le(119896+1)120575

|119883 (119905)|120579 le (119896120575)1+120572+1205792 (51)


ln 119905 le ln (119896120575)1+120572+1205792ln 119905 le ln (119896120575)1+120572+1205792

ln (119896120575) (52)

which implies

ln |119883 (119905)|120579ln 119905 le 1 + 120572 + 1205792 (53)



sup ln |119883 (119905)|ln 119905 le 1 + 120572 + 1205792120579 (54)


sup ln |119883 (119905)|ln 119905 le 12 + 1120579 (55)


ln |119883 (119905)|ln 119905 le 12 + 1120579 + 120578 (56)

which means



sup 11990512+1120579+120578119905 = 0 (57)

Together with



inf

100381610038161003816100381610038161003816int1199050 (119878 (119904) minus 1205821) d1198611 (119904)100381610038161003816100381610038161003816119905ge 0

(58)

we derive


|119883 (119905)|119905 = 0 119886119904 (59)

and


int1199050(119878 (119904) minus 1205821) d1198611 (119904)119905 = 0 119886119904 (60)





int1199050119909119894 (119905) d119861119894+1 (119904)119905 = 0 119886119904

119894 = 2 3 119899(61)



d120601 (119905) = [(1198780 minus 119878 (119905)) (119889 + 119899sum119894=1

119896119894119909119894)] d119905+ 1205901120601 (119905) d1198611 (119905)

(62)


0119911V(d119911) =


119901 (119911) = exp [2 (119889 + sum119899119894=1 119896119894119909119894) 119878012059021119911 ]


8 Complexity



gt 119899sum119894=1

1198961198941198780 + (120579 minus 12 or 0) (12059021 or or 1205902119899) 11988611989412058211 + 1198871198941205821 +

1198861198942radic119887119894 (1 + 1198871198941205821) ( 119867119889 + 119873sum119899119894=1 119896119894)12

le (119889119894 + 119873 119899sum119894=1

119896119894) + 1205902119894+12 (119894 = 2 3 119899)


119909V (d119909)1 + 1198871119909 gt 120590222 + (119889 + 119872 119899sum119894=1

119896119894)

(64)



supln119909119894 (119905)119905 le 0 119886119904 (119894 = 2 119899) (65)




(66)



(1 + 1198871198941205821) 119909119894

(67)


d119881 = 119878 minus 1205821119878 [(1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1

1198861198941198781 + 119887119894119878119909119894] d119905 + 1205901119878d1198611 (119905) + 12 12058211198782 120590211198782d119905

+ [(1 + 11988711205821) minus (1 + 11988711205821) 119909lowast11199091 ]

sdot [minus(1198891 + 119899sum119894=1

119896119894119909119894)1199091 + 11988611198781 + 11988711198781199091] d119905

+ 12059021199091d1198612 (119905) + 12(1 + 11988711205821) 119909lowast111990921 1205902211990921d119905 + 119899sum

119894=2

(1

+ 1198871198941205821) [minus(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894 + 1198861198941198781 + 119887119894119878119909119894] d119905

+ 120590119894+1119909119894d119861119894+1 (119905) = 119871119881d119905 + (119878 minus 1205821) 1205901d1198611 (119905)


(1 + 1198871198941205821) 120590119894+1119909119894d119861119894+1 (119905) (68)

where

119871119881 = 119878 minus 1205821119878 [(1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1


sdot [minus(1198891 + 119899sum119894=1

119896119894119909119894)1199091 + 11988611198781 + 11988711198781199091]

+ 119899sum119894=2

(1 + 1198871198941205821) [minus(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894 + 1198861198941198781 + 119887119894119878119909119894]

(69)



(70)



Complexity 9


119899sum119894=2

119896119894119909119894+ 119899sum119894=2

119909119894 [ 11988611989412058211 + 1198871198941205821 minus (119889119894 + 119899sum119894=1

119896119894119909119894)] + 12120582112059021+ 12 (1 + 11988711205821) 119909lowast112059022


119899sum119894=2

119896119894119909119894


119909119894 [minus 119899sum119894=1


(71)

Set

119872 = max 1199091 119909119899 119873 = min 1199091 119909119899 (72)

then


119894=2

119896119894


119872 119899sum119894=1


(73)

where


119894=2

119896119894


+ (119899 minus 1)119872 119899sum119894=1

119896119894119909lowast119894 + 12120582112059021+ 12 (1 + 11988711205821) 119909lowast112059022

(74)



0

(119878 (119904) minus 1205821)2119878 (119904) d119904 + 119867+ 1205901 1119905 int119905

0(119878 (119904) minus 1205821) d1198611 (119904)


+ 119899sum119894=2

(1 + 1198871198941205821) 120590119894+1 int1199050119909119894 (119904) d119861119894+1 (119905)

(75)



sup 1119905 int1199050

(119878 (119904) minus 1205821)2119878 (119904) d119904 le 119867119889 + 119873sum119899119894=1 119896119894 (76)

Define

119881119894 (119909119894 (119905)) = ln119909119894 (119905) 119894 = 2 3 119899 (77)


d119881119894 = [minus(119889119894 + 119899sum119894=1

119896119894119909119894) + 1198861198941198781 + 119887119894119878 minus 1205902119894+12 ] d119905


119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12 ] d119905

+ 120590119894+1d119861119894+1 (119905)

(78)


119881119894 (119905) minus 119881119894 (0)119905le 1198861198941 + 1198871198941205821 ⟨

119878 (119905) minus 12058211 + 119887119894119878 (119905)⟩ minus (119889119894 + 119873 119899sum119894=1

119896119894)

10 Complexity

+ 11988611989412058211 + 1198871198941205821 minus1205902119894+12 + 120590119894+1119861119894+1 (119905)119905


119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 120590119894+1119861119894+1 (119905)119905le 1198861198942radic119887119894 (1 + 1198871198941205821) [1119905 int119905

0

(119878 (119905) minus 1205821)2119878 (119905) d119904]12

minus (119889119894 + 119873 119899sum119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 120590119894+1119861119894+1 (119905)119905 (79)

which meansln119909119894 (119905)119905

le ln119909119894 (0)119905+ 1198861198942radic119887119894 (1 + 1198871198941205821) [1119905 int119905

0

(119878 (119905) minus 1205821)2119878 (119905) d119904]12

minus (119889119894 + 119873 119899sum119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 120590119894+1119861119894+1 (119905)119905

(80)



supln119909119894 (119905)119905

le minus(119889119894 + 119873 119899sum119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 1198861198942radic119887119894 (1 + 1198871198941205821) ( 119867119889 + 119873sum119899119894=1 119896119894)12

(119894 = 2 119899)

(81)

If

11988611989412058211 + 1198871198941205821 +1198861198942radic119887119894 (1 + 1198871198941205821) ( 119867119889 + 119873sum119899119894=1 119896119894)

12

le (119889119894 + 119873 119899sum119894=1

119896119894) + 1205902119894+12 (82)

then we have


supln119909119894 (119905)119905 le 0 119886119904 (119894 = 2 119899) (83)




1198861198941199091198941 + 119887119894119878 minus 120590212 ] d119905+ 1205901d1198611 (119905)


(84)



119894=1

1198861198941199091198941 + 119887119894119878] d119905

minus 120590212 + 1119905 int11990501205901d1198611 (119905)

= (119889 + 119899sum119894=1

119896119894119909119894)1198780 ⟨1119878⟩ minus 1198861⟨ 11990911 + 1198871119878⟩

minus 119899sum119894=2

119886119894⟨ 1199091198941 + 119887119894119878⟩ minus (119889 + 119899sum119894=1

119896119894119909119894) minus 120590212+ 12059011198611 (119905)119905 + ln 119878 (0)119905

ln120601 (119905)119905= ln120601 (0)119905 + 1119905 int119905

0

(1198780 minus 120601) (119889 + sum119899119894=1 119896119894119909119894)120601 d119905 minus 120590212+ 12059011198611 (119905)119905

= (119889 + 119899sum119894=1

119896119894119909119894)1198780⟨1120601⟩ minus (119889 + 119899sum119894=1

119896119894119909119894) minus 120590212+ 12059011198611 (119905)119905 + ln120601 (0)119905

(85)

Complexity 11

Therefore

0 ge ln 119878 (119905) minus ln120601 (119905)119905= (119889 + 119899sum

119894=1

119896119894119909119894)1198780⟨1119878 minus 1120601⟩ minus 1198861⟨ 11990911 + 1198871119878⟩

minus 119899sum119894=2

119886119894⟨ 1199091198941 + 119887119894119878⟩

ge (119889 + 119873 119899sum119894=1

119896119894)1198780⟨1119878 minus 1120601⟩ minus 1198861 ⟨1199091⟩

minus 119899sum119894=2

119886119894 ⟨119909119894⟩

(86)

which implies that

⟨1119878 minus 1120601⟩

le 1(119889 + 119873sum119899119894=1 119896119894) 1198780 (1198861 ⟨1199091⟩ + 119899sum119894=2

119886119894 ⟨119909119894⟩) (87)


119894=1

119896119894119909119894) + 11988611198781 + 1198871119878 minus 120590222 ] d119905+ 1205902d1198612 (119905)

ge [minus(1198891 + 119872 119899sum119894=1

119896119894) + 11988611198781 + 1198871119878 minus 120590222 ] d119905+ 1205902d1198612 (119905)

(88)



119894=1

119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩ minus 1198861⟨ 120601 minus 119878(1 + 1198871120601) (1 + 1198871119878)⟩


119894=1

119896119894)+ 1198861⟨ 1206011 + 1198871120601⟩ minus 119886111988721 ⟨1119878 minus 1120601⟩ + 12059021198612 (119905)119905


119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩

minus 119886111988721 (119889 + 119873sum119899119894=1 119896119894) 1198780 (1198861 ⟨1199091⟩ + 119899sum119894=2

119886119894 ⟨119909119894⟩)+ 12059021198612 (119905)119905


119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩ minus 11988621 ⟨1199091⟩11988721 (119889 + 119873sum119899119894=1 119896119894) 1198780+ 12059021198612 (119905)119905

(89)



sup ln 1199091 (119905)119905 le 0 (90)

which yields


119894=1

119896119894)+ 1198861⟨ 1206011 + 1198871120601⟩ + 12059021198612 (119905)119905

(91)

that is



119894=1

119896119894) + 1198861 intinfin0

119909V (d119909)1 + 1198871119909 ] (92)

Therefore if

1198861 intinfin0

119909V (d119909)1 + 1198871119909 gt 120590222 + (119889 + 119872 119899sum119894=1

119896119894) (93)

then


inf ⟨1199091 (119905)⟩ gt 0 119886119904 (94)



119886119894 intinfin0

119909V (d119909)1 + 119887119894119909 lt 119889119894 + 119873 119899sum119894=1

119896119894 + 1205902119894+12 119894 = 1 2 119899 (95)

12 Complexity

then


int1199050119878 (119904) d119904119905 = 1198780 119886119904


supln 119909119894 (119905)119905 lt 0 119886119904 119894 = 1 2 119899

(96)




119896119894119909119894 + 1198861198941198781 + 119887119894119878 minus 1205902119894+12 )d119905+ 120590119894+1d119861119894+1 (119905)


119896119894 + 1198861198941206011 + 119887119894120601 minus 1205902119894+12 )d119905+ 120590119894+1d119861119894+1 (119905)

(97)


119894=1

119896119894 minus 1205902119894+12 ) + 119886119894⟨ 1206011 + 119887119894120601⟩+ 120590119894+1119861119894+1 (119905)119905 + ln 119909119894 (0)119905

(98)


119886119894 intinfin0

119909V (d119909)1 + 119887119894119909 lt 119889119894 + 119873 119899sum119894=1

119896119894 + 1205902119894+12 119894 = 1 2 119899 (99)

we know that


supln119909119894 (119905)119905 le 0 119894 = 1 2 119899 (100)



d119884 = minus(119889 + 119899sum119894=1

119896119894119909119894)119884d119905 + 1205901 (1198780 + 119884) d1198611 (119905) (101)



119894=1

119896119894119909119894) minus 120576119878120576] d119905+ 1205901119878120576d1198611 (119905)

(102)

Then

d (119878 minus 119878120576) = [(120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878) 119878

minus (119889 + 120576 + 119899sum119894=1

119896119894119909119894) (119878 minus 119878120576)] d119905 + 1205901 (119878minus 119878120576) d1198611 (119905)

(103)



119894=1

119896119894119909119894 + 120576 + 120590212 ) 119905]

sdot int1199050exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904

(104)


120576 gt 119899sum119894=1

119886119894119909119894 (119905)1 + 119887119894119878 (119905) forall119905 ge 119879 (105)


119894=1

119896119894119909119894 + 120576

+ 120590212 ) 119905]

sdot int1198790exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904

+ int119905119879exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

Complexity 13

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904 ge exp[12059011198611 (119905)

minus (119889 + 119872 119899sum119894=1

119896119894 + 120576 + 120590212 ) 119905]

sdot int1198790exp[(119889 + 119873 119899sum

119894=1

119896119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904(106)


exp[12059011198611 (119905) minus (119889 + 119872 119899sum119894=1

119896119894 + 120576 + 120590212 ) 119905] 997888rarr 01003816100381610038161003816100381610038161003816100381610038161003816int119879

0exp[(119889 + 119873 119899sum

119894=1

119896119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d1199041003816100381610038161003816100381610038161003816100381610038161003816 lt infin

(107)

which implies that



inf (119878 (119905) minus 119878120576 (119905))ge 0 119886119904 (108)

Now we consider

d (120601 (119905) minus 119878120576 (119905))= [120576119878120576 (119905) minus (119889 + 119899sum

119894=1

119896119894119909119894) (120601 (119905) minus 119878120576 (119905))] d119905+ 1205901 (120601 (119905) minus 119878120576 (119905)) d1198611 (119905)

le [120576119878120576 (119905) minus (119889 + 119873 119899sum119894=1

119896119894)(120601 (119905) minus 119878120576 (119905))] d119905+ 1205901 (120601 (119905) minus 119878120576 (119905)) d1198611 (119905)

(109)



119894=1


(110)


119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816le 119864[int119905

0120576119878120576 (119904) minus (119889 + 119873 119899sum

119894=1

119896119894) 1003816100381610038161003816120601 (119904) minus 119878120576 (119904)1003816100381610038161003816 d119904]

le 119864[int1199050120576120601 (119904) minus (119889 + 119873 119899sum

119894=1

119896119894) 1003816100381610038161003816120601 (119904) minus 119878120576 (119904)1003816100381610038161003816 d119904] (111)

then

119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816 le 120576 sup0le119904

119864120601 (119904) (112)

which leads to

lim120576997888rarr0


119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816 = 0 (113)



(120601 (119905) minus 119878 (119905)) = 0 (114)









14 Complexity

200 400 600 800 1000 1200 1400 1600 1800 20000Time

0

01

02

03

04

05

06

07

08

Valu

es


0 100806040200

02

04x2(t) path

(a)

0

01

02

03

04

05

06

07

08

09

1

Valu

es

1 2 3 4 5 6 7 8 9 100Time


times104

(b)


S(t) path

01

02

03

04

05

06

07

S(t)

200 400 600 800 1000 1200 1400 1600 1800 20000Time

(a)

0

5

10

15

20

25

30


(b)



d1199091 (119905) = [minus (032 + 00071199091 (119905) + 0011199092 (119905)) 1199091 (119905)

+ 2119878 (119905)1 + 3119878 (119905)1199091 (119905)] d119905 + 011199091 (119905) d1198612 (119905) d1199092 (119905) = [minus (031 + 00071199091 (119905) + 0011199092 (119905)) 1199092 (119905)

+ 2119878 (119905)1 + 5119878 (119905)1199092 (119905)] d119905 + 021199092 (119905) d1198613 (119905) (116)


Complexity 15

01

02

03

04

05

06

07

08

09

1

S(t)

S(t) path

1 2 3 4 5 6 7 8 9 100Time times10

4

(a)


0

5

10

15

20

25

30

(b)


x1(t) path

005

01

015

02

025

03

035

04

x1(t)

200 400 600 800 1000 1200 1400 1600 1800 20000Time

(a)


0

5

10

15

20

25

30

35

40

45

50

(b)





16 Complexity

1 2 3 4 5 6 7 8 9 100Time

005

01

015

02

025

03

035

04

045

05

x1(t)

x1(t) path

times104

(a)


0

5

10

15

20

25

30

35

40

45

50

(b)






d1199091 (119905) = [minus (09 + 021199091 (119905) + 021199092 (119905)) 1199091 (119905)+ 119878 (119905)1 + 04119878 (119905)1199091 (119905)] d119905 + 081199091 (119905) d1198612 (119905)

d1199092 (119905) = [minus (09 + 021199091 (119905) + 021199092 (119905)) 1199092 (119905)+ 119878 (119905)1 + 06119878 (119905)1199092 (119905)] d119905 + 081199092 (119905) d1198613 (119905)

(117)





Data Availability




Acknowledgments


Complexity 17

0 20018016014012010080604020Time

0

02

04

06

08

1

12

14

16

18

2Va

lues


0 109876543210

01

02


(a)

0 10000900080007000600050004000300020001000Time

0

05

1

15

2

25

Valu

es



0

02

04

0 87654321

(b)



References






















18 Complexity






























Journal of












Journal of







Volume 2018






Volume 2018








6 Complexity

sdot 1198780 (119889 + 119899sum119894=1

119896119894) (1 + 119906) d1199051003816100381610038161003816100381610038161003816100381610038161003816] le 1198780 (119889

+ 119899sum119894=1

119896119894)120579119864int(119896+1)120575119896120575

(1 + 119906)120579 d119905

le 1198621120575119864[ sup119896120575le119905le(119896+1)120575

(1 + 119906 (119905))120579] (35)

here

1198621 = 1198780(119889 + 119899sum119894=1

119896119894)120579 (36)


1198682 = 119864[ sup119896120575le119905le(119896+1)120575

1003816100381610038161003816100381610038161003816100381610038161003816int119905

119896120575120579 (1 + 119906)120579minus1

sdot 1205901119878d1198611 (119905) + 119899sum119894=1

120590119894+1119909119894d119861119894+1 (119905)1003816100381610038161003816100381610038161003816100381610038161003816]

le radic32119864[int(119896+1)120575119896120575

1205792 (1 + 119906)2120579minus2(120590211198782

+ 119899sum119894=1

1205902119894+11199092119894) d119905]12

le radic32119864[int(119896+1)120575119896120575

1205792 (1

+ 119906)2120579minus2 (120590211199062 +119899sum119894=1

1205902119894+11199062) d119905]12

le radic3212057912057512119864[ sup119896120575le119905le(119896+1)120575

(12059021 or or 1205902119899) (1

+ 119906)2120579]12 le radic3212057912057512 (12059021 or or 1205902119899)12

sdot 119864 [ sup119896120575le119905le(119896+1)120575

(1 + 119906)120579]

(37)

Hence we have

119864[ sup119896120575le119905le(119896+1)120575

(1 + 119906)120579]le 119864 [(1 + 119906 (119896120575))120579]


(1 + 119906)120579]

(38)


1198621120575 + radic3212057912057512 (12059021 or or 1205902119899)12 le 12 (39)

which yields

119864[ sup119896120575le119905le(119896+1)120575

(1 + 119906)120579] le 2119864 [(1 + 119906 (119896120575))120579] le 2119872 (40)


119875 sup119896120575le119905le(119896+1)120575

(1 + 119906 (119905))120579 ge (119896120575)1+120573

le 119864 [sup119896120575le119905le(119896+1)120575 (1 + 119906 (119905))120579](119896120575)1+120573 le 2119872

(119896120575)1+120573 (41)


sup119896120575le119905le(119896+1)120575

(1 + 119906 (119905))120579 le (119896120575)1+120573 (42)


ln (1 + 119906 (119905))120579ln 119905 le ln (119896120575)1+120573

ln 119905 le ln (119896120575)1+120573ln (119896120575) = 1 + 120573 (43)

which means


sup ln (1 + 119906 (119905))120579ln 119905 le 1 + 120573 119886119904 (44)


sup ln (1 + 119906 (119905))ln 119905 le 1120579 119886119904 (45)



supln [119878 (119905) + sum119899119894=1 119909119894 (119905)]

ln 119905 = lim119905997888rarrinfin

sup ln 119906 (119905)ln 119905


sup ln (1 + 119906 (119905))ln 119905 le 1120579 119886119904

(46)




int1199050(119878 (119904) minus 1205821) d1198611 (119904)119905 = 0 119886119904




int1199050119909119894 (119904) d119861119894+1 (119904)119905 = 0 119886119904

(119894 = 2 3 119899)

(47)


Complexity 7

Proof Define

119883 (119905) = int1199050(119878 (119904) minus 1205821)d1198611 (119904)


119885119894 (119905) = int1199050119909119894 (119904) d119861119894+1 (119904) 119894 = 2 3 119899

(48)


119864[ sup0le119904le119905

|119883 (119904)|120579] le 119862120579119864[int1199050(119878 (119904) minus 1205821)2 d119904]

12

le 1198621205791199051205792119864[ sup0le119904le119905

(119878 (119904) minus 1205821)120579]le 1198621205791199051205792119864[ sup

0le119904le119905(1 + 119906 (119905))120579]

le 21198721198621205791199051205792

(49)


119875120596 sup119896120575le119905le(119896+1)120575

|119883 (119905)|120579 gt (119896120575)1+120572+1205792

le 119864 [|119883 ((119896 + 1) 120575)|120579](119896120575)1+120572+1205792 le 2119872119862120579(119896120575)1+120572 (

(119896 + 1) 120575119896120575 )1205792

le 21+1205792119872119862120579(119896120575)1+120572

(50)


sup119896120575le119905le(119896+1)120575

|119883 (119905)|120579 le (119896120575)1+120572+1205792 (51)


ln 119905 le ln (119896120575)1+120572+1205792ln 119905 le ln (119896120575)1+120572+1205792

ln (119896120575) (52)

which implies

ln |119883 (119905)|120579ln 119905 le 1 + 120572 + 1205792 (53)



sup ln |119883 (119905)|ln 119905 le 1 + 120572 + 1205792120579 (54)


sup ln |119883 (119905)|ln 119905 le 12 + 1120579 (55)


ln |119883 (119905)|ln 119905 le 12 + 1120579 + 120578 (56)

which means



sup 11990512+1120579+120578119905 = 0 (57)

Together with



inf

100381610038161003816100381610038161003816int1199050 (119878 (119904) minus 1205821) d1198611 (119904)100381610038161003816100381610038161003816119905ge 0

(58)

we derive


|119883 (119905)|119905 = 0 119886119904 (59)

and


int1199050(119878 (119904) minus 1205821) d1198611 (119904)119905 = 0 119886119904 (60)





int1199050119909119894 (119905) d119861119894+1 (119904)119905 = 0 119886119904

119894 = 2 3 119899(61)



d120601 (119905) = [(1198780 minus 119878 (119905)) (119889 + 119899sum119894=1

119896119894119909119894)] d119905+ 1205901120601 (119905) d1198611 (119905)

(62)


0119911V(d119911) =


119901 (119911) = exp [2 (119889 + sum119899119894=1 119896119894119909119894) 119878012059021119911 ]


8 Complexity



gt 119899sum119894=1

1198961198941198780 + (120579 minus 12 or 0) (12059021 or or 1205902119899) 11988611989412058211 + 1198871198941205821 +

1198861198942radic119887119894 (1 + 1198871198941205821) ( 119867119889 + 119873sum119899119894=1 119896119894)12

le (119889119894 + 119873 119899sum119894=1

119896119894) + 1205902119894+12 (119894 = 2 3 119899)


119909V (d119909)1 + 1198871119909 gt 120590222 + (119889 + 119872 119899sum119894=1

119896119894)

(64)



supln119909119894 (119905)119905 le 0 119886119904 (119894 = 2 119899) (65)




(66)



(1 + 1198871198941205821) 119909119894

(67)


d119881 = 119878 minus 1205821119878 [(1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1

1198861198941198781 + 119887119894119878119909119894] d119905 + 1205901119878d1198611 (119905) + 12 12058211198782 120590211198782d119905

+ [(1 + 11988711205821) minus (1 + 11988711205821) 119909lowast11199091 ]

sdot [minus(1198891 + 119899sum119894=1

119896119894119909119894)1199091 + 11988611198781 + 11988711198781199091] d119905

+ 12059021199091d1198612 (119905) + 12(1 + 11988711205821) 119909lowast111990921 1205902211990921d119905 + 119899sum

119894=2

(1

+ 1198871198941205821) [minus(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894 + 1198861198941198781 + 119887119894119878119909119894] d119905

+ 120590119894+1119909119894d119861119894+1 (119905) = 119871119881d119905 + (119878 minus 1205821) 1205901d1198611 (119905)


(1 + 1198871198941205821) 120590119894+1119909119894d119861119894+1 (119905) (68)

where

119871119881 = 119878 minus 1205821119878 [(1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1


sdot [minus(1198891 + 119899sum119894=1

119896119894119909119894)1199091 + 11988611198781 + 11988711198781199091]

+ 119899sum119894=2

(1 + 1198871198941205821) [minus(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894 + 1198861198941198781 + 119887119894119878119909119894]

(69)



(70)



Complexity 9


119899sum119894=2

119896119894119909119894+ 119899sum119894=2

119909119894 [ 11988611989412058211 + 1198871198941205821 minus (119889119894 + 119899sum119894=1

119896119894119909119894)] + 12120582112059021+ 12 (1 + 11988711205821) 119909lowast112059022


119899sum119894=2

119896119894119909119894


119909119894 [minus 119899sum119894=1


(71)

Set

119872 = max 1199091 119909119899 119873 = min 1199091 119909119899 (72)

then


119894=2

119896119894


119872 119899sum119894=1


(73)

where


119894=2

119896119894


+ (119899 minus 1)119872 119899sum119894=1

119896119894119909lowast119894 + 12120582112059021+ 12 (1 + 11988711205821) 119909lowast112059022

(74)



0

(119878 (119904) minus 1205821)2119878 (119904) d119904 + 119867+ 1205901 1119905 int119905

0(119878 (119904) minus 1205821) d1198611 (119904)


+ 119899sum119894=2

(1 + 1198871198941205821) 120590119894+1 int1199050119909119894 (119904) d119861119894+1 (119905)

(75)



sup 1119905 int1199050

(119878 (119904) minus 1205821)2119878 (119904) d119904 le 119867119889 + 119873sum119899119894=1 119896119894 (76)

Define

119881119894 (119909119894 (119905)) = ln119909119894 (119905) 119894 = 2 3 119899 (77)


d119881119894 = [minus(119889119894 + 119899sum119894=1

119896119894119909119894) + 1198861198941198781 + 119887119894119878 minus 1205902119894+12 ] d119905


119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12 ] d119905

+ 120590119894+1d119861119894+1 (119905)

(78)


119881119894 (119905) minus 119881119894 (0)119905le 1198861198941 + 1198871198941205821 ⟨

119878 (119905) minus 12058211 + 119887119894119878 (119905)⟩ minus (119889119894 + 119873 119899sum119894=1

119896119894)

10 Complexity

+ 11988611989412058211 + 1198871198941205821 minus1205902119894+12 + 120590119894+1119861119894+1 (119905)119905


119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 120590119894+1119861119894+1 (119905)119905le 1198861198942radic119887119894 (1 + 1198871198941205821) [1119905 int119905

0

(119878 (119905) minus 1205821)2119878 (119905) d119904]12

minus (119889119894 + 119873 119899sum119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 120590119894+1119861119894+1 (119905)119905 (79)

which meansln119909119894 (119905)119905

le ln119909119894 (0)119905+ 1198861198942radic119887119894 (1 + 1198871198941205821) [1119905 int119905

0

(119878 (119905) minus 1205821)2119878 (119905) d119904]12

minus (119889119894 + 119873 119899sum119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 120590119894+1119861119894+1 (119905)119905

(80)



supln119909119894 (119905)119905

le minus(119889119894 + 119873 119899sum119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 1198861198942radic119887119894 (1 + 1198871198941205821) ( 119867119889 + 119873sum119899119894=1 119896119894)12

(119894 = 2 119899)

(81)

If

11988611989412058211 + 1198871198941205821 +1198861198942radic119887119894 (1 + 1198871198941205821) ( 119867119889 + 119873sum119899119894=1 119896119894)

12

le (119889119894 + 119873 119899sum119894=1

119896119894) + 1205902119894+12 (82)

then we have


supln119909119894 (119905)119905 le 0 119886119904 (119894 = 2 119899) (83)




1198861198941199091198941 + 119887119894119878 minus 120590212 ] d119905+ 1205901d1198611 (119905)


(84)



119894=1

1198861198941199091198941 + 119887119894119878] d119905

minus 120590212 + 1119905 int11990501205901d1198611 (119905)

= (119889 + 119899sum119894=1

119896119894119909119894)1198780 ⟨1119878⟩ minus 1198861⟨ 11990911 + 1198871119878⟩

minus 119899sum119894=2

119886119894⟨ 1199091198941 + 119887119894119878⟩ minus (119889 + 119899sum119894=1

119896119894119909119894) minus 120590212+ 12059011198611 (119905)119905 + ln 119878 (0)119905

ln120601 (119905)119905= ln120601 (0)119905 + 1119905 int119905

0

(1198780 minus 120601) (119889 + sum119899119894=1 119896119894119909119894)120601 d119905 minus 120590212+ 12059011198611 (119905)119905

= (119889 + 119899sum119894=1

119896119894119909119894)1198780⟨1120601⟩ minus (119889 + 119899sum119894=1

119896119894119909119894) minus 120590212+ 12059011198611 (119905)119905 + ln120601 (0)119905

(85)

Complexity 11

Therefore

0 ge ln 119878 (119905) minus ln120601 (119905)119905= (119889 + 119899sum

119894=1

119896119894119909119894)1198780⟨1119878 minus 1120601⟩ minus 1198861⟨ 11990911 + 1198871119878⟩

minus 119899sum119894=2

119886119894⟨ 1199091198941 + 119887119894119878⟩

ge (119889 + 119873 119899sum119894=1

119896119894)1198780⟨1119878 minus 1120601⟩ minus 1198861 ⟨1199091⟩

minus 119899sum119894=2

119886119894 ⟨119909119894⟩

(86)

which implies that

⟨1119878 minus 1120601⟩

le 1(119889 + 119873sum119899119894=1 119896119894) 1198780 (1198861 ⟨1199091⟩ + 119899sum119894=2

119886119894 ⟨119909119894⟩) (87)


119894=1

119896119894119909119894) + 11988611198781 + 1198871119878 minus 120590222 ] d119905+ 1205902d1198612 (119905)

ge [minus(1198891 + 119872 119899sum119894=1

119896119894) + 11988611198781 + 1198871119878 minus 120590222 ] d119905+ 1205902d1198612 (119905)

(88)



119894=1

119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩ minus 1198861⟨ 120601 minus 119878(1 + 1198871120601) (1 + 1198871119878)⟩


119894=1

119896119894)+ 1198861⟨ 1206011 + 1198871120601⟩ minus 119886111988721 ⟨1119878 minus 1120601⟩ + 12059021198612 (119905)119905


119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩

minus 119886111988721 (119889 + 119873sum119899119894=1 119896119894) 1198780 (1198861 ⟨1199091⟩ + 119899sum119894=2

119886119894 ⟨119909119894⟩)+ 12059021198612 (119905)119905


119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩ minus 11988621 ⟨1199091⟩11988721 (119889 + 119873sum119899119894=1 119896119894) 1198780+ 12059021198612 (119905)119905

(89)



sup ln 1199091 (119905)119905 le 0 (90)

which yields


119894=1

119896119894)+ 1198861⟨ 1206011 + 1198871120601⟩ + 12059021198612 (119905)119905

(91)

that is



119894=1

119896119894) + 1198861 intinfin0

119909V (d119909)1 + 1198871119909 ] (92)

Therefore if

1198861 intinfin0

119909V (d119909)1 + 1198871119909 gt 120590222 + (119889 + 119872 119899sum119894=1

119896119894) (93)

then


inf ⟨1199091 (119905)⟩ gt 0 119886119904 (94)



119886119894 intinfin0

119909V (d119909)1 + 119887119894119909 lt 119889119894 + 119873 119899sum119894=1

119896119894 + 1205902119894+12 119894 = 1 2 119899 (95)

12 Complexity

then


int1199050119878 (119904) d119904119905 = 1198780 119886119904


supln 119909119894 (119905)119905 lt 0 119886119904 119894 = 1 2 119899

(96)




119896119894119909119894 + 1198861198941198781 + 119887119894119878 minus 1205902119894+12 )d119905+ 120590119894+1d119861119894+1 (119905)


119896119894 + 1198861198941206011 + 119887119894120601 minus 1205902119894+12 )d119905+ 120590119894+1d119861119894+1 (119905)

(97)


119894=1

119896119894 minus 1205902119894+12 ) + 119886119894⟨ 1206011 + 119887119894120601⟩+ 120590119894+1119861119894+1 (119905)119905 + ln 119909119894 (0)119905

(98)


119886119894 intinfin0

119909V (d119909)1 + 119887119894119909 lt 119889119894 + 119873 119899sum119894=1

119896119894 + 1205902119894+12 119894 = 1 2 119899 (99)

we know that


supln119909119894 (119905)119905 le 0 119894 = 1 2 119899 (100)



d119884 = minus(119889 + 119899sum119894=1

119896119894119909119894)119884d119905 + 1205901 (1198780 + 119884) d1198611 (119905) (101)



119894=1

119896119894119909119894) minus 120576119878120576] d119905+ 1205901119878120576d1198611 (119905)

(102)

Then

d (119878 minus 119878120576) = [(120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878) 119878

minus (119889 + 120576 + 119899sum119894=1

119896119894119909119894) (119878 minus 119878120576)] d119905 + 1205901 (119878minus 119878120576) d1198611 (119905)

(103)



119894=1

119896119894119909119894 + 120576 + 120590212 ) 119905]

sdot int1199050exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904

(104)


120576 gt 119899sum119894=1

119886119894119909119894 (119905)1 + 119887119894119878 (119905) forall119905 ge 119879 (105)


119894=1

119896119894119909119894 + 120576

+ 120590212 ) 119905]

sdot int1198790exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904

+ int119905119879exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

Complexity 13

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904 ge exp[12059011198611 (119905)

minus (119889 + 119872 119899sum119894=1

119896119894 + 120576 + 120590212 ) 119905]

sdot int1198790exp[(119889 + 119873 119899sum

119894=1

119896119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904(106)


exp[12059011198611 (119905) minus (119889 + 119872 119899sum119894=1

119896119894 + 120576 + 120590212 ) 119905] 997888rarr 01003816100381610038161003816100381610038161003816100381610038161003816int119879

0exp[(119889 + 119873 119899sum

119894=1

119896119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d1199041003816100381610038161003816100381610038161003816100381610038161003816 lt infin

(107)

which implies that



inf (119878 (119905) minus 119878120576 (119905))ge 0 119886119904 (108)

Now we consider

d (120601 (119905) minus 119878120576 (119905))= [120576119878120576 (119905) minus (119889 + 119899sum

119894=1

119896119894119909119894) (120601 (119905) minus 119878120576 (119905))] d119905+ 1205901 (120601 (119905) minus 119878120576 (119905)) d1198611 (119905)

le [120576119878120576 (119905) minus (119889 + 119873 119899sum119894=1

119896119894)(120601 (119905) minus 119878120576 (119905))] d119905+ 1205901 (120601 (119905) minus 119878120576 (119905)) d1198611 (119905)

(109)



119894=1


(110)


119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816le 119864[int119905

0120576119878120576 (119904) minus (119889 + 119873 119899sum

119894=1

119896119894) 1003816100381610038161003816120601 (119904) minus 119878120576 (119904)1003816100381610038161003816 d119904]

le 119864[int1199050120576120601 (119904) minus (119889 + 119873 119899sum

119894=1

119896119894) 1003816100381610038161003816120601 (119904) minus 119878120576 (119904)1003816100381610038161003816 d119904] (111)

then

119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816 le 120576 sup0le119904

119864120601 (119904) (112)

which leads to

lim120576997888rarr0


119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816 = 0 (113)



(120601 (119905) minus 119878 (119905)) = 0 (114)









14 Complexity

200 400 600 800 1000 1200 1400 1600 1800 20000Time

0

01

02

03

04

05

06

07

08

Valu

es


0 100806040200

02

04x2(t) path

(a)

0

01

02

03

04

05

06

07

08

09

1

Valu

es

1 2 3 4 5 6 7 8 9 100Time


times104

(b)


S(t) path

01

02

03

04

05

06

07

S(t)

200 400 600 800 1000 1200 1400 1600 1800 20000Time

(a)

0

5

10

15

20

25

30


(b)



d1199091 (119905) = [minus (032 + 00071199091 (119905) + 0011199092 (119905)) 1199091 (119905)

+ 2119878 (119905)1 + 3119878 (119905)1199091 (119905)] d119905 + 011199091 (119905) d1198612 (119905) d1199092 (119905) = [minus (031 + 00071199091 (119905) + 0011199092 (119905)) 1199092 (119905)

+ 2119878 (119905)1 + 5119878 (119905)1199092 (119905)] d119905 + 021199092 (119905) d1198613 (119905) (116)


Complexity 15

01

02

03

04

05

06

07

08

09

1

S(t)

S(t) path

1 2 3 4 5 6 7 8 9 100Time times10

4

(a)


0

5

10

15

20

25

30

(b)


x1(t) path

005

01

015

02

025

03

035

04

x1(t)

200 400 600 800 1000 1200 1400 1600 1800 20000Time

(a)


0

5

10

15

20

25

30

35

40

45

50

(b)





16 Complexity

1 2 3 4 5 6 7 8 9 100Time

005

01

015

02

025

03

035

04

045

05

x1(t)

x1(t) path

times104

(a)


0

5

10

15

20

25

30

35

40

45

50

(b)






d1199091 (119905) = [minus (09 + 021199091 (119905) + 021199092 (119905)) 1199091 (119905)+ 119878 (119905)1 + 04119878 (119905)1199091 (119905)] d119905 + 081199091 (119905) d1198612 (119905)

d1199092 (119905) = [minus (09 + 021199091 (119905) + 021199092 (119905)) 1199092 (119905)+ 119878 (119905)1 + 06119878 (119905)1199092 (119905)] d119905 + 081199092 (119905) d1198613 (119905)

(117)





Data Availability




Acknowledgments


Complexity 17

0 20018016014012010080604020Time

0

02

04

06

08

1

12

14

16

18

2Va

lues


0 109876543210

01

02


(a)

0 10000900080007000600050004000300020001000Time

0

05

1

15

2

25

Valu

es



0

02

04

0 87654321

(b)



References






















18 Complexity






























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 7

Proof Define

119883 (119905) = int1199050(119878 (119904) minus 1205821)d1198611 (119904)


119885119894 (119905) = int1199050119909119894 (119904) d119861119894+1 (119904) 119894 = 2 3 119899

(48)


119864[ sup0le119904le119905

|119883 (119904)|120579] le 119862120579119864[int1199050(119878 (119904) minus 1205821)2 d119904]

12

le 1198621205791199051205792119864[ sup0le119904le119905

(119878 (119904) minus 1205821)120579]le 1198621205791199051205792119864[ sup

0le119904le119905(1 + 119906 (119905))120579]

le 21198721198621205791199051205792

(49)


119875120596 sup119896120575le119905le(119896+1)120575

|119883 (119905)|120579 gt (119896120575)1+120572+1205792

le 119864 [|119883 ((119896 + 1) 120575)|120579](119896120575)1+120572+1205792 le 2119872119862120579(119896120575)1+120572 (

(119896 + 1) 120575119896120575 )1205792

le 21+1205792119872119862120579(119896120575)1+120572

(50)


sup119896120575le119905le(119896+1)120575

|119883 (119905)|120579 le (119896120575)1+120572+1205792 (51)


ln 119905 le ln (119896120575)1+120572+1205792ln 119905 le ln (119896120575)1+120572+1205792

ln (119896120575) (52)

which implies

ln |119883 (119905)|120579ln 119905 le 1 + 120572 + 1205792 (53)



sup ln |119883 (119905)|ln 119905 le 1 + 120572 + 1205792120579 (54)


sup ln |119883 (119905)|ln 119905 le 12 + 1120579 (55)


ln |119883 (119905)|ln 119905 le 12 + 1120579 + 120578 (56)

which means



sup 11990512+1120579+120578119905 = 0 (57)

Together with



inf

100381610038161003816100381610038161003816int1199050 (119878 (119904) minus 1205821) d1198611 (119904)100381610038161003816100381610038161003816119905ge 0

(58)

we derive


|119883 (119905)|119905 = 0 119886119904 (59)

and


int1199050(119878 (119904) minus 1205821) d1198611 (119904)119905 = 0 119886119904 (60)





int1199050119909119894 (119905) d119861119894+1 (119904)119905 = 0 119886119904

119894 = 2 3 119899(61)



d120601 (119905) = [(1198780 minus 119878 (119905)) (119889 + 119899sum119894=1

119896119894119909119894)] d119905+ 1205901120601 (119905) d1198611 (119905)

(62)


0119911V(d119911) =


119901 (119911) = exp [2 (119889 + sum119899119894=1 119896119894119909119894) 119878012059021119911 ]


8 Complexity



gt 119899sum119894=1

1198961198941198780 + (120579 minus 12 or 0) (12059021 or or 1205902119899) 11988611989412058211 + 1198871198941205821 +

1198861198942radic119887119894 (1 + 1198871198941205821) ( 119867119889 + 119873sum119899119894=1 119896119894)12

le (119889119894 + 119873 119899sum119894=1

119896119894) + 1205902119894+12 (119894 = 2 3 119899)


119909V (d119909)1 + 1198871119909 gt 120590222 + (119889 + 119872 119899sum119894=1

119896119894)

(64)



supln119909119894 (119905)119905 le 0 119886119904 (119894 = 2 119899) (65)




(66)



(1 + 1198871198941205821) 119909119894

(67)


d119881 = 119878 minus 1205821119878 [(1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1

1198861198941198781 + 119887119894119878119909119894] d119905 + 1205901119878d1198611 (119905) + 12 12058211198782 120590211198782d119905

+ [(1 + 11988711205821) minus (1 + 11988711205821) 119909lowast11199091 ]

sdot [minus(1198891 + 119899sum119894=1

119896119894119909119894)1199091 + 11988611198781 + 11988711198781199091] d119905

+ 12059021199091d1198612 (119905) + 12(1 + 11988711205821) 119909lowast111990921 1205902211990921d119905 + 119899sum

119894=2

(1

+ 1198871198941205821) [minus(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894 + 1198861198941198781 + 119887119894119878119909119894] d119905

+ 120590119894+1119909119894d119861119894+1 (119905) = 119871119881d119905 + (119878 minus 1205821) 1205901d1198611 (119905)


(1 + 1198871198941205821) 120590119894+1119909119894d119861119894+1 (119905) (68)

where

119871119881 = 119878 minus 1205821119878 [(1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1


sdot [minus(1198891 + 119899sum119894=1

119896119894119909119894)1199091 + 11988611198781 + 11988711198781199091]

+ 119899sum119894=2

(1 + 1198871198941205821) [minus(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894 + 1198861198941198781 + 119887119894119878119909119894]

(69)



(70)



Complexity 9


119899sum119894=2

119896119894119909119894+ 119899sum119894=2

119909119894 [ 11988611989412058211 + 1198871198941205821 minus (119889119894 + 119899sum119894=1

119896119894119909119894)] + 12120582112059021+ 12 (1 + 11988711205821) 119909lowast112059022


119899sum119894=2

119896119894119909119894


119909119894 [minus 119899sum119894=1


(71)

Set

119872 = max 1199091 119909119899 119873 = min 1199091 119909119899 (72)

then


119894=2

119896119894


119872 119899sum119894=1


(73)

where


119894=2

119896119894


+ (119899 minus 1)119872 119899sum119894=1

119896119894119909lowast119894 + 12120582112059021+ 12 (1 + 11988711205821) 119909lowast112059022

(74)



0

(119878 (119904) minus 1205821)2119878 (119904) d119904 + 119867+ 1205901 1119905 int119905

0(119878 (119904) minus 1205821) d1198611 (119904)


+ 119899sum119894=2

(1 + 1198871198941205821) 120590119894+1 int1199050119909119894 (119904) d119861119894+1 (119905)

(75)



sup 1119905 int1199050

(119878 (119904) minus 1205821)2119878 (119904) d119904 le 119867119889 + 119873sum119899119894=1 119896119894 (76)

Define

119881119894 (119909119894 (119905)) = ln119909119894 (119905) 119894 = 2 3 119899 (77)


d119881119894 = [minus(119889119894 + 119899sum119894=1

119896119894119909119894) + 1198861198941198781 + 119887119894119878 minus 1205902119894+12 ] d119905


119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12 ] d119905

+ 120590119894+1d119861119894+1 (119905)

(78)


119881119894 (119905) minus 119881119894 (0)119905le 1198861198941 + 1198871198941205821 ⟨

119878 (119905) minus 12058211 + 119887119894119878 (119905)⟩ minus (119889119894 + 119873 119899sum119894=1

119896119894)

10 Complexity

+ 11988611989412058211 + 1198871198941205821 minus1205902119894+12 + 120590119894+1119861119894+1 (119905)119905


119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 120590119894+1119861119894+1 (119905)119905le 1198861198942radic119887119894 (1 + 1198871198941205821) [1119905 int119905

0

(119878 (119905) minus 1205821)2119878 (119905) d119904]12

minus (119889119894 + 119873 119899sum119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 120590119894+1119861119894+1 (119905)119905 (79)

which meansln119909119894 (119905)119905

le ln119909119894 (0)119905+ 1198861198942radic119887119894 (1 + 1198871198941205821) [1119905 int119905

0

(119878 (119905) minus 1205821)2119878 (119905) d119904]12

minus (119889119894 + 119873 119899sum119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 120590119894+1119861119894+1 (119905)119905

(80)



supln119909119894 (119905)119905

le minus(119889119894 + 119873 119899sum119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 1198861198942radic119887119894 (1 + 1198871198941205821) ( 119867119889 + 119873sum119899119894=1 119896119894)12

(119894 = 2 119899)

(81)

If

11988611989412058211 + 1198871198941205821 +1198861198942radic119887119894 (1 + 1198871198941205821) ( 119867119889 + 119873sum119899119894=1 119896119894)

12

le (119889119894 + 119873 119899sum119894=1

119896119894) + 1205902119894+12 (82)

then we have


supln119909119894 (119905)119905 le 0 119886119904 (119894 = 2 119899) (83)




1198861198941199091198941 + 119887119894119878 minus 120590212 ] d119905+ 1205901d1198611 (119905)


(84)



119894=1

1198861198941199091198941 + 119887119894119878] d119905

minus 120590212 + 1119905 int11990501205901d1198611 (119905)

= (119889 + 119899sum119894=1

119896119894119909119894)1198780 ⟨1119878⟩ minus 1198861⟨ 11990911 + 1198871119878⟩

minus 119899sum119894=2

119886119894⟨ 1199091198941 + 119887119894119878⟩ minus (119889 + 119899sum119894=1

119896119894119909119894) minus 120590212+ 12059011198611 (119905)119905 + ln 119878 (0)119905

ln120601 (119905)119905= ln120601 (0)119905 + 1119905 int119905

0

(1198780 minus 120601) (119889 + sum119899119894=1 119896119894119909119894)120601 d119905 minus 120590212+ 12059011198611 (119905)119905

= (119889 + 119899sum119894=1

119896119894119909119894)1198780⟨1120601⟩ minus (119889 + 119899sum119894=1

119896119894119909119894) minus 120590212+ 12059011198611 (119905)119905 + ln120601 (0)119905

(85)

Complexity 11

Therefore

0 ge ln 119878 (119905) minus ln120601 (119905)119905= (119889 + 119899sum

119894=1

119896119894119909119894)1198780⟨1119878 minus 1120601⟩ minus 1198861⟨ 11990911 + 1198871119878⟩

minus 119899sum119894=2

119886119894⟨ 1199091198941 + 119887119894119878⟩

ge (119889 + 119873 119899sum119894=1

119896119894)1198780⟨1119878 minus 1120601⟩ minus 1198861 ⟨1199091⟩

minus 119899sum119894=2

119886119894 ⟨119909119894⟩

(86)

which implies that

⟨1119878 minus 1120601⟩

le 1(119889 + 119873sum119899119894=1 119896119894) 1198780 (1198861 ⟨1199091⟩ + 119899sum119894=2

119886119894 ⟨119909119894⟩) (87)


119894=1

119896119894119909119894) + 11988611198781 + 1198871119878 minus 120590222 ] d119905+ 1205902d1198612 (119905)

ge [minus(1198891 + 119872 119899sum119894=1

119896119894) + 11988611198781 + 1198871119878 minus 120590222 ] d119905+ 1205902d1198612 (119905)

(88)



119894=1

119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩ minus 1198861⟨ 120601 minus 119878(1 + 1198871120601) (1 + 1198871119878)⟩


119894=1

119896119894)+ 1198861⟨ 1206011 + 1198871120601⟩ minus 119886111988721 ⟨1119878 minus 1120601⟩ + 12059021198612 (119905)119905


119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩

minus 119886111988721 (119889 + 119873sum119899119894=1 119896119894) 1198780 (1198861 ⟨1199091⟩ + 119899sum119894=2

119886119894 ⟨119909119894⟩)+ 12059021198612 (119905)119905


119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩ minus 11988621 ⟨1199091⟩11988721 (119889 + 119873sum119899119894=1 119896119894) 1198780+ 12059021198612 (119905)119905

(89)



sup ln 1199091 (119905)119905 le 0 (90)

which yields


119894=1

119896119894)+ 1198861⟨ 1206011 + 1198871120601⟩ + 12059021198612 (119905)119905

(91)

that is



119894=1

119896119894) + 1198861 intinfin0

119909V (d119909)1 + 1198871119909 ] (92)

Therefore if

1198861 intinfin0

119909V (d119909)1 + 1198871119909 gt 120590222 + (119889 + 119872 119899sum119894=1

119896119894) (93)

then


inf ⟨1199091 (119905)⟩ gt 0 119886119904 (94)



119886119894 intinfin0

119909V (d119909)1 + 119887119894119909 lt 119889119894 + 119873 119899sum119894=1

119896119894 + 1205902119894+12 119894 = 1 2 119899 (95)

12 Complexity

then


int1199050119878 (119904) d119904119905 = 1198780 119886119904


supln 119909119894 (119905)119905 lt 0 119886119904 119894 = 1 2 119899

(96)




119896119894119909119894 + 1198861198941198781 + 119887119894119878 minus 1205902119894+12 )d119905+ 120590119894+1d119861119894+1 (119905)


119896119894 + 1198861198941206011 + 119887119894120601 minus 1205902119894+12 )d119905+ 120590119894+1d119861119894+1 (119905)

(97)


119894=1

119896119894 minus 1205902119894+12 ) + 119886119894⟨ 1206011 + 119887119894120601⟩+ 120590119894+1119861119894+1 (119905)119905 + ln 119909119894 (0)119905

(98)


119886119894 intinfin0

119909V (d119909)1 + 119887119894119909 lt 119889119894 + 119873 119899sum119894=1

119896119894 + 1205902119894+12 119894 = 1 2 119899 (99)

we know that


supln119909119894 (119905)119905 le 0 119894 = 1 2 119899 (100)



d119884 = minus(119889 + 119899sum119894=1

119896119894119909119894)119884d119905 + 1205901 (1198780 + 119884) d1198611 (119905) (101)



119894=1

119896119894119909119894) minus 120576119878120576] d119905+ 1205901119878120576d1198611 (119905)

(102)

Then

d (119878 minus 119878120576) = [(120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878) 119878

minus (119889 + 120576 + 119899sum119894=1

119896119894119909119894) (119878 minus 119878120576)] d119905 + 1205901 (119878minus 119878120576) d1198611 (119905)

(103)



119894=1

119896119894119909119894 + 120576 + 120590212 ) 119905]

sdot int1199050exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904

(104)


120576 gt 119899sum119894=1

119886119894119909119894 (119905)1 + 119887119894119878 (119905) forall119905 ge 119879 (105)


119894=1

119896119894119909119894 + 120576

+ 120590212 ) 119905]

sdot int1198790exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904

+ int119905119879exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

Complexity 13

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904 ge exp[12059011198611 (119905)

minus (119889 + 119872 119899sum119894=1

119896119894 + 120576 + 120590212 ) 119905]

sdot int1198790exp[(119889 + 119873 119899sum

119894=1

119896119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904(106)


exp[12059011198611 (119905) minus (119889 + 119872 119899sum119894=1

119896119894 + 120576 + 120590212 ) 119905] 997888rarr 01003816100381610038161003816100381610038161003816100381610038161003816int119879

0exp[(119889 + 119873 119899sum

119894=1

119896119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d1199041003816100381610038161003816100381610038161003816100381610038161003816 lt infin

(107)

which implies that



inf (119878 (119905) minus 119878120576 (119905))ge 0 119886119904 (108)

Now we consider

d (120601 (119905) minus 119878120576 (119905))= [120576119878120576 (119905) minus (119889 + 119899sum

119894=1

119896119894119909119894) (120601 (119905) minus 119878120576 (119905))] d119905+ 1205901 (120601 (119905) minus 119878120576 (119905)) d1198611 (119905)

le [120576119878120576 (119905) minus (119889 + 119873 119899sum119894=1

119896119894)(120601 (119905) minus 119878120576 (119905))] d119905+ 1205901 (120601 (119905) minus 119878120576 (119905)) d1198611 (119905)

(109)



119894=1


(110)


119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816le 119864[int119905

0120576119878120576 (119904) minus (119889 + 119873 119899sum

119894=1

119896119894) 1003816100381610038161003816120601 (119904) minus 119878120576 (119904)1003816100381610038161003816 d119904]

le 119864[int1199050120576120601 (119904) minus (119889 + 119873 119899sum

119894=1

119896119894) 1003816100381610038161003816120601 (119904) minus 119878120576 (119904)1003816100381610038161003816 d119904] (111)

then

119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816 le 120576 sup0le119904

119864120601 (119904) (112)

which leads to

lim120576997888rarr0


119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816 = 0 (113)



(120601 (119905) minus 119878 (119905)) = 0 (114)









14 Complexity

200 400 600 800 1000 1200 1400 1600 1800 20000Time

0

01

02

03

04

05

06

07

08

Valu

es


0 100806040200

02

04x2(t) path

(a)

0

01

02

03

04

05

06

07

08

09

1

Valu

es

1 2 3 4 5 6 7 8 9 100Time


times104

(b)


S(t) path

01

02

03

04

05

06

07

S(t)

200 400 600 800 1000 1200 1400 1600 1800 20000Time

(a)

0

5

10

15

20

25

30


(b)



d1199091 (119905) = [minus (032 + 00071199091 (119905) + 0011199092 (119905)) 1199091 (119905)

+ 2119878 (119905)1 + 3119878 (119905)1199091 (119905)] d119905 + 011199091 (119905) d1198612 (119905) d1199092 (119905) = [minus (031 + 00071199091 (119905) + 0011199092 (119905)) 1199092 (119905)

+ 2119878 (119905)1 + 5119878 (119905)1199092 (119905)] d119905 + 021199092 (119905) d1198613 (119905) (116)


Complexity 15

01

02

03

04

05

06

07

08

09

1

S(t)

S(t) path

1 2 3 4 5 6 7 8 9 100Time times10

4

(a)


0

5

10

15

20

25

30

(b)


x1(t) path

005

01

015

02

025

03

035

04

x1(t)

200 400 600 800 1000 1200 1400 1600 1800 20000Time

(a)


0

5

10

15

20

25

30

35

40

45

50

(b)





16 Complexity

1 2 3 4 5 6 7 8 9 100Time

005

01

015

02

025

03

035

04

045

05

x1(t)

x1(t) path

times104

(a)


0

5

10

15

20

25

30

35

40

45

50

(b)






d1199091 (119905) = [minus (09 + 021199091 (119905) + 021199092 (119905)) 1199091 (119905)+ 119878 (119905)1 + 04119878 (119905)1199091 (119905)] d119905 + 081199091 (119905) d1198612 (119905)

d1199092 (119905) = [minus (09 + 021199091 (119905) + 021199092 (119905)) 1199092 (119905)+ 119878 (119905)1 + 06119878 (119905)1199092 (119905)] d119905 + 081199092 (119905) d1198613 (119905)

(117)





Data Availability




Acknowledgments


Complexity 17

0 20018016014012010080604020Time

0

02

04

06

08

1

12

14

16

18

2Va

lues


0 109876543210

01

02


(a)

0 10000900080007000600050004000300020001000Time

0

05

1

15

2

25

Valu

es



0

02

04

0 87654321

(b)



References






















18 Complexity






























Journal of












Journal of







Volume 2018






Volume 2018








8 Complexity



gt 119899sum119894=1

1198961198941198780 + (120579 minus 12 or 0) (12059021 or or 1205902119899) 11988611989412058211 + 1198871198941205821 +

1198861198942radic119887119894 (1 + 1198871198941205821) ( 119867119889 + 119873sum119899119894=1 119896119894)12

le (119889119894 + 119873 119899sum119894=1

119896119894) + 1205902119894+12 (119894 = 2 3 119899)


119909V (d119909)1 + 1198871119909 gt 120590222 + (119889 + 119872 119899sum119894=1

119896119894)

(64)



supln119909119894 (119905)119905 le 0 119886119904 (119894 = 2 119899) (65)




(66)



(1 + 1198871198941205821) 119909119894

(67)


d119881 = 119878 minus 1205821119878 [(1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1

1198861198941198781 + 119887119894119878119909119894] d119905 + 1205901119878d1198611 (119905) + 12 12058211198782 120590211198782d119905

+ [(1 + 11988711205821) minus (1 + 11988711205821) 119909lowast11199091 ]

sdot [minus(1198891 + 119899sum119894=1

119896119894119909119894)1199091 + 11988611198781 + 11988711198781199091] d119905

+ 12059021199091d1198612 (119905) + 12(1 + 11988711205821) 119909lowast111990921 1205902211990921d119905 + 119899sum

119894=2

(1

+ 1198871198941205821) [minus(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894 + 1198861198941198781 + 119887119894119878119909119894] d119905

+ 120590119894+1119909119894d119861119894+1 (119905) = 119871119881d119905 + (119878 minus 1205821) 1205901d1198611 (119905)


(1 + 1198871198941205821) 120590119894+1119909119894d119861119894+1 (119905) (68)

where

119871119881 = 119878 minus 1205821119878 [(1198780 minus 119878)(119889 + 119899sum119894=1

119896119894119909119894)

minus 119899sum119894=1


sdot [minus(1198891 + 119899sum119894=1

119896119894119909119894)1199091 + 11988611198781 + 11988711198781199091]

+ 119899sum119894=2

(1 + 1198871198941205821) [minus(119889119894 + 119899sum119894=1

119896119894119909119894)119909119894 + 1198861198941198781 + 119887119894119878119909119894]

(69)



(70)



Complexity 9


119899sum119894=2

119896119894119909119894+ 119899sum119894=2

119909119894 [ 11988611989412058211 + 1198871198941205821 minus (119889119894 + 119899sum119894=1

119896119894119909119894)] + 12120582112059021+ 12 (1 + 11988711205821) 119909lowast112059022


119899sum119894=2

119896119894119909119894


119909119894 [minus 119899sum119894=1


(71)

Set

119872 = max 1199091 119909119899 119873 = min 1199091 119909119899 (72)

then


119894=2

119896119894


119872 119899sum119894=1


(73)

where


119894=2

119896119894


+ (119899 minus 1)119872 119899sum119894=1

119896119894119909lowast119894 + 12120582112059021+ 12 (1 + 11988711205821) 119909lowast112059022

(74)



0

(119878 (119904) minus 1205821)2119878 (119904) d119904 + 119867+ 1205901 1119905 int119905

0(119878 (119904) minus 1205821) d1198611 (119904)


+ 119899sum119894=2

(1 + 1198871198941205821) 120590119894+1 int1199050119909119894 (119904) d119861119894+1 (119905)

(75)



sup 1119905 int1199050

(119878 (119904) minus 1205821)2119878 (119904) d119904 le 119867119889 + 119873sum119899119894=1 119896119894 (76)

Define

119881119894 (119909119894 (119905)) = ln119909119894 (119905) 119894 = 2 3 119899 (77)


d119881119894 = [minus(119889119894 + 119899sum119894=1

119896119894119909119894) + 1198861198941198781 + 119887119894119878 minus 1205902119894+12 ] d119905


119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12 ] d119905

+ 120590119894+1d119861119894+1 (119905)

(78)


119881119894 (119905) minus 119881119894 (0)119905le 1198861198941 + 1198871198941205821 ⟨

119878 (119905) minus 12058211 + 119887119894119878 (119905)⟩ minus (119889119894 + 119873 119899sum119894=1

119896119894)

10 Complexity

+ 11988611989412058211 + 1198871198941205821 minus1205902119894+12 + 120590119894+1119861119894+1 (119905)119905


119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 120590119894+1119861119894+1 (119905)119905le 1198861198942radic119887119894 (1 + 1198871198941205821) [1119905 int119905

0

(119878 (119905) minus 1205821)2119878 (119905) d119904]12

minus (119889119894 + 119873 119899sum119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 120590119894+1119861119894+1 (119905)119905 (79)

which meansln119909119894 (119905)119905

le ln119909119894 (0)119905+ 1198861198942radic119887119894 (1 + 1198871198941205821) [1119905 int119905

0

(119878 (119905) minus 1205821)2119878 (119905) d119904]12

minus (119889119894 + 119873 119899sum119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 120590119894+1119861119894+1 (119905)119905

(80)



supln119909119894 (119905)119905

le minus(119889119894 + 119873 119899sum119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 1198861198942radic119887119894 (1 + 1198871198941205821) ( 119867119889 + 119873sum119899119894=1 119896119894)12

(119894 = 2 119899)

(81)

If

11988611989412058211 + 1198871198941205821 +1198861198942radic119887119894 (1 + 1198871198941205821) ( 119867119889 + 119873sum119899119894=1 119896119894)

12

le (119889119894 + 119873 119899sum119894=1

119896119894) + 1205902119894+12 (82)

then we have


supln119909119894 (119905)119905 le 0 119886119904 (119894 = 2 119899) (83)




1198861198941199091198941 + 119887119894119878 minus 120590212 ] d119905+ 1205901d1198611 (119905)


(84)



119894=1

1198861198941199091198941 + 119887119894119878] d119905

minus 120590212 + 1119905 int11990501205901d1198611 (119905)

= (119889 + 119899sum119894=1

119896119894119909119894)1198780 ⟨1119878⟩ minus 1198861⟨ 11990911 + 1198871119878⟩

minus 119899sum119894=2

119886119894⟨ 1199091198941 + 119887119894119878⟩ minus (119889 + 119899sum119894=1

119896119894119909119894) minus 120590212+ 12059011198611 (119905)119905 + ln 119878 (0)119905

ln120601 (119905)119905= ln120601 (0)119905 + 1119905 int119905

0

(1198780 minus 120601) (119889 + sum119899119894=1 119896119894119909119894)120601 d119905 minus 120590212+ 12059011198611 (119905)119905

= (119889 + 119899sum119894=1

119896119894119909119894)1198780⟨1120601⟩ minus (119889 + 119899sum119894=1

119896119894119909119894) minus 120590212+ 12059011198611 (119905)119905 + ln120601 (0)119905

(85)

Complexity 11

Therefore

0 ge ln 119878 (119905) minus ln120601 (119905)119905= (119889 + 119899sum

119894=1

119896119894119909119894)1198780⟨1119878 minus 1120601⟩ minus 1198861⟨ 11990911 + 1198871119878⟩

minus 119899sum119894=2

119886119894⟨ 1199091198941 + 119887119894119878⟩

ge (119889 + 119873 119899sum119894=1

119896119894)1198780⟨1119878 minus 1120601⟩ minus 1198861 ⟨1199091⟩

minus 119899sum119894=2

119886119894 ⟨119909119894⟩

(86)

which implies that

⟨1119878 minus 1120601⟩

le 1(119889 + 119873sum119899119894=1 119896119894) 1198780 (1198861 ⟨1199091⟩ + 119899sum119894=2

119886119894 ⟨119909119894⟩) (87)


119894=1

119896119894119909119894) + 11988611198781 + 1198871119878 minus 120590222 ] d119905+ 1205902d1198612 (119905)

ge [minus(1198891 + 119872 119899sum119894=1

119896119894) + 11988611198781 + 1198871119878 minus 120590222 ] d119905+ 1205902d1198612 (119905)

(88)



119894=1

119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩ minus 1198861⟨ 120601 minus 119878(1 + 1198871120601) (1 + 1198871119878)⟩


119894=1

119896119894)+ 1198861⟨ 1206011 + 1198871120601⟩ minus 119886111988721 ⟨1119878 minus 1120601⟩ + 12059021198612 (119905)119905


119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩

minus 119886111988721 (119889 + 119873sum119899119894=1 119896119894) 1198780 (1198861 ⟨1199091⟩ + 119899sum119894=2

119886119894 ⟨119909119894⟩)+ 12059021198612 (119905)119905


119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩ minus 11988621 ⟨1199091⟩11988721 (119889 + 119873sum119899119894=1 119896119894) 1198780+ 12059021198612 (119905)119905

(89)



sup ln 1199091 (119905)119905 le 0 (90)

which yields


119894=1

119896119894)+ 1198861⟨ 1206011 + 1198871120601⟩ + 12059021198612 (119905)119905

(91)

that is



119894=1

119896119894) + 1198861 intinfin0

119909V (d119909)1 + 1198871119909 ] (92)

Therefore if

1198861 intinfin0

119909V (d119909)1 + 1198871119909 gt 120590222 + (119889 + 119872 119899sum119894=1

119896119894) (93)

then


inf ⟨1199091 (119905)⟩ gt 0 119886119904 (94)



119886119894 intinfin0

119909V (d119909)1 + 119887119894119909 lt 119889119894 + 119873 119899sum119894=1

119896119894 + 1205902119894+12 119894 = 1 2 119899 (95)

12 Complexity

then


int1199050119878 (119904) d119904119905 = 1198780 119886119904


supln 119909119894 (119905)119905 lt 0 119886119904 119894 = 1 2 119899

(96)




119896119894119909119894 + 1198861198941198781 + 119887119894119878 minus 1205902119894+12 )d119905+ 120590119894+1d119861119894+1 (119905)


119896119894 + 1198861198941206011 + 119887119894120601 minus 1205902119894+12 )d119905+ 120590119894+1d119861119894+1 (119905)

(97)


119894=1

119896119894 minus 1205902119894+12 ) + 119886119894⟨ 1206011 + 119887119894120601⟩+ 120590119894+1119861119894+1 (119905)119905 + ln 119909119894 (0)119905

(98)


119886119894 intinfin0

119909V (d119909)1 + 119887119894119909 lt 119889119894 + 119873 119899sum119894=1

119896119894 + 1205902119894+12 119894 = 1 2 119899 (99)

we know that


supln119909119894 (119905)119905 le 0 119894 = 1 2 119899 (100)



d119884 = minus(119889 + 119899sum119894=1

119896119894119909119894)119884d119905 + 1205901 (1198780 + 119884) d1198611 (119905) (101)



119894=1

119896119894119909119894) minus 120576119878120576] d119905+ 1205901119878120576d1198611 (119905)

(102)

Then

d (119878 minus 119878120576) = [(120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878) 119878

minus (119889 + 120576 + 119899sum119894=1

119896119894119909119894) (119878 minus 119878120576)] d119905 + 1205901 (119878minus 119878120576) d1198611 (119905)

(103)



119894=1

119896119894119909119894 + 120576 + 120590212 ) 119905]

sdot int1199050exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904

(104)


120576 gt 119899sum119894=1

119886119894119909119894 (119905)1 + 119887119894119878 (119905) forall119905 ge 119879 (105)


119894=1

119896119894119909119894 + 120576

+ 120590212 ) 119905]

sdot int1198790exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904

+ int119905119879exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

Complexity 13

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904 ge exp[12059011198611 (119905)

minus (119889 + 119872 119899sum119894=1

119896119894 + 120576 + 120590212 ) 119905]

sdot int1198790exp[(119889 + 119873 119899sum

119894=1

119896119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904(106)


exp[12059011198611 (119905) minus (119889 + 119872 119899sum119894=1

119896119894 + 120576 + 120590212 ) 119905] 997888rarr 01003816100381610038161003816100381610038161003816100381610038161003816int119879

0exp[(119889 + 119873 119899sum

119894=1

119896119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d1199041003816100381610038161003816100381610038161003816100381610038161003816 lt infin

(107)

which implies that



inf (119878 (119905) minus 119878120576 (119905))ge 0 119886119904 (108)

Now we consider

d (120601 (119905) minus 119878120576 (119905))= [120576119878120576 (119905) minus (119889 + 119899sum

119894=1

119896119894119909119894) (120601 (119905) minus 119878120576 (119905))] d119905+ 1205901 (120601 (119905) minus 119878120576 (119905)) d1198611 (119905)

le [120576119878120576 (119905) minus (119889 + 119873 119899sum119894=1

119896119894)(120601 (119905) minus 119878120576 (119905))] d119905+ 1205901 (120601 (119905) minus 119878120576 (119905)) d1198611 (119905)

(109)



119894=1


(110)


119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816le 119864[int119905

0120576119878120576 (119904) minus (119889 + 119873 119899sum

119894=1

119896119894) 1003816100381610038161003816120601 (119904) minus 119878120576 (119904)1003816100381610038161003816 d119904]

le 119864[int1199050120576120601 (119904) minus (119889 + 119873 119899sum

119894=1

119896119894) 1003816100381610038161003816120601 (119904) minus 119878120576 (119904)1003816100381610038161003816 d119904] (111)

then

119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816 le 120576 sup0le119904

119864120601 (119904) (112)

which leads to

lim120576997888rarr0


119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816 = 0 (113)



(120601 (119905) minus 119878 (119905)) = 0 (114)









14 Complexity

200 400 600 800 1000 1200 1400 1600 1800 20000Time

0

01

02

03

04

05

06

07

08

Valu

es


0 100806040200

02

04x2(t) path

(a)

0

01

02

03

04

05

06

07

08

09

1

Valu

es

1 2 3 4 5 6 7 8 9 100Time


times104

(b)


S(t) path

01

02

03

04

05

06

07

S(t)

200 400 600 800 1000 1200 1400 1600 1800 20000Time

(a)

0

5

10

15

20

25

30


(b)



d1199091 (119905) = [minus (032 + 00071199091 (119905) + 0011199092 (119905)) 1199091 (119905)

+ 2119878 (119905)1 + 3119878 (119905)1199091 (119905)] d119905 + 011199091 (119905) d1198612 (119905) d1199092 (119905) = [minus (031 + 00071199091 (119905) + 0011199092 (119905)) 1199092 (119905)

+ 2119878 (119905)1 + 5119878 (119905)1199092 (119905)] d119905 + 021199092 (119905) d1198613 (119905) (116)


Complexity 15

01

02

03

04

05

06

07

08

09

1

S(t)

S(t) path

1 2 3 4 5 6 7 8 9 100Time times10

4

(a)


0

5

10

15

20

25

30

(b)


x1(t) path

005

01

015

02

025

03

035

04

x1(t)

200 400 600 800 1000 1200 1400 1600 1800 20000Time

(a)


0

5

10

15

20

25

30

35

40

45

50

(b)





16 Complexity

1 2 3 4 5 6 7 8 9 100Time

005

01

015

02

025

03

035

04

045

05

x1(t)

x1(t) path

times104

(a)


0

5

10

15

20

25

30

35

40

45

50

(b)






d1199091 (119905) = [minus (09 + 021199091 (119905) + 021199092 (119905)) 1199091 (119905)+ 119878 (119905)1 + 04119878 (119905)1199091 (119905)] d119905 + 081199091 (119905) d1198612 (119905)

d1199092 (119905) = [minus (09 + 021199091 (119905) + 021199092 (119905)) 1199092 (119905)+ 119878 (119905)1 + 06119878 (119905)1199092 (119905)] d119905 + 081199092 (119905) d1198613 (119905)

(117)





Data Availability




Acknowledgments


Complexity 17

0 20018016014012010080604020Time

0

02

04

06

08

1

12

14

16

18

2Va

lues


0 109876543210

01

02


(a)

0 10000900080007000600050004000300020001000Time

0

05

1

15

2

25

Valu

es



0

02

04

0 87654321

(b)



References






















18 Complexity






























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 9


119899sum119894=2

119896119894119909119894+ 119899sum119894=2

119909119894 [ 11988611989412058211 + 1198871198941205821 minus (119889119894 + 119899sum119894=1

119896119894119909119894)] + 12120582112059021+ 12 (1 + 11988711205821) 119909lowast112059022


119899sum119894=2

119896119894119909119894


119909119894 [minus 119899sum119894=1


(71)

Set

119872 = max 1199091 119909119899 119873 = min 1199091 119909119899 (72)

then


119894=2

119896119894


119872 119899sum119894=1


(73)

where


119894=2

119896119894


+ (119899 minus 1)119872 119899sum119894=1

119896119894119909lowast119894 + 12120582112059021+ 12 (1 + 11988711205821) 119909lowast112059022

(74)



0

(119878 (119904) minus 1205821)2119878 (119904) d119904 + 119867+ 1205901 1119905 int119905

0(119878 (119904) minus 1205821) d1198611 (119904)


+ 119899sum119894=2

(1 + 1198871198941205821) 120590119894+1 int1199050119909119894 (119904) d119861119894+1 (119905)

(75)



sup 1119905 int1199050

(119878 (119904) minus 1205821)2119878 (119904) d119904 le 119867119889 + 119873sum119899119894=1 119896119894 (76)

Define

119881119894 (119909119894 (119905)) = ln119909119894 (119905) 119894 = 2 3 119899 (77)


d119881119894 = [minus(119889119894 + 119899sum119894=1

119896119894119909119894) + 1198861198941198781 + 119887119894119878 minus 1205902119894+12 ] d119905


119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12 ] d119905

+ 120590119894+1d119861119894+1 (119905)

(78)


119881119894 (119905) minus 119881119894 (0)119905le 1198861198941 + 1198871198941205821 ⟨

119878 (119905) minus 12058211 + 119887119894119878 (119905)⟩ minus (119889119894 + 119873 119899sum119894=1

119896119894)

10 Complexity

+ 11988611989412058211 + 1198871198941205821 minus1205902119894+12 + 120590119894+1119861119894+1 (119905)119905


119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 120590119894+1119861119894+1 (119905)119905le 1198861198942radic119887119894 (1 + 1198871198941205821) [1119905 int119905

0

(119878 (119905) minus 1205821)2119878 (119905) d119904]12

minus (119889119894 + 119873 119899sum119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 120590119894+1119861119894+1 (119905)119905 (79)

which meansln119909119894 (119905)119905

le ln119909119894 (0)119905+ 1198861198942radic119887119894 (1 + 1198871198941205821) [1119905 int119905

0

(119878 (119905) minus 1205821)2119878 (119905) d119904]12

minus (119889119894 + 119873 119899sum119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 120590119894+1119861119894+1 (119905)119905

(80)



supln119909119894 (119905)119905

le minus(119889119894 + 119873 119899sum119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 1198861198942radic119887119894 (1 + 1198871198941205821) ( 119867119889 + 119873sum119899119894=1 119896119894)12

(119894 = 2 119899)

(81)

If

11988611989412058211 + 1198871198941205821 +1198861198942radic119887119894 (1 + 1198871198941205821) ( 119867119889 + 119873sum119899119894=1 119896119894)

12

le (119889119894 + 119873 119899sum119894=1

119896119894) + 1205902119894+12 (82)

then we have


supln119909119894 (119905)119905 le 0 119886119904 (119894 = 2 119899) (83)




1198861198941199091198941 + 119887119894119878 minus 120590212 ] d119905+ 1205901d1198611 (119905)


(84)



119894=1

1198861198941199091198941 + 119887119894119878] d119905

minus 120590212 + 1119905 int11990501205901d1198611 (119905)

= (119889 + 119899sum119894=1

119896119894119909119894)1198780 ⟨1119878⟩ minus 1198861⟨ 11990911 + 1198871119878⟩

minus 119899sum119894=2

119886119894⟨ 1199091198941 + 119887119894119878⟩ minus (119889 + 119899sum119894=1

119896119894119909119894) minus 120590212+ 12059011198611 (119905)119905 + ln 119878 (0)119905

ln120601 (119905)119905= ln120601 (0)119905 + 1119905 int119905

0

(1198780 minus 120601) (119889 + sum119899119894=1 119896119894119909119894)120601 d119905 minus 120590212+ 12059011198611 (119905)119905

= (119889 + 119899sum119894=1

119896119894119909119894)1198780⟨1120601⟩ minus (119889 + 119899sum119894=1

119896119894119909119894) minus 120590212+ 12059011198611 (119905)119905 + ln120601 (0)119905

(85)

Complexity 11

Therefore

0 ge ln 119878 (119905) minus ln120601 (119905)119905= (119889 + 119899sum

119894=1

119896119894119909119894)1198780⟨1119878 minus 1120601⟩ minus 1198861⟨ 11990911 + 1198871119878⟩

minus 119899sum119894=2

119886119894⟨ 1199091198941 + 119887119894119878⟩

ge (119889 + 119873 119899sum119894=1

119896119894)1198780⟨1119878 minus 1120601⟩ minus 1198861 ⟨1199091⟩

minus 119899sum119894=2

119886119894 ⟨119909119894⟩

(86)

which implies that

⟨1119878 minus 1120601⟩

le 1(119889 + 119873sum119899119894=1 119896119894) 1198780 (1198861 ⟨1199091⟩ + 119899sum119894=2

119886119894 ⟨119909119894⟩) (87)


119894=1

119896119894119909119894) + 11988611198781 + 1198871119878 minus 120590222 ] d119905+ 1205902d1198612 (119905)

ge [minus(1198891 + 119872 119899sum119894=1

119896119894) + 11988611198781 + 1198871119878 minus 120590222 ] d119905+ 1205902d1198612 (119905)

(88)



119894=1

119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩ minus 1198861⟨ 120601 minus 119878(1 + 1198871120601) (1 + 1198871119878)⟩


119894=1

119896119894)+ 1198861⟨ 1206011 + 1198871120601⟩ minus 119886111988721 ⟨1119878 minus 1120601⟩ + 12059021198612 (119905)119905


119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩

minus 119886111988721 (119889 + 119873sum119899119894=1 119896119894) 1198780 (1198861 ⟨1199091⟩ + 119899sum119894=2

119886119894 ⟨119909119894⟩)+ 12059021198612 (119905)119905


119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩ minus 11988621 ⟨1199091⟩11988721 (119889 + 119873sum119899119894=1 119896119894) 1198780+ 12059021198612 (119905)119905

(89)



sup ln 1199091 (119905)119905 le 0 (90)

which yields


119894=1

119896119894)+ 1198861⟨ 1206011 + 1198871120601⟩ + 12059021198612 (119905)119905

(91)

that is



119894=1

119896119894) + 1198861 intinfin0

119909V (d119909)1 + 1198871119909 ] (92)

Therefore if

1198861 intinfin0

119909V (d119909)1 + 1198871119909 gt 120590222 + (119889 + 119872 119899sum119894=1

119896119894) (93)

then


inf ⟨1199091 (119905)⟩ gt 0 119886119904 (94)



119886119894 intinfin0

119909V (d119909)1 + 119887119894119909 lt 119889119894 + 119873 119899sum119894=1

119896119894 + 1205902119894+12 119894 = 1 2 119899 (95)

12 Complexity

then


int1199050119878 (119904) d119904119905 = 1198780 119886119904


supln 119909119894 (119905)119905 lt 0 119886119904 119894 = 1 2 119899

(96)




119896119894119909119894 + 1198861198941198781 + 119887119894119878 minus 1205902119894+12 )d119905+ 120590119894+1d119861119894+1 (119905)


119896119894 + 1198861198941206011 + 119887119894120601 minus 1205902119894+12 )d119905+ 120590119894+1d119861119894+1 (119905)

(97)


119894=1

119896119894 minus 1205902119894+12 ) + 119886119894⟨ 1206011 + 119887119894120601⟩+ 120590119894+1119861119894+1 (119905)119905 + ln 119909119894 (0)119905

(98)


119886119894 intinfin0

119909V (d119909)1 + 119887119894119909 lt 119889119894 + 119873 119899sum119894=1

119896119894 + 1205902119894+12 119894 = 1 2 119899 (99)

we know that


supln119909119894 (119905)119905 le 0 119894 = 1 2 119899 (100)



d119884 = minus(119889 + 119899sum119894=1

119896119894119909119894)119884d119905 + 1205901 (1198780 + 119884) d1198611 (119905) (101)



119894=1

119896119894119909119894) minus 120576119878120576] d119905+ 1205901119878120576d1198611 (119905)

(102)

Then

d (119878 minus 119878120576) = [(120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878) 119878

minus (119889 + 120576 + 119899sum119894=1

119896119894119909119894) (119878 minus 119878120576)] d119905 + 1205901 (119878minus 119878120576) d1198611 (119905)

(103)



119894=1

119896119894119909119894 + 120576 + 120590212 ) 119905]

sdot int1199050exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904

(104)


120576 gt 119899sum119894=1

119886119894119909119894 (119905)1 + 119887119894119878 (119905) forall119905 ge 119879 (105)


119894=1

119896119894119909119894 + 120576

+ 120590212 ) 119905]

sdot int1198790exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904

+ int119905119879exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

Complexity 13

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904 ge exp[12059011198611 (119905)

minus (119889 + 119872 119899sum119894=1

119896119894 + 120576 + 120590212 ) 119905]

sdot int1198790exp[(119889 + 119873 119899sum

119894=1

119896119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904(106)


exp[12059011198611 (119905) minus (119889 + 119872 119899sum119894=1

119896119894 + 120576 + 120590212 ) 119905] 997888rarr 01003816100381610038161003816100381610038161003816100381610038161003816int119879

0exp[(119889 + 119873 119899sum

119894=1

119896119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d1199041003816100381610038161003816100381610038161003816100381610038161003816 lt infin

(107)

which implies that



inf (119878 (119905) minus 119878120576 (119905))ge 0 119886119904 (108)

Now we consider

d (120601 (119905) minus 119878120576 (119905))= [120576119878120576 (119905) minus (119889 + 119899sum

119894=1

119896119894119909119894) (120601 (119905) minus 119878120576 (119905))] d119905+ 1205901 (120601 (119905) minus 119878120576 (119905)) d1198611 (119905)

le [120576119878120576 (119905) minus (119889 + 119873 119899sum119894=1

119896119894)(120601 (119905) minus 119878120576 (119905))] d119905+ 1205901 (120601 (119905) minus 119878120576 (119905)) d1198611 (119905)

(109)



119894=1


(110)


119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816le 119864[int119905

0120576119878120576 (119904) minus (119889 + 119873 119899sum

119894=1

119896119894) 1003816100381610038161003816120601 (119904) minus 119878120576 (119904)1003816100381610038161003816 d119904]

le 119864[int1199050120576120601 (119904) minus (119889 + 119873 119899sum

119894=1

119896119894) 1003816100381610038161003816120601 (119904) minus 119878120576 (119904)1003816100381610038161003816 d119904] (111)

then

119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816 le 120576 sup0le119904

119864120601 (119904) (112)

which leads to

lim120576997888rarr0


119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816 = 0 (113)



(120601 (119905) minus 119878 (119905)) = 0 (114)









14 Complexity

200 400 600 800 1000 1200 1400 1600 1800 20000Time

0

01

02

03

04

05

06

07

08

Valu

es


0 100806040200

02

04x2(t) path

(a)

0

01

02

03

04

05

06

07

08

09

1

Valu

es

1 2 3 4 5 6 7 8 9 100Time


times104

(b)


S(t) path

01

02

03

04

05

06

07

S(t)

200 400 600 800 1000 1200 1400 1600 1800 20000Time

(a)

0

5

10

15

20

25

30


(b)



d1199091 (119905) = [minus (032 + 00071199091 (119905) + 0011199092 (119905)) 1199091 (119905)

+ 2119878 (119905)1 + 3119878 (119905)1199091 (119905)] d119905 + 011199091 (119905) d1198612 (119905) d1199092 (119905) = [minus (031 + 00071199091 (119905) + 0011199092 (119905)) 1199092 (119905)

+ 2119878 (119905)1 + 5119878 (119905)1199092 (119905)] d119905 + 021199092 (119905) d1198613 (119905) (116)


Complexity 15

01

02

03

04

05

06

07

08

09

1

S(t)

S(t) path

1 2 3 4 5 6 7 8 9 100Time times10

4

(a)


0

5

10

15

20

25

30

(b)


x1(t) path

005

01

015

02

025

03

035

04

x1(t)

200 400 600 800 1000 1200 1400 1600 1800 20000Time

(a)


0

5

10

15

20

25

30

35

40

45

50

(b)





16 Complexity

1 2 3 4 5 6 7 8 9 100Time

005

01

015

02

025

03

035

04

045

05

x1(t)

x1(t) path

times104

(a)


0

5

10

15

20

25

30

35

40

45

50

(b)






d1199091 (119905) = [minus (09 + 021199091 (119905) + 021199092 (119905)) 1199091 (119905)+ 119878 (119905)1 + 04119878 (119905)1199091 (119905)] d119905 + 081199091 (119905) d1198612 (119905)

d1199092 (119905) = [minus (09 + 021199091 (119905) + 021199092 (119905)) 1199092 (119905)+ 119878 (119905)1 + 06119878 (119905)1199092 (119905)] d119905 + 081199092 (119905) d1198613 (119905)

(117)





Data Availability




Acknowledgments


Complexity 17

0 20018016014012010080604020Time

0

02

04

06

08

1

12

14

16

18

2Va

lues


0 109876543210

01

02


(a)

0 10000900080007000600050004000300020001000Time

0

05

1

15

2

25

Valu

es



0

02

04

0 87654321

(b)



References






















18 Complexity






























Journal of












Journal of







Volume 2018






Volume 2018








10 Complexity

+ 11988611989412058211 + 1198871198941205821 minus1205902119894+12 + 120590119894+1119861119894+1 (119905)119905


119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 120590119894+1119861119894+1 (119905)119905le 1198861198942radic119887119894 (1 + 1198871198941205821) [1119905 int119905

0

(119878 (119905) minus 1205821)2119878 (119905) d119904]12

minus (119889119894 + 119873 119899sum119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 120590119894+1119861119894+1 (119905)119905 (79)

which meansln119909119894 (119905)119905

le ln119909119894 (0)119905+ 1198861198942radic119887119894 (1 + 1198871198941205821) [1119905 int119905

0

(119878 (119905) minus 1205821)2119878 (119905) d119904]12

minus (119889119894 + 119873 119899sum119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 120590119894+1119861119894+1 (119905)119905

(80)



supln119909119894 (119905)119905

le minus(119889119894 + 119873 119899sum119894=1

119896119894) + 11988611989412058211 + 1198871198941205821 minus1205902119894+12

+ 1198861198942radic119887119894 (1 + 1198871198941205821) ( 119867119889 + 119873sum119899119894=1 119896119894)12

(119894 = 2 119899)

(81)

If

11988611989412058211 + 1198871198941205821 +1198861198942radic119887119894 (1 + 1198871198941205821) ( 119867119889 + 119873sum119899119894=1 119896119894)

12

le (119889119894 + 119873 119899sum119894=1

119896119894) + 1205902119894+12 (82)

then we have


supln119909119894 (119905)119905 le 0 119886119904 (119894 = 2 119899) (83)




1198861198941199091198941 + 119887119894119878 minus 120590212 ] d119905+ 1205901d1198611 (119905)


(84)



119894=1

1198861198941199091198941 + 119887119894119878] d119905

minus 120590212 + 1119905 int11990501205901d1198611 (119905)

= (119889 + 119899sum119894=1

119896119894119909119894)1198780 ⟨1119878⟩ minus 1198861⟨ 11990911 + 1198871119878⟩

minus 119899sum119894=2

119886119894⟨ 1199091198941 + 119887119894119878⟩ minus (119889 + 119899sum119894=1

119896119894119909119894) minus 120590212+ 12059011198611 (119905)119905 + ln 119878 (0)119905

ln120601 (119905)119905= ln120601 (0)119905 + 1119905 int119905

0

(1198780 minus 120601) (119889 + sum119899119894=1 119896119894119909119894)120601 d119905 minus 120590212+ 12059011198611 (119905)119905

= (119889 + 119899sum119894=1

119896119894119909119894)1198780⟨1120601⟩ minus (119889 + 119899sum119894=1

119896119894119909119894) minus 120590212+ 12059011198611 (119905)119905 + ln120601 (0)119905

(85)

Complexity 11

Therefore

0 ge ln 119878 (119905) minus ln120601 (119905)119905= (119889 + 119899sum

119894=1

119896119894119909119894)1198780⟨1119878 minus 1120601⟩ minus 1198861⟨ 11990911 + 1198871119878⟩

minus 119899sum119894=2

119886119894⟨ 1199091198941 + 119887119894119878⟩

ge (119889 + 119873 119899sum119894=1

119896119894)1198780⟨1119878 minus 1120601⟩ minus 1198861 ⟨1199091⟩

minus 119899sum119894=2

119886119894 ⟨119909119894⟩

(86)

which implies that

⟨1119878 minus 1120601⟩

le 1(119889 + 119873sum119899119894=1 119896119894) 1198780 (1198861 ⟨1199091⟩ + 119899sum119894=2

119886119894 ⟨119909119894⟩) (87)


119894=1

119896119894119909119894) + 11988611198781 + 1198871119878 minus 120590222 ] d119905+ 1205902d1198612 (119905)

ge [minus(1198891 + 119872 119899sum119894=1

119896119894) + 11988611198781 + 1198871119878 minus 120590222 ] d119905+ 1205902d1198612 (119905)

(88)



119894=1

119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩ minus 1198861⟨ 120601 minus 119878(1 + 1198871120601) (1 + 1198871119878)⟩


119894=1

119896119894)+ 1198861⟨ 1206011 + 1198871120601⟩ minus 119886111988721 ⟨1119878 minus 1120601⟩ + 12059021198612 (119905)119905


119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩

minus 119886111988721 (119889 + 119873sum119899119894=1 119896119894) 1198780 (1198861 ⟨1199091⟩ + 119899sum119894=2

119886119894 ⟨119909119894⟩)+ 12059021198612 (119905)119905


119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩ minus 11988621 ⟨1199091⟩11988721 (119889 + 119873sum119899119894=1 119896119894) 1198780+ 12059021198612 (119905)119905

(89)



sup ln 1199091 (119905)119905 le 0 (90)

which yields


119894=1

119896119894)+ 1198861⟨ 1206011 + 1198871120601⟩ + 12059021198612 (119905)119905

(91)

that is



119894=1

119896119894) + 1198861 intinfin0

119909V (d119909)1 + 1198871119909 ] (92)

Therefore if

1198861 intinfin0

119909V (d119909)1 + 1198871119909 gt 120590222 + (119889 + 119872 119899sum119894=1

119896119894) (93)

then


inf ⟨1199091 (119905)⟩ gt 0 119886119904 (94)



119886119894 intinfin0

119909V (d119909)1 + 119887119894119909 lt 119889119894 + 119873 119899sum119894=1

119896119894 + 1205902119894+12 119894 = 1 2 119899 (95)

12 Complexity

then


int1199050119878 (119904) d119904119905 = 1198780 119886119904


supln 119909119894 (119905)119905 lt 0 119886119904 119894 = 1 2 119899

(96)




119896119894119909119894 + 1198861198941198781 + 119887119894119878 minus 1205902119894+12 )d119905+ 120590119894+1d119861119894+1 (119905)


119896119894 + 1198861198941206011 + 119887119894120601 minus 1205902119894+12 )d119905+ 120590119894+1d119861119894+1 (119905)

(97)


119894=1

119896119894 minus 1205902119894+12 ) + 119886119894⟨ 1206011 + 119887119894120601⟩+ 120590119894+1119861119894+1 (119905)119905 + ln 119909119894 (0)119905

(98)


119886119894 intinfin0

119909V (d119909)1 + 119887119894119909 lt 119889119894 + 119873 119899sum119894=1

119896119894 + 1205902119894+12 119894 = 1 2 119899 (99)

we know that


supln119909119894 (119905)119905 le 0 119894 = 1 2 119899 (100)



d119884 = minus(119889 + 119899sum119894=1

119896119894119909119894)119884d119905 + 1205901 (1198780 + 119884) d1198611 (119905) (101)



119894=1

119896119894119909119894) minus 120576119878120576] d119905+ 1205901119878120576d1198611 (119905)

(102)

Then

d (119878 minus 119878120576) = [(120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878) 119878

minus (119889 + 120576 + 119899sum119894=1

119896119894119909119894) (119878 minus 119878120576)] d119905 + 1205901 (119878minus 119878120576) d1198611 (119905)

(103)



119894=1

119896119894119909119894 + 120576 + 120590212 ) 119905]

sdot int1199050exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904

(104)


120576 gt 119899sum119894=1

119886119894119909119894 (119905)1 + 119887119894119878 (119905) forall119905 ge 119879 (105)


119894=1

119896119894119909119894 + 120576

+ 120590212 ) 119905]

sdot int1198790exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904

+ int119905119879exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

Complexity 13

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904 ge exp[12059011198611 (119905)

minus (119889 + 119872 119899sum119894=1

119896119894 + 120576 + 120590212 ) 119905]

sdot int1198790exp[(119889 + 119873 119899sum

119894=1

119896119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904(106)


exp[12059011198611 (119905) minus (119889 + 119872 119899sum119894=1

119896119894 + 120576 + 120590212 ) 119905] 997888rarr 01003816100381610038161003816100381610038161003816100381610038161003816int119879

0exp[(119889 + 119873 119899sum

119894=1

119896119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d1199041003816100381610038161003816100381610038161003816100381610038161003816 lt infin

(107)

which implies that



inf (119878 (119905) minus 119878120576 (119905))ge 0 119886119904 (108)

Now we consider

d (120601 (119905) minus 119878120576 (119905))= [120576119878120576 (119905) minus (119889 + 119899sum

119894=1

119896119894119909119894) (120601 (119905) minus 119878120576 (119905))] d119905+ 1205901 (120601 (119905) minus 119878120576 (119905)) d1198611 (119905)

le [120576119878120576 (119905) minus (119889 + 119873 119899sum119894=1

119896119894)(120601 (119905) minus 119878120576 (119905))] d119905+ 1205901 (120601 (119905) minus 119878120576 (119905)) d1198611 (119905)

(109)



119894=1


(110)


119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816le 119864[int119905

0120576119878120576 (119904) minus (119889 + 119873 119899sum

119894=1

119896119894) 1003816100381610038161003816120601 (119904) minus 119878120576 (119904)1003816100381610038161003816 d119904]

le 119864[int1199050120576120601 (119904) minus (119889 + 119873 119899sum

119894=1

119896119894) 1003816100381610038161003816120601 (119904) minus 119878120576 (119904)1003816100381610038161003816 d119904] (111)

then

119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816 le 120576 sup0le119904

119864120601 (119904) (112)

which leads to

lim120576997888rarr0


119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816 = 0 (113)



(120601 (119905) minus 119878 (119905)) = 0 (114)









14 Complexity

200 400 600 800 1000 1200 1400 1600 1800 20000Time

0

01

02

03

04

05

06

07

08

Valu

es


0 100806040200

02

04x2(t) path

(a)

0

01

02

03

04

05

06

07

08

09

1

Valu

es

1 2 3 4 5 6 7 8 9 100Time


times104

(b)


S(t) path

01

02

03

04

05

06

07

S(t)

200 400 600 800 1000 1200 1400 1600 1800 20000Time

(a)

0

5

10

15

20

25

30


(b)



d1199091 (119905) = [minus (032 + 00071199091 (119905) + 0011199092 (119905)) 1199091 (119905)

+ 2119878 (119905)1 + 3119878 (119905)1199091 (119905)] d119905 + 011199091 (119905) d1198612 (119905) d1199092 (119905) = [minus (031 + 00071199091 (119905) + 0011199092 (119905)) 1199092 (119905)

+ 2119878 (119905)1 + 5119878 (119905)1199092 (119905)] d119905 + 021199092 (119905) d1198613 (119905) (116)


Complexity 15

01

02

03

04

05

06

07

08

09

1

S(t)

S(t) path

1 2 3 4 5 6 7 8 9 100Time times10

4

(a)


0

5

10

15

20

25

30

(b)


x1(t) path

005

01

015

02

025

03

035

04

x1(t)

200 400 600 800 1000 1200 1400 1600 1800 20000Time

(a)


0

5

10

15

20

25

30

35

40

45

50

(b)





16 Complexity

1 2 3 4 5 6 7 8 9 100Time

005

01

015

02

025

03

035

04

045

05

x1(t)

x1(t) path

times104

(a)


0

5

10

15

20

25

30

35

40

45

50

(b)






d1199091 (119905) = [minus (09 + 021199091 (119905) + 021199092 (119905)) 1199091 (119905)+ 119878 (119905)1 + 04119878 (119905)1199091 (119905)] d119905 + 081199091 (119905) d1198612 (119905)

d1199092 (119905) = [minus (09 + 021199091 (119905) + 021199092 (119905)) 1199092 (119905)+ 119878 (119905)1 + 06119878 (119905)1199092 (119905)] d119905 + 081199092 (119905) d1198613 (119905)

(117)





Data Availability




Acknowledgments


Complexity 17

0 20018016014012010080604020Time

0

02

04

06

08

1

12

14

16

18

2Va

lues


0 109876543210

01

02


(a)

0 10000900080007000600050004000300020001000Time

0

05

1

15

2

25

Valu

es



0

02

04

0 87654321

(b)



References






















18 Complexity






























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 11

Therefore

0 ge ln 119878 (119905) minus ln120601 (119905)119905= (119889 + 119899sum

119894=1

119896119894119909119894)1198780⟨1119878 minus 1120601⟩ minus 1198861⟨ 11990911 + 1198871119878⟩

minus 119899sum119894=2

119886119894⟨ 1199091198941 + 119887119894119878⟩

ge (119889 + 119873 119899sum119894=1

119896119894)1198780⟨1119878 minus 1120601⟩ minus 1198861 ⟨1199091⟩

minus 119899sum119894=2

119886119894 ⟨119909119894⟩

(86)

which implies that

⟨1119878 minus 1120601⟩

le 1(119889 + 119873sum119899119894=1 119896119894) 1198780 (1198861 ⟨1199091⟩ + 119899sum119894=2

119886119894 ⟨119909119894⟩) (87)


119894=1

119896119894119909119894) + 11988611198781 + 1198871119878 minus 120590222 ] d119905+ 1205902d1198612 (119905)

ge [minus(1198891 + 119872 119899sum119894=1

119896119894) + 11988611198781 + 1198871119878 minus 120590222 ] d119905+ 1205902d1198612 (119905)

(88)



119894=1

119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩ minus 1198861⟨ 120601 minus 119878(1 + 1198871120601) (1 + 1198871119878)⟩


119894=1

119896119894)+ 1198861⟨ 1206011 + 1198871120601⟩ minus 119886111988721 ⟨1119878 minus 1120601⟩ + 12059021198612 (119905)119905


119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩

minus 119886111988721 (119889 + 119873sum119899119894=1 119896119894) 1198780 (1198861 ⟨1199091⟩ + 119899sum119894=2

119886119894 ⟨119909119894⟩)+ 12059021198612 (119905)119905


119896119894)

+ 1198861⟨ 1206011 + 1198871120601⟩ minus 11988621 ⟨1199091⟩11988721 (119889 + 119873sum119899119894=1 119896119894) 1198780+ 12059021198612 (119905)119905

(89)



sup ln 1199091 (119905)119905 le 0 (90)

which yields


119894=1

119896119894)+ 1198861⟨ 1206011 + 1198871120601⟩ + 12059021198612 (119905)119905

(91)

that is



119894=1

119896119894) + 1198861 intinfin0

119909V (d119909)1 + 1198871119909 ] (92)

Therefore if

1198861 intinfin0

119909V (d119909)1 + 1198871119909 gt 120590222 + (119889 + 119872 119899sum119894=1

119896119894) (93)

then


inf ⟨1199091 (119905)⟩ gt 0 119886119904 (94)



119886119894 intinfin0

119909V (d119909)1 + 119887119894119909 lt 119889119894 + 119873 119899sum119894=1

119896119894 + 1205902119894+12 119894 = 1 2 119899 (95)

12 Complexity

then


int1199050119878 (119904) d119904119905 = 1198780 119886119904


supln 119909119894 (119905)119905 lt 0 119886119904 119894 = 1 2 119899

(96)




119896119894119909119894 + 1198861198941198781 + 119887119894119878 minus 1205902119894+12 )d119905+ 120590119894+1d119861119894+1 (119905)


119896119894 + 1198861198941206011 + 119887119894120601 minus 1205902119894+12 )d119905+ 120590119894+1d119861119894+1 (119905)

(97)


119894=1

119896119894 minus 1205902119894+12 ) + 119886119894⟨ 1206011 + 119887119894120601⟩+ 120590119894+1119861119894+1 (119905)119905 + ln 119909119894 (0)119905

(98)


119886119894 intinfin0

119909V (d119909)1 + 119887119894119909 lt 119889119894 + 119873 119899sum119894=1

119896119894 + 1205902119894+12 119894 = 1 2 119899 (99)

we know that


supln119909119894 (119905)119905 le 0 119894 = 1 2 119899 (100)



d119884 = minus(119889 + 119899sum119894=1

119896119894119909119894)119884d119905 + 1205901 (1198780 + 119884) d1198611 (119905) (101)



119894=1

119896119894119909119894) minus 120576119878120576] d119905+ 1205901119878120576d1198611 (119905)

(102)

Then

d (119878 minus 119878120576) = [(120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878) 119878

minus (119889 + 120576 + 119899sum119894=1

119896119894119909119894) (119878 minus 119878120576)] d119905 + 1205901 (119878minus 119878120576) d1198611 (119905)

(103)



119894=1

119896119894119909119894 + 120576 + 120590212 ) 119905]

sdot int1199050exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904

(104)


120576 gt 119899sum119894=1

119886119894119909119894 (119905)1 + 119887119894119878 (119905) forall119905 ge 119879 (105)


119894=1

119896119894119909119894 + 120576

+ 120590212 ) 119905]

sdot int1198790exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904

+ int119905119879exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

Complexity 13

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904 ge exp[12059011198611 (119905)

minus (119889 + 119872 119899sum119894=1

119896119894 + 120576 + 120590212 ) 119905]

sdot int1198790exp[(119889 + 119873 119899sum

119894=1

119896119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904(106)


exp[12059011198611 (119905) minus (119889 + 119872 119899sum119894=1

119896119894 + 120576 + 120590212 ) 119905] 997888rarr 01003816100381610038161003816100381610038161003816100381610038161003816int119879

0exp[(119889 + 119873 119899sum

119894=1

119896119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d1199041003816100381610038161003816100381610038161003816100381610038161003816 lt infin

(107)

which implies that



inf (119878 (119905) minus 119878120576 (119905))ge 0 119886119904 (108)

Now we consider

d (120601 (119905) minus 119878120576 (119905))= [120576119878120576 (119905) minus (119889 + 119899sum

119894=1

119896119894119909119894) (120601 (119905) minus 119878120576 (119905))] d119905+ 1205901 (120601 (119905) minus 119878120576 (119905)) d1198611 (119905)

le [120576119878120576 (119905) minus (119889 + 119873 119899sum119894=1

119896119894)(120601 (119905) minus 119878120576 (119905))] d119905+ 1205901 (120601 (119905) minus 119878120576 (119905)) d1198611 (119905)

(109)



119894=1


(110)


119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816le 119864[int119905

0120576119878120576 (119904) minus (119889 + 119873 119899sum

119894=1

119896119894) 1003816100381610038161003816120601 (119904) minus 119878120576 (119904)1003816100381610038161003816 d119904]

le 119864[int1199050120576120601 (119904) minus (119889 + 119873 119899sum

119894=1

119896119894) 1003816100381610038161003816120601 (119904) minus 119878120576 (119904)1003816100381610038161003816 d119904] (111)

then

119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816 le 120576 sup0le119904

119864120601 (119904) (112)

which leads to

lim120576997888rarr0


119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816 = 0 (113)



(120601 (119905) minus 119878 (119905)) = 0 (114)









14 Complexity

200 400 600 800 1000 1200 1400 1600 1800 20000Time

0

01

02

03

04

05

06

07

08

Valu

es


0 100806040200

02

04x2(t) path

(a)

0

01

02

03

04

05

06

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08

09

1

Valu

es

1 2 3 4 5 6 7 8 9 100Time


times104

(b)


S(t) path

01

02

03

04

05

06

07

S(t)

200 400 600 800 1000 1200 1400 1600 1800 20000Time

(a)

0

5

10

15

20

25

30


(b)



d1199091 (119905) = [minus (032 + 00071199091 (119905) + 0011199092 (119905)) 1199091 (119905)

+ 2119878 (119905)1 + 3119878 (119905)1199091 (119905)] d119905 + 011199091 (119905) d1198612 (119905) d1199092 (119905) = [minus (031 + 00071199091 (119905) + 0011199092 (119905)) 1199092 (119905)

+ 2119878 (119905)1 + 5119878 (119905)1199092 (119905)] d119905 + 021199092 (119905) d1198613 (119905) (116)


Complexity 15

01

02

03

04

05

06

07

08

09

1

S(t)

S(t) path

1 2 3 4 5 6 7 8 9 100Time times10

4

(a)


0

5

10

15

20

25

30

(b)


x1(t) path

005

01

015

02

025

03

035

04

x1(t)

200 400 600 800 1000 1200 1400 1600 1800 20000Time

(a)


0

5

10

15

20

25

30

35

40

45

50

(b)





16 Complexity

1 2 3 4 5 6 7 8 9 100Time

005

01

015

02

025

03

035

04

045

05

x1(t)

x1(t) path

times104

(a)


0

5

10

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50

(b)






d1199091 (119905) = [minus (09 + 021199091 (119905) + 021199092 (119905)) 1199091 (119905)+ 119878 (119905)1 + 04119878 (119905)1199091 (119905)] d119905 + 081199091 (119905) d1198612 (119905)

d1199092 (119905) = [minus (09 + 021199091 (119905) + 021199092 (119905)) 1199092 (119905)+ 119878 (119905)1 + 06119878 (119905)1199092 (119905)] d119905 + 081199092 (119905) d1198613 (119905)

(117)





Data Availability




Acknowledgments


Complexity 17

0 20018016014012010080604020Time

0

02

04

06

08

1

12

14

16

18

2Va

lues


0 109876543210

01

02


(a)

0 10000900080007000600050004000300020001000Time

0

05

1

15

2

25

Valu

es



0

02

04

0 87654321

(b)



References






















18 Complexity






























Journal of












Journal of







Volume 2018






Volume 2018








12 Complexity

then


int1199050119878 (119904) d119904119905 = 1198780 119886119904


supln 119909119894 (119905)119905 lt 0 119886119904 119894 = 1 2 119899

(96)




119896119894119909119894 + 1198861198941198781 + 119887119894119878 minus 1205902119894+12 )d119905+ 120590119894+1d119861119894+1 (119905)


119896119894 + 1198861198941206011 + 119887119894120601 minus 1205902119894+12 )d119905+ 120590119894+1d119861119894+1 (119905)

(97)


119894=1

119896119894 minus 1205902119894+12 ) + 119886119894⟨ 1206011 + 119887119894120601⟩+ 120590119894+1119861119894+1 (119905)119905 + ln 119909119894 (0)119905

(98)


119886119894 intinfin0

119909V (d119909)1 + 119887119894119909 lt 119889119894 + 119873 119899sum119894=1

119896119894 + 1205902119894+12 119894 = 1 2 119899 (99)

we know that


supln119909119894 (119905)119905 le 0 119894 = 1 2 119899 (100)



d119884 = minus(119889 + 119899sum119894=1

119896119894119909119894)119884d119905 + 1205901 (1198780 + 119884) d1198611 (119905) (101)



119894=1

119896119894119909119894) minus 120576119878120576] d119905+ 1205901119878120576d1198611 (119905)

(102)

Then

d (119878 minus 119878120576) = [(120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878) 119878

minus (119889 + 120576 + 119899sum119894=1

119896119894119909119894) (119878 minus 119878120576)] d119905 + 1205901 (119878minus 119878120576) d1198611 (119905)

(103)



119894=1

119896119894119909119894 + 120576 + 120590212 ) 119905]

sdot int1199050exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904

(104)


120576 gt 119899sum119894=1

119886119894119909119894 (119905)1 + 119887119894119878 (119905) forall119905 ge 119879 (105)


119894=1

119896119894119909119894 + 120576

+ 120590212 ) 119905]

sdot int1198790exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904

+ int119905119879exp[(119889 + 119899sum

119894=1

119896119894119909119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

Complexity 13

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904 ge exp[12059011198611 (119905)

minus (119889 + 119872 119899sum119894=1

119896119894 + 120576 + 120590212 ) 119905]

sdot int1198790exp[(119889 + 119873 119899sum

119894=1

119896119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904(106)


exp[12059011198611 (119905) minus (119889 + 119872 119899sum119894=1

119896119894 + 120576 + 120590212 ) 119905] 997888rarr 01003816100381610038161003816100381610038161003816100381610038161003816int119879

0exp[(119889 + 119873 119899sum

119894=1

119896119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d1199041003816100381610038161003816100381610038161003816100381610038161003816 lt infin

(107)

which implies that



inf (119878 (119905) minus 119878120576 (119905))ge 0 119886119904 (108)

Now we consider

d (120601 (119905) minus 119878120576 (119905))= [120576119878120576 (119905) minus (119889 + 119899sum

119894=1

119896119894119909119894) (120601 (119905) minus 119878120576 (119905))] d119905+ 1205901 (120601 (119905) minus 119878120576 (119905)) d1198611 (119905)

le [120576119878120576 (119905) minus (119889 + 119873 119899sum119894=1

119896119894)(120601 (119905) minus 119878120576 (119905))] d119905+ 1205901 (120601 (119905) minus 119878120576 (119905)) d1198611 (119905)

(109)



119894=1


(110)


119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816le 119864[int119905

0120576119878120576 (119904) minus (119889 + 119873 119899sum

119894=1

119896119894) 1003816100381610038161003816120601 (119904) minus 119878120576 (119904)1003816100381610038161003816 d119904]

le 119864[int1199050120576120601 (119904) minus (119889 + 119873 119899sum

119894=1

119896119894) 1003816100381610038161003816120601 (119904) minus 119878120576 (119904)1003816100381610038161003816 d119904] (111)

then

119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816 le 120576 sup0le119904

119864120601 (119904) (112)

which leads to

lim120576997888rarr0


119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816 = 0 (113)



(120601 (119905) minus 119878 (119905)) = 0 (114)









14 Complexity

200 400 600 800 1000 1200 1400 1600 1800 20000Time

0

01

02

03

04

05

06

07

08

Valu

es


0 100806040200

02

04x2(t) path

(a)

0

01

02

03

04

05

06

07

08

09

1

Valu

es

1 2 3 4 5 6 7 8 9 100Time


times104

(b)


S(t) path

01

02

03

04

05

06

07

S(t)

200 400 600 800 1000 1200 1400 1600 1800 20000Time

(a)

0

5

10

15

20

25

30


(b)



d1199091 (119905) = [minus (032 + 00071199091 (119905) + 0011199092 (119905)) 1199091 (119905)

+ 2119878 (119905)1 + 3119878 (119905)1199091 (119905)] d119905 + 011199091 (119905) d1198612 (119905) d1199092 (119905) = [minus (031 + 00071199091 (119905) + 0011199092 (119905)) 1199092 (119905)

+ 2119878 (119905)1 + 5119878 (119905)1199092 (119905)] d119905 + 021199092 (119905) d1198613 (119905) (116)


Complexity 15

01

02

03

04

05

06

07

08

09

1

S(t)

S(t) path

1 2 3 4 5 6 7 8 9 100Time times10

4

(a)


0

5

10

15

20

25

30

(b)


x1(t) path

005

01

015

02

025

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035

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x1(t)

200 400 600 800 1000 1200 1400 1600 1800 20000Time

(a)


0

5

10

15

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30

35

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45

50

(b)





16 Complexity

1 2 3 4 5 6 7 8 9 100Time

005

01

015

02

025

03

035

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045

05

x1(t)

x1(t) path

times104

(a)


0

5

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(b)






d1199091 (119905) = [minus (09 + 021199091 (119905) + 021199092 (119905)) 1199091 (119905)+ 119878 (119905)1 + 04119878 (119905)1199091 (119905)] d119905 + 081199091 (119905) d1198612 (119905)

d1199092 (119905) = [minus (09 + 021199091 (119905) + 021199092 (119905)) 1199092 (119905)+ 119878 (119905)1 + 06119878 (119905)1199092 (119905)] d119905 + 081199092 (119905) d1198613 (119905)

(117)





Data Availability




Acknowledgments


Complexity 17

0 20018016014012010080604020Time

0

02

04

06

08

1

12

14

16

18

2Va

lues


0 109876543210

01

02


(a)

0 10000900080007000600050004000300020001000Time

0

05

1

15

2

25

Valu

es



0

02

04

0 87654321

(b)



References






















18 Complexity






























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 13

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904 ge exp[12059011198611 (119905)

minus (119889 + 119872 119899sum119894=1

119896119894 + 120576 + 120590212 ) 119905]

sdot int1198790exp[(119889 + 119873 119899sum

119894=1

119896119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d119904(106)


exp[12059011198611 (119905) minus (119889 + 119872 119899sum119894=1

119896119894 + 120576 + 120590212 ) 119905] 997888rarr 01003816100381610038161003816100381610038161003816100381610038161003816int119879

0exp[(119889 + 119873 119899sum

119894=1

119896119894 + 120576 + 120590212 ) 119904 minus 12059011198611 (119904)]

sdot (120576 minus 119899sum119894=1

1198861198941199091198941 + 119887119894119878 (119904)) 119878 (119904) d1199041003816100381610038161003816100381610038161003816100381610038161003816 lt infin

(107)

which implies that



inf (119878 (119905) minus 119878120576 (119905))ge 0 119886119904 (108)

Now we consider

d (120601 (119905) minus 119878120576 (119905))= [120576119878120576 (119905) minus (119889 + 119899sum

119894=1

119896119894119909119894) (120601 (119905) minus 119878120576 (119905))] d119905+ 1205901 (120601 (119905) minus 119878120576 (119905)) d1198611 (119905)

le [120576119878120576 (119905) minus (119889 + 119873 119899sum119894=1

119896119894)(120601 (119905) minus 119878120576 (119905))] d119905+ 1205901 (120601 (119905) minus 119878120576 (119905)) d1198611 (119905)

(109)



119894=1


(110)


119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816le 119864[int119905

0120576119878120576 (119904) minus (119889 + 119873 119899sum

119894=1

119896119894) 1003816100381610038161003816120601 (119904) minus 119878120576 (119904)1003816100381610038161003816 d119904]

le 119864[int1199050120576120601 (119904) minus (119889 + 119873 119899sum

119894=1

119896119894) 1003816100381610038161003816120601 (119904) minus 119878120576 (119904)1003816100381610038161003816 d119904] (111)

then

119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816 le 120576 sup0le119904

119864120601 (119904) (112)

which leads to

lim120576997888rarr0


119864 1003816100381610038161003816120601 (119905) minus 119878120576 (119905)1003816100381610038161003816 = 0 (113)



(120601 (119905) minus 119878 (119905)) = 0 (114)









14 Complexity

200 400 600 800 1000 1200 1400 1600 1800 20000Time

0

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03

04

05

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Valu

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0 100806040200

02

04x2(t) path

(a)

0

01

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04

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es

1 2 3 4 5 6 7 8 9 100Time


times104

(b)


S(t) path

01

02

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S(t)

200 400 600 800 1000 1200 1400 1600 1800 20000Time

(a)

0

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(b)



d1199091 (119905) = [minus (032 + 00071199091 (119905) + 0011199092 (119905)) 1199091 (119905)

+ 2119878 (119905)1 + 3119878 (119905)1199091 (119905)] d119905 + 011199091 (119905) d1198612 (119905) d1199092 (119905) = [minus (031 + 00071199091 (119905) + 0011199092 (119905)) 1199092 (119905)

+ 2119878 (119905)1 + 5119878 (119905)1199092 (119905)] d119905 + 021199092 (119905) d1198613 (119905) (116)


Complexity 15

01

02

03

04

05

06

07

08

09

1

S(t)

S(t) path

1 2 3 4 5 6 7 8 9 100Time times10

4

(a)


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30

(b)


x1(t) path

005

01

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x1(t)

200 400 600 800 1000 1200 1400 1600 1800 20000Time

(a)


0

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(b)





16 Complexity

1 2 3 4 5 6 7 8 9 100Time

005

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025

03

035

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045

05

x1(t)

x1(t) path

times104

(a)


0

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(b)






d1199091 (119905) = [minus (09 + 021199091 (119905) + 021199092 (119905)) 1199091 (119905)+ 119878 (119905)1 + 04119878 (119905)1199091 (119905)] d119905 + 081199091 (119905) d1198612 (119905)

d1199092 (119905) = [minus (09 + 021199091 (119905) + 021199092 (119905)) 1199092 (119905)+ 119878 (119905)1 + 06119878 (119905)1199092 (119905)] d119905 + 081199092 (119905) d1198613 (119905)

(117)





Data Availability




Acknowledgments


Complexity 17

0 20018016014012010080604020Time

0

02

04

06

08

1

12

14

16

18

2Va

lues


0 109876543210

01

02


(a)

0 10000900080007000600050004000300020001000Time

0

05

1

15

2

25

Valu

es



0

02

04

0 87654321

(b)



References






















18 Complexity






























Journal of












Journal of







Volume 2018






Volume 2018








14 Complexity

200 400 600 800 1000 1200 1400 1600 1800 20000Time

0

01

02

03

04

05

06

07

08

Valu

es


0 100806040200

02

04x2(t) path

(a)

0

01

02

03

04

05

06

07

08

09

1

Valu

es

1 2 3 4 5 6 7 8 9 100Time


times104

(b)


S(t) path

01

02

03

04

05

06

07

S(t)

200 400 600 800 1000 1200 1400 1600 1800 20000Time

(a)

0

5

10

15

20

25

30


(b)



d1199091 (119905) = [minus (032 + 00071199091 (119905) + 0011199092 (119905)) 1199091 (119905)

+ 2119878 (119905)1 + 3119878 (119905)1199091 (119905)] d119905 + 011199091 (119905) d1198612 (119905) d1199092 (119905) = [minus (031 + 00071199091 (119905) + 0011199092 (119905)) 1199092 (119905)

+ 2119878 (119905)1 + 5119878 (119905)1199092 (119905)] d119905 + 021199092 (119905) d1198613 (119905) (116)


Complexity 15

01

02

03

04

05

06

07

08

09

1

S(t)

S(t) path

1 2 3 4 5 6 7 8 9 100Time times10

4

(a)


0

5

10

15

20

25

30

(b)


x1(t) path

005

01

015

02

025

03

035

04

x1(t)

200 400 600 800 1000 1200 1400 1600 1800 20000Time

(a)


0

5

10

15

20

25

30

35

40

45

50

(b)





16 Complexity

1 2 3 4 5 6 7 8 9 100Time

005

01

015

02

025

03

035

04

045

05

x1(t)

x1(t) path

times104

(a)


0

5

10

15

20

25

30

35

40

45

50

(b)






d1199091 (119905) = [minus (09 + 021199091 (119905) + 021199092 (119905)) 1199091 (119905)+ 119878 (119905)1 + 04119878 (119905)1199091 (119905)] d119905 + 081199091 (119905) d1198612 (119905)

d1199092 (119905) = [minus (09 + 021199091 (119905) + 021199092 (119905)) 1199092 (119905)+ 119878 (119905)1 + 06119878 (119905)1199092 (119905)] d119905 + 081199092 (119905) d1198613 (119905)

(117)





Data Availability




Acknowledgments


Complexity 17

0 20018016014012010080604020Time

0

02

04

06

08

1

12

14

16

18

2Va

lues


0 109876543210

01

02


(a)

0 10000900080007000600050004000300020001000Time

0

05

1

15

2

25

Valu

es



0

02

04

0 87654321

(b)



References






















18 Complexity






























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 15

01

02

03

04

05

06

07

08

09

1

S(t)

S(t) path

1 2 3 4 5 6 7 8 9 100Time times10

4

(a)


0

5

10

15

20

25

30

(b)


x1(t) path

005

01

015

02

025

03

035

04

x1(t)

200 400 600 800 1000 1200 1400 1600 1800 20000Time

(a)


0

5

10

15

20

25

30

35

40

45

50

(b)





16 Complexity

1 2 3 4 5 6 7 8 9 100Time

005

01

015

02

025

03

035

04

045

05

x1(t)

x1(t) path

times104

(a)


0

5

10

15

20

25

30

35

40

45

50

(b)






d1199091 (119905) = [minus (09 + 021199091 (119905) + 021199092 (119905)) 1199091 (119905)+ 119878 (119905)1 + 04119878 (119905)1199091 (119905)] d119905 + 081199091 (119905) d1198612 (119905)

d1199092 (119905) = [minus (09 + 021199091 (119905) + 021199092 (119905)) 1199092 (119905)+ 119878 (119905)1 + 06119878 (119905)1199092 (119905)] d119905 + 081199092 (119905) d1198613 (119905)

(117)





Data Availability




Acknowledgments


Complexity 17

0 20018016014012010080604020Time

0

02

04

06

08

1

12

14

16

18

2Va

lues


0 109876543210

01

02


(a)

0 10000900080007000600050004000300020001000Time

0

05

1

15

2

25

Valu

es



0

02

04

0 87654321

(b)



References






















18 Complexity






























Journal of












Journal of







Volume 2018






Volume 2018








16 Complexity

1 2 3 4 5 6 7 8 9 100Time

005

01

015

02

025

03

035

04

045

05

x1(t)

x1(t) path

times104

(a)


0

5

10

15

20

25

30

35

40

45

50

(b)






d1199091 (119905) = [minus (09 + 021199091 (119905) + 021199092 (119905)) 1199091 (119905)+ 119878 (119905)1 + 04119878 (119905)1199091 (119905)] d119905 + 081199091 (119905) d1198612 (119905)

d1199092 (119905) = [minus (09 + 021199091 (119905) + 021199092 (119905)) 1199092 (119905)+ 119878 (119905)1 + 06119878 (119905)1199092 (119905)] d119905 + 081199092 (119905) d1198613 (119905)

(117)





Data Availability




Acknowledgments


Complexity 17

0 20018016014012010080604020Time

0

02

04

06

08

1

12

14

16

18

2Va

lues


0 109876543210

01

02


(a)

0 10000900080007000600050004000300020001000Time

0

05

1

15

2

25

Valu

es



0

02

04

0 87654321

(b)



References






















18 Complexity






























Journal of












Journal of







Volume 2018






Volume 2018








Complexity 17

0 20018016014012010080604020Time

0

02

04

06

08

1

12

14

16

18

2Va

lues


0 109876543210

01

02


(a)

0 10000900080007000600050004000300020001000Time

0

05

1

15

2

25

Valu

es



0

02

04

0 87654321

(b)



References






















18 Complexity






























Journal of












Journal of







Volume 2018






Volume 2018








18 Complexity






























Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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