Hindawi Publishing CorporationThe Scientific World JournalVolume 2013 Article ID 645080 13 pageshttpdxdoiorg1011552013645080

Research ArticleStability Analysis of Distributed Order Fractional Chen System

H Aminikhah1 A Refahi Sheikhani2 and H Rezazadeh1

1 Department of Applied Mathematics School of Mathematical Sciences University of Guilan PO Box 1914 Rasht Iran2Department of Applied Mathematics Faculty of Mathematical Sciences Islamic Azad University Lahijan BranchPO Box 1616 Lahijan Iran

Correspondence should be addressed to H Aminikhah hosseinaminikhahgmailcom

Received 31 August 2013 Accepted 3 October 2013

Academic Editors C Li F Liu and R Magin

Copyright copy 2013 H Aminikhah et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We first investigate sufficient and necessary conditions of stability of nonlinear distributed order fractional system and then wegeneralize the integer-order Chen system into the distributed order fractional domain Based on the asymptotic stability theory ofnonlinear distributed order fractional systems the stability of distributed order fractional Chen system is discussed In additionwe have found that chaos exists in the double fractional order Chen system Numerical solutions are used to verify the analyticalresults

1 Introduction

Thehistory of fractional calculus is more than three centuriesold yet only in the past 20 years has the field receivedmuch attention and interest The reader may refer to [12] The generalization of dynamical equations using frac-tional derivatives proved to be useful and more accuratein mathematical modeling related to many interdisciplinaryareas Applications of fractional calculus and fractional-orderdifferential equations include dielectric relaxation phenom-ena in polymeric materials [3] transport of passive tracerscarried by fluid flow in a porous medium in groundwaterhydrology [4] transport dynamics in systems governed byanomalous diffusion [5 6] long-time memory in financialtime series [7] and so on [8 9] Stability analysis and controlsystems are two of the most important problems such that in1996 Matignon [10] firstly studied stability of 119899-dimensionallinear fractional systems from a point of view of controlSince then many researchers have completed further studieson the stability of linear fractional differential systems [11ndash14] For the nonlinear fractional differential systems thestability analysis is much more difficult and only a feware available Some authors [15 16] studied the nonlinearfractional differential system

Nonlinear systems are very interesting to engineersphysicists and mathematicians because most real physi-cal systems are inherently nonlinear in nature Nonlinear

dynamical systems can exhibit completely an unpredictablebehavior the so-called deterministic chaos Chaos is animportant dynamical phenomenon that has been extensivelystudied and developed by scientists and engineers in thelast decades in various fields such as physical [17] chemical[18] and ecological systems [19] Since chaos is useful andhas potential applications in many technological disciplinesthe discovery and the creation of chaos are important In1963 Lorenz found the first chaotic attractor in a three-dimensional autonomous system [20] Later several dynam-ical systems exhibiting chaos have been presented in variousbranches of science [21] For example in 1999 Chen andUetafound another simple three-dimensional autonomous sys-tem which is not topologically equivalent to Lorenzrsquos systemand which has a chaotic attractor [22 23] Similar to integer-order chaotic systems the fractional order chaotic systemshave also interested several researchers I Grigorenko and EGrigorenko extended the study of this prototypical system toequations of fractional order but unfortunately the resultspresented in this paper are not correct [24] In [25 26] thechaotic behavior and its control in the fractional order Chensystem are investigated Also chaotic behaviors have beenfound in the fractional order systems of Chuna [27] Rossler[28] Coullet [29]modifiedVan der Pol-Duffing [30] and Liu[31] In addition the fractional order Chen system has beenstudied with time-delay in [32]

2 The Scientific World Journal

The idea of fractional derivative of distributed order isstated by Caputo [33] and later developed by Caputo himself[34 35] and Bagley and Torvik [36 37] Other researchersused this idea and interesting reviews appeared to describethe related mathematical models of partial fractional differ-ential equation of distributed order For example Diethelmand Ford [38] used a numerical technique along with itserror analysis to solve the distributed order differential equa-tion and analyze the physical phenomena and engineeringproblems see [38] and references therein Recently SaberiNajafi et al [39 40] studied stability analysis of distributedorder fractional differential equations with respect to thenonnegative density function The aim of the present workis twofold First we consider the stability of 119899-dimensionalnonlinear distributed order fractional differential systemwith respect to the nonnegative density function and thenwe study the stability of distributed order fractional Chensystem

This paper is organized as follows In the next sectionwe recall some basic definitions of the Caputo fractionalderivative operator systems with fractional derivatives ofdistributed order Section 3 contains the main definitionsand theorems for checking the stability analysis of dis-tributed order fractional system In Section 4 we presentthe distributed order fractional Chen system and afterwardsbased on the stability theorem of distributed order fractionalsystems the stability of the distributed order fractionalChen system is discussed Finally the numerical solution toillustrate the validity of the results is presented in Section 5

2 Elementary Definitions

In this section we consider the main definitions and proper-ties of fractional derivative operators of single and distributedorder

The differential and integral operator in fractional calcu-lus is denoted by

119886119863120572

119905 where 119886 and 119905 are the bounds of the

operation and 120572 is the fractional order which can be rationalirrational or even complex For simplicity and without lossof generality in the following we assume that 119886 = 0 and119863120572

119905=0119863120572

119905 The continuous integrodifferential operator is

defined as follows

119863120572

119905=

119889120572

119889119905120572 120572 gt 0

1 120572 = 0

int

119905

0

(119889120591)minus120572

120572 lt 0

(1)

There are many definitions of fractional derivatives of order120572 gt 0 [1 2] such as Grunwald-Letnikovrsquos definition (GL)Riemann-Liouvillersquos definition (RL) and Caputorsquos fractionalderivative The RL definition is given as

RL119863120572

119905119891 (119905) =

1

Γ (119899 minus 120572)

119889119899

119889119905119899int

119905

0

(119905 minus 120591)119899minus120572minus1

119891 (120591) 119889120591 (2)

where 119899 is the first integer which is not less than 120572 that is119899 minus 1 lt 120572 lt 119899 and Γ(sdot) is a Gamma function The Caputofractional derivative of 119891(119905) is defined as

119862

119863

120572

119905119891 (119905) =

1

Γ (119899 minus 120572)

int

119905

0

(119905 minus 120591)119899minus120572minus1

119891(119899)

(120591) 119889120591 (3)

Finally Grunwald-Letnikov definition is given by

GL119863

120572

119905119891 (119905) = lim

ℎrarr0

ℎminus120572

[(119905minus119886)ℎ]

sum

119894=0

(minus1)119894

(

120572

119894)119891 (119905 minus 119895ℎ) (4)

Fortunately the Laplace transform of the Caputo fractionalderivative satisfies

L 119888

119863

120572

119905119891 (119905) = 119904

120572

L 119891 (119905) minus

119898minus1

sum

119896=0

119891(119896)

(0+

) 119904120572minus1minus119896

(5)

where 119898 minus 1 lt 120572 le 119898 and 119904 is the Laplace variable TheLaplace transform of Caputo fractional derivative requiresthe knowledge of the initial values of the function and itsinteger derivatives of order 119896 = 1 2 119898 minus 1 When 120572 isin

(0 1] is given by

L 119888

119863

120572

119905119891 (119905) = 119904

120572

L 119891 (119905) minus 119891 (0+

) 119904120572minus1

(6)

When the fractional calculus operators act on 119891(119905) and weintegrate

119886119863120572

119905119891(119905) with respect to the order then distributed

order fractional differential equations can be obtained In thisbrief the following distributed order fractional differentialoperator notation is adopted

119888

119863

119887(120572)

119905119891 (119905) = int

119898

119898minus1

119887 (120572)119888

119863

120572

119905119891 (119905) 119889120572

119898 minus 1 le 120572 le 119898 119898 isin N

(7)

where the derivative 119888119863120572119905is taken to be a fractional derivative

of Caputo type of order 120572 with respect to the nonnegativedensity function 119887(120572)

The idea of distributed order is stated by Caputo [3334] Further the Laplace transform of the Caputo distributedorder satisfies

L 119888

119863

119887(120572)

119905119891 (119905) = int

119898

119898minus1

119887 (120572)

times [119904120572

119865 (119904) minus

119898minus1

sum

119896=0

119891(119896)

(0+

) 119904120572minus1minus119896

]119889120572

= 119861 (119904) 119865 (119904) minus

119898minus1

sum

119896=0

1

119904119896+1

119861 (119904) 119891(119896)

(0+

)

(8)where 119865(119904) is the Laplace transform of 119891(119905) and

119861 (119904) = int

119898

119898minus1

119887 (120572) 119904120572

119889120572 (9)

A distributed order fractional equations can be defined by thefollowing model

119888

119863

119887(120572)

119905119909 (119905) = 119891 (119905 119909 (119905))

119909 (0) = 1199090

(10)

where 119909(119905) isin R and 119888119863119887(120572)119905

119909(119905) = int

1

0

119887(120572)119888

119863

120572

119905119909(119905)119889120572

The Scientific World Journal 3

In [38] the following results about the existence anduniqueness of solutions for (10) are further presented

Theorem 1 Let the function 119887 be absolutely integrable on theinterval [0 1] and satisfy int1

0

119887(120572)119904120572

119889120572 = 0 for Re(119904) gt 0 and119891 isin L1[0infin) and 119909 is such that 119888119863120572

119905119909(119905) lt M for 119905 isin [0infin]

for every120572 isin [0 1] then initial value problem (10) has a uniquesolution

Furthermore the above definition in one dimension cannaturally be generalized to the case of multiple dimensionsthat is let 119883(119905) = (119909

1(119905) 1199092(119905) 119909

119899(119905))119879

isin R119899 and 119887(120572) =

(1198871(120572) 1198872(120572) 119887

119899(120572))119879 0 lt 120572 lt 1 The 119899-dimension

distributed order fractional system is described as follows

119888

119863

119887(120572)

119905119883(119905) = int

1

0

119887 (120572)119888

119863

120572

119905119883(119905) 119889120572 = 119865 (119905 119883 (119905)) (11)

where119888

119863

119887(120572)

119905119883 (119905)

= (119888

119863

1198871(120572)

1199051199091(119905) 119888

119863

1198872(120572)

1199051199092(119905)

119888

119863

119887119899(120572)

119905119909119899(119905))

119865 (119905 119883 (119905)) = (

1198911(119905 1199091(119905) 1199092(119905) 119909

119899(119905))

1198912(119905 1199091(119905) 1199092(119905) 119909

119899(119905))

119891119899(119905 1199091(119905) 1199092(119905) 119909

119899(119905))

)

(12)

The result ofTheorem 1 can be easily generalized to the initialvalue problem of (11)

3 Stability Analysis of Distributed OrderFractional Systems

In this section we generalize the main stability properties forsystems of distributed order fractionalThe linear distributedorder fractional systems are expressed as

119888

119863

119887(120572)

119905119883(119905) = 119860119883 (119905)

119883 (0) = 1198830

(13)

where 119883(119905) = (1199091(119905) 1199092(119905) 119909

119899(119905))119879

isin R119899 the matrix 119860 isin

R119899times119899 and 119887(120572) = (1198871(120572) 1198872(120572) 119887

119899(120572))119879 0 lt 120572 lt 1 Then

Saberi Najafi et al [39] have obtained the general solution ofthe distributed order fractional systems (13) which is writtenby

119883(119905)

= 119883 (0) +

1

120587

int

119905

0

int

infin

0

int

infin

0

119890minus119903119905+119860120591minus120588 cos(120587120574)

times sin (120588 sin (120587120574)) sin119860119883 (0) 119889119903119889120591119889119905

(14)

where 119861(119904) = 120588 cos(120587120574) + 119894120588 sin(120587120574) 120588 = |119861(119904)| 120574 = (1

120587) arg[119861(119904)] and 119903 = 119890119894120587

Now we will recall some theorems and definitions aboutlinear distributed order fractional equations and then we willshow this theorem for nonlinear distributed order fractionalequations as well

Theorem 2 (see [39]) The distributed order fractional systemof (13) is asymptotically stable if and only if all roots of119889119890119905(119861(119904)119868 minus 119860) = 0 have negative real parts

Definition 3 (see [39]) The value of det(119861(119904)119868 minus 119860) = 0 isthe characteristic function of the matrix 119860 with respect tothe distributed function 119861(119904) where 119861(119904) = int

1

0

119887(120572)119904120572

119889120572 isthe distributed function with respect to the density function119887(120572) ge 0

The inertia of a matrix is the triplet of the numbers ofeigenvalues of 119860 with positive negative and zero real partsAs pointed out in [39] authors have generalized the inertiaconcept for analyzing the stability of linear distributed orderfractional systems

Definition 4 The inertia of the system (13) is the triple

119868119899119861(119904)

(119860) = (120587119861(119904)

(119860) 120592119861(119904)

(119860) 120575119861(119904)

(119860)) (15)

where 120587119861(119904)

(119860) 120592119861(119904)

(119860) and 120575119861(119904)

(119860) are respectively thenumber of roots of det(119861(119904)119868minus119860) = 0with positive negativeand zero real parts where 119861(119904) = (119861

1(119904) 1198612(119904) 119861

119899(119904))119879 is

the distributed function with respect to the density function119887(120572) = (119887

1(120572) 1198872(120572) 119887

119899(120572))119879

Theorem 5 (see [39]) The linear distributed order fractionalsystem (13) is asymptotically stable if and only if any of thefollowing equivalent conditions holds

(1) 120587119899119861(119904)

(119860) = 120575119899119861(119904)

(119860) = 0(2) all roots 119904 of the characteristic function of119860with respect

to 119861(119904) = (1198611(119904) 1198612(119904) 119861

119899(119904))119879 satisfy |119886119903119892(119904)| gt

1205872

Next we will mainly discuss the stability of a nonlinearautonomous distributed order fractional system which canbe described by

119888

119863

119887(120572)

119905119883 (119905) = 119865 (119883 (119905)) (16)

with the initial value119883(0) = 1198830 where

119865 (119883 (119905)) = (

1198911(1199091(119905) 1199092(119905) 119909

119899(119905))

1198912(1199091(119905) 1199092(119905) 119909

119899(119905))

119891119899(1199091(119905) 1199092(119905) 119909

119899(119905))

) (17)

119883(119905) = (1199091(119905) 1199092(119905) 119909

119899(119905))119879

isin R119899 and 119887(120572) = (1198871(120572)

1198872(120572) 119887

119899(120572))119879 0 lt 120572 lt 1

Theorem 6 Let 119883 = (1199091 1199092 119909

119899)119879 be the equilibrium of

system (16) that is 119888119863119887(120572)119905

119883 = 119865(119883) = 0 and J = (120597119865

120597119883)|119883=

is the Jacobian matrix at the point 119883 then the point119883 is asymptotically stable if and only if all roots 119904 of the

4 The Scientific World Journal

minus50

0

50

minus40minus2002040

0

10

20

30

40

50

x(t)

z(t)

y(t)

(a)

minus30 minus20 minus10 0 10 20 30minus30

minus20

minus10

0

10

20

30

40

x(t)y(t)

(b)

minus30 minus20 minus10 0 10 20 30

0

5

10

15

20

25

30

35

40

45

50

x(t)

z(t)

(c)

minus30 minus20 minus10 0 10 20 30 40

0

5

10

15

20

25

30

35

40

45

50

z(t)

y(t)

(d)Figure 1 Chaotic attractor of the integer-order Chen system (21) with (119886 119887 119888) = (35 3 28)

characteristic function of J with respect to 119861(119904) = (1198611(119904)

1198612(119904) 119861

119899(119904))119879 satisfy |119886119903119892(119904)| gt 1205872

Proof Let 120577(119905) = 119883(119905)minus119883 where 120577(119905) = (1205771(119905) 1205772(119905) 120577

119899(119905))

is a small disturbance from a fixed point Therefore

119888

119863

119887(120572)

119905120577 (119905) =

119888

119863

119887(120572)

119905(119883 (119905) minus 119883) (18)

since 119888119863119887(120572)119905

(119883(119905)minus119883) =119888

119863

119887(120572)

119905119883(119905)minus

119888

119863

119887(120572)

119905119883 and 119888119863119887(120572)

119905119883 =

0 thus we have

119888

119863

119887(120572)

119905120577 (119905) =

119888

119863119887(120572)

119905119883 (119905) = 119865 (119883 (119905)) = 119865 (120577 (119905) + 119883)

= 119865 (119883) + J120577 (119905) + higher order terms asymp J120577 (119905) (19)

The Scientific World Journal 5

Table 1 Stability analysis of system (22) for various density functions

119887119894(120572) = 120575(120572 minus 119902

119894) 119894 = 1 2 3 119887

119894(120572) = 120575(120572 minus 119902

1119894) + 120575(120572 minus 119902

2119894) 119894 = 1 2 3

(119886 119887 119888) 119902 = (1199021 1199022 1199023) 119868

119899119861(119904)(J) (119886 119887 119888)

(11990211 11990212)

(11990221 11990222)

(11990231 11990232)

119868119899119861(119904)

(J)

(35 6 11) (095 09 09)

1199091997888rarr (0 2 0)

1199092= 1199093997888rarr (1 2 0)

(35 3 29)

(085 03)

(065 03)

(095 03)

1199091997888rarr (1 0 0)

1199092= 1199093997888rarr (2 0 0)

(30 3 28) (07 08 09)

1199091997888rarr (1 0 0)

1199092997888rarr (0 2 0)

(35 3 28)

(095 01)

(095 01)

(095 01)

1199091997888rarr (1 0 0)

1199092= 1199093997888rarr (20 18 0)

(35 3 28) (085 09 095)

1199091997888rarr (1 0 0)

1199092= 1199093997888rarr (2 0 0)

(50 3 20)

(025 085)

(025 095)

(025 075)

1199091997888rarr (0 4 0)

11990921199093997888rarr (1 0 0)

System (18) can be written as119888

119863

119887(120572)

119905120577 (119905) asymp J120577 (119905) (20)

with the initial value 120577(0) = 1198830minus 119883

The analytical procedure of linearization is based on thefact that if the matrix J has no purely imaginary eigenvaluesthen the trajectories of the nonlinear system in the neigh-borhood of the equilibrium point have the same form asthe trajectories of the linear system [41] Hence by applyingTheorem 2 the linear system (20) is asymptotically stable ifand only if all roots 119904 of the characteristic function of J withrespect to 119861(119904) = (119861

1(119904) 1198612(119904) 119861

119899(119904))119879 satisfy | arg(119904)| gt

1205872 which implies that the equilibrium 119883 of the nonlineardistributed order fractional system (16) is as asymptoticallystable

Remark 7 The nonlinear distributed order fractional system(16) in the point 119883 is asymptotically stable if and only if120587119899119861(119904)

(119869) = 120575119899119861(119904)

(119869) = 0

4 Distributed Order Fractional Chen System

The Chen system is described by the following nonlineardifferential equations on R3 [22 23]

(119905) = 119886 (119910 (119905) minus 119909 (119905))

119910 (119905) = (119888 minus 119886) 119909 (119905) minus 119909 (119905) 119911 (119905) + 119888119910 (119905)

(119905) = 119909 (119905) 119910 (119905) minus 119887119911 (119905)

(21)

where 119909 119910 and 119911 are the state variables and 119886 119887 and 119888

are three system parameters The above system has a chaoticattractor when 119886 = 35 119887 = 3 and 119888 = 28 as shown in Figure 1The corresponding distributed order fractional Chen system(21) can be written in the form119888

1198631198871(120572)

119905119909 (119905) = 119886 (119910 (119905) minus 119909 (119905))

119888

1198631198872(120572)

119905119910 (119905) = (119888 minus 119886) 119909 (119905) minus 119909 (119905) 119911 (119905) + 119888119910 (119905)

119888

1198631198873(120572)

119905119911 (119905) = 119909 (119905) 119910 (119905) minus 119887119911 (119905)

(22)

where 119887119894(120572) for 119894 = 1 2 3 denote the nonnegative density

function of order 120572 isin (0 1] As a generalization of nonlinearfractional order differential equation into nonlinear dis-tributed order fractional differential equation the linearizedform of the system (22) at the equilibrium points 119909

1=

(0 0 0) 1199092

= (radic119887(2119888 minus 119886) radic119887(2119888 minus 119886) 2119888 minus 119886) and 1199093

=

(minusradic119887(2119888 minus 119886) minusradic119887(2119888 minus 119886) 2119888 minus 119886) that is 1198881198631198871(120572)119905

119909(119905) =

119865(119909) = 0 can be written in the form

119888

119863

119887(120572)

119905119883 (119905) = J119883 (119905) (23)

where 119883(119905) = (119909(119905) 119910(119905) 119911(119905))119879 119887(120572) = (119887

1(120572) 1198872(120572) 1198873(120572))119879

0 lt 120572 le 1 and J = (120597119865120597119883)|119883=119909119894

for 119894 = 1 2 3 The Jacobianmatrix of distributed order fractional Chen system (22) at theequilibrium point119883lowast = (119909

lowast

119910lowast

119911lowast

) is given by

J = [

[

minus119886 119886 0

119888 minus 119886 minus 119911lowast

119888 minus119909lowast

119910lowast

119909lowast

minus119887

]

]

(24)

Remark 8 If 119887119894(120572) = 120575(120572 minus 119902

119894) where 0 lt 119902

119894le 1 for 119894 = 1 2 3

and 120575(120572) is the Dirac delta function then we have the follow-ing fractional incommensurate-order Chen system [25]

119888

1198631205721

119905119909 (119905) = 119886 (119910 (119905) minus 119909 (119905))

119888

1198631205722

119905119910 (119905) = (119888 minus 119886) 119909 (119905) minus 119909 (119905) 119911 (119905) + 119888119910 (119905)

119888

1198631205723

119905119911 (119905) = 119909 (119905) 119910 (119905) minus 119887119911 (119905)

(25)

Based upon Theorem 6 the stability of the distributedorder fractional Chen system can be reached with ease Foranalyzing the stability of the distributed order fractionalChen system we compute 119868

119899119861(119904)

(J) in the case that the density

6 The Scientific World Journal

02

015

01

005

0

02

02

0

0

minus02

minus02minus04

minus04minus06x(t)

z(t)

y(t)

(a)

0 500 1000 1500 2000

04

03

02

01

0

minus01

minus02

minus03

minus04

tx(t)

(b)

0 500 1000 1500 2000

t

06

04

02

0

minus02

minus04

minus06

y(t)

(c)Figure 2 The equilibrium point 119909

1of the distributed order fractional Chen system (22) with (119902

1 1199022 1199023) = (095 09 09) and (119886 119887 119888) =

(35 6 11) is asymptotically stable

function varies The results are shown in Table 1 for someparameters 119886 119887 and 119888

5 Numerical Methods

As pointed out in [38] distributed order fractional differen-tial equations may be regarded as a generalization of single-term fractional differential equations

119888

119863

120572

119905119909 (119905) = 119891 (119905 119909 (119905)) (26)

or multiterm fractional differential equations119899

sum

119894=1

120574119894

119888

119863

120572119894

119905119909 (119905) = 119891 (119905 119909 (119905)) 0 lt 120572

1lt 1205722lt sdot sdot sdot lt 120572

119899 (27)

Therefore in this section numerical method to solving of(26) and (27) is presented The approximate Grunwald-Letnikovrsquos definition is given below where the step size of ℎis assumed to be very small [2 42 43]

119888

119863

120572

119905119896

119910 (119905) asymp ℎminus120572

119896

sum

119894=0

119888(120572)

119894119910 (119905119896minus119894

) (28)

The Scientific World Journal 7

minus40

minus20

0

20

minus40

minus20

0

20

0

10

20

30

40

50

x(t)

z(t)

y(t)

(a)

0 500 1000 1500 2000

t

minus40

minus30

minus20

minus10

0

10

20

x(t)

(b)

minus40

minus30

minus20

minus10

0

10

20

0 500 1000 1500 2000

t

y(t)

(c)

0

5

10

15

20

25

30

35

40

45

50

0 500 1000 1500 2000

t

z(t)

(d)

Figure 3 The equilibrium point 1199092of the distributed order fractional Chen system (22) with (119902

1 1199022 1199023) = (07 08 09) and (119886 119887 119888) =

(30 3 28) is asymptotically stable

where 119905119896

= 119896ℎ (119896 = 0 1 ) and 119888(120572)

119894(119894 = 0 1 ) are

binomial coefficients which can be computed as [43]

119888(120572)

0= 1 119888

(120572)

119894= (1 minus

1 + 120572

119894

) 119888(120572)

119894minus1 (29)

Then general numerical solution of (26) and (27) can beexpressed as

ℎminus120572

119896

sum

119894=0

119888(120572)

119894119909 (119905119896minus119894

) = 119891 (119905 119909 (119905))

1205741ℎminus1205721

119896

sum

119894=0

119888(1205721)

119894119909 (119905119896minus119894

) + sdot sdot sdot + 120574119899ℎminus120572119899

119896

sum

119894=0

119888(120572119899)

119894119909 (119905119896minus119894

)

= 119891 (119905 119909 (119905))

(30)

where 119888(120572119895)119894

for 119895 = 1 119899 are binomial coefficients calculatedaccording to (29) Equations in (30) can be rewritten as thefollowing forms

8 The Scientific World Journal

z(t)

y(t)

x(t)

minus30 minus20 minus10 0 10 20

minus50

0

50

0

10

20

30

40

50

(a)

minus30

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

0 500 1000 1500 2000

tx(t)

(b)

0 500 1000 1500 2000

t

minus40

minus30

minus20

minus10

0

10

20

y(t)

(c)

0

5

10

15

20

25

30

35

40

45

0 500 1000 1500 2000

t

z(t)

(d)Figure 4 Chaotic attractor of the distributed order fractional Chen system (22) with (119902

1 1199022 1199023) = (085 09 095) and (119886 119887 119888) = (35 3 28)

119909 (119905119896) = 119891 (119909 (119905

119896minus1) 119905119896minus1

) ℎ120572

minus

119896

sum

119894=1

119888(120572)

119894119909 (119905119896minus119894

)

119909 (119905119896)

=

1

sum119899

119895=1119886119895ℎminus120572119895

times[

[

119891 (119909 (119905119896minus1

) 119905119896minus1

) minus

119899

sum

119895=1

120574119895ℎminus120572119895

119896

sum

119894=1

119888

(120572119895)

119894119909 (119905119896minus119894

)]

]

(31)

Based on the previous algorithmwe can obtain the numericalsolution of the fractional differential equations (26) andmultiterm fractional differential equations (27) where 119896 =

1 2 119873 for 119873 = 119879ℎ and where 119879 is the total time of thecalculation

To verify the efficiency of the obtained results in Table 1the numerical solution for the distributed order fractionalChen system has been computed In the following calcula-tions let 119879 = 10 ℎ = 0005 with the initial conditions(02 minus05 02)

The Scientific World Journal 9

x(t)

z(t)

y(t)

minus50

0

50

minus100minus50

050

0

10

20

30

40

50

60

(a)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

0 500 1000 1500 2000

tx(t)

(b)

minus80

minus60

minus40

minus20

0

20

40

0 500 1000 1500 2000

t

y(t)

(c)

0

10

20

30

40

50

60

z(t)

0 500 1000 1500 2000

t

(d)

Figure 5 Chaotic attractor of the distributed order fractional Chen system (22) with (11990211 11990212 11990221 11990222 11990231 11990232) = (085 03 085 03 085 03)

and (119886 119887 119888) = (35 3 29)

Figure 2 shows that the system (22) with parameters(119886 119887 119888) = (35 6 11) and (119902

1 1199022 1199023) = (095 09 09) is

asymptotically stable in the equilibrium 1199091 Figure 3 dem-

onstrates that the system (22) with parameters (119886 119887 119888) =

(30 3 28) and (1199021 1199022 1199023) = (07 08 09) is asymptotically

stable in the equilibrium 1199092 From Figure 4 we can see that

for the distributed order Chen system (21) with parameters(119886 119887 119888) = (35 3 28) and (119902

1 1199022 1199023) = (085 09 095)

is chaotic Figure 5 demonstrates that the system (22)with parameters (119886 119887 119888) = (35 3 29) and (119902

11 11990212 11990221

11990222 11990231 11990232) = (085 03 085 03 085 03) has chaotic

attractor Figure 6 shows the chaotic attractor for the

10 The Scientific World Journal

minus50

0

50

minus40minus2002040

0

10

20

30

40

50

x(t)

z(t)

y(t)

(a)

minus30

minus20

minus10

0

10

20

30

0 500 1000 1500 2000

tx(t)

(b)

minus40

minus30

minus20

minus10

10

20

30

0

0 500 1000 1500 2000

t

y(t)

(c)

0 500 1000 1500 2000

t

z(t)

0

5

10

15

20

25

30

35

40

45

(d)

Figure 6 Chaotic attractor of the distributed order fractional Chen system (22) with (11990211 11990212 11990221 11990222 11990231 11990232) = (095 01 095 01 095 01)

and (119886 119887 119888) = (35 3 28)

distributed order Chen system (22) for parameters(119886 119887 119888) = (35 3 28) and (119902

11 11990212 11990221 11990222 11990231 11990232) =

(095 01 095 01 095 01) From Figure 7 we cansee that the system (22) is asymptotically stable in theequilibrium 119909

1with parameters (119886 119887 119888) = (50 3 20)

and (11990211 11990212 11990221 11990222 11990231 11990232) = (025 085 025 095

025 075)

6 Conclusion

In this paper we introduced the nonlinear distributed orderfractional differential equations with respect to a nonnegativedensity function hence the asymptotical stability for suchsystems has been investigated In addition we presented thedistributed order fractional Chen system and then in two

The Scientific World Journal 11

02

015

01

005

0

0

minus02

minus02

minus04

minus04

minus06

minus08

0

02

x(t)

z(t)

y(t)

(a)

minus03

minus04

02

03

01

minus02

minus01

0

0 500 1000 1500 2000

t

x(t)

(b)

02

01

0 500 1000 1500 2000

t

minus025

minus015

minus005

0

005

015

025

minus01

minus02

y(t)

(c)

minus008

minus006

minus004

minus002

0

002

004

006

008

01

minus01

0 500 1000 1500 2000

t

z(t)

(d)

Figure 7 The equilibrium point 1199091of the distributed order fractional Chen system (22) with (119902

11 11990212 11990221 11990222 11990231 11990232) = (025 085 025

095 025 075) and (119886 119887 119888) = (50 3 20) is asymptotically stable

special cases the stability for the distributed order fractionalChen is discussed Numerical solutions were coincident withresults of Table 1 described in Section 4 All numerical resultsare obtained using MATLAB 78

References

[1] A A Kilbas H M Srivastava and J J Trujillo Theory andApplication of Fractional Differential Equations Elsevier NewYork NY USA 2006

[2] K B Oldham and J SpanierThe Fractional Calculus AcadmicPress New York NY USA 1974

[3] E Reyes-Melo J Martinez-Vega C Guerrero-Salazar andU Ortiz-Mendez ldquoApplication of fractional calculus to themodeling of dielectric relaxation phenomena in polymericmaterialsrdquo Journal of Applied Polymer Science vol 98 no 2 pp923ndash935 2005

[4] R Schumer D A Benson M M Meerschaert and SW Wheatcraft ldquoEulerian derivation of the fractional

12 The Scientific World Journal

advection-dispersion equationrdquo Journal of ContaminantHydrology vol 48 no 1-2 pp 69ndash88 2001

[5] B I Henry and S L Wearne ldquoExistence of turing instabilitiesin a two-species fractional reaction-diffusion systemrdquo SIAMJournal on Applied Mathematics vol 62 no 3 pp 870ndash8872002

[6] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportvol 339 no 1 pp 1ndash77 2000

[7] S Picozzi andB JWest ldquoFractional Langevinmodel ofmemoryin financialmarketsrdquo Physical Review E vol 66 no 4 Article ID046118 pp 0461181ndash04611812 2002

[8] A Ansari A Refahi Sheikhani and S Kordrostami ldquoOn thegenerating function 119890119909119905+119910120593(119905) and its fractional calculusrdquo CentralEuropean Journal of Physics 2013

[9] A Ansari A Refahi Sheikhani and H Saberi Najafi ldquoSolutionto system of partial fractional differential equations using thefractional exponential operatorsrdquoMathematical Methods in theApplied Sciences vol 35 no 1 pp 119ndash123 2012

[10] D Matignon ldquoStability results for fractional differential equa-tions with applications to control processingrdquo in Proceedingsof the Computational Engineering in Systems and ApplicationMulticonference (IMACS IEEE-SMC rsquo96) vol 2 pp 963ndash968Lille France 1996

[11] M S Tavazoei and M Haeri ldquoA note on the stability offractional order systemsrdquoMathematics and Computers in Simu-lation vol 79 no 5 pp 1566ndash1576 2009

[12] V Daftardar-Gejji and A Babakhani ldquoAnalysis of a systemof fractional differential equationsrdquo Journal of MathematicalAnalysis and Applications vol 293 no 2 pp 511ndash522 2004

[13] K Diethelm and N J Ford ldquoAnalysis of fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 265 no 2 pp 229ndash248 2002

[14] J Sabatier M Moze and C Farges ldquoLMI stability conditionsfor fractional order systemsrdquo Computers and Mathematics withApplications vol 59 no 5 pp 1594ndash1609 2010

[15] Y Li Y Chen and I Podlubny ldquoMittag-Leffler stability offractional order nonlinear dynamic systemsrdquo Automatica vol45 no 8 pp 1965ndash1969 2009

[16] Y Li Y Chen and I Podlubny ldquoStability of fractional-ordernonlinear dynamic systems lyapunov directmethod and gener-alizedMittag-Leffler stabilityrdquoComputers andMathematics withApplications vol 59 no 5 pp 1810ndash1821 2010

[17] M Lakshmanan and K Murali Chaos in Nonlinear OscillatorsControlling and Synchronization World Scientific Singapore1996

[18] S KHan C Kurrer andY Kuramoto ldquoDephasing and burstingin coupled neural oscillatorsrdquo Physical Review Letters vol 75no 17 pp 3190ndash3193 1995

[19] B Blasius A Huppert and L Stone ldquoComplex dynamics andphase synchronization in spatially extended ecological systemsrdquoNature vol 399 no 6734 pp 354ndash359 1999

[20] C Sparrow The Lorenz Equations Bifurcations Chaos andStrange Attractors Springer New York NY USA 1982

[21] K T Alligood T D Sauer and J A Yorke Chaos An Intro-duction to Dynamical Systems Springer New York NY USA2008

[22] G Chen and T Ueta ldquoYet another chaotic attractorrdquo Interna-tional Journal of Bifurcation and Chaos vol 9 no 7 pp 1465ndash1466 1999

[23] T Ueta and G Chen ldquoBifurcation analysis of Chenrsquos equationrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 10 no 8 pp 1917ndash1931 2000

[24] I Grigorenko and E Grigorenko ldquoChaotic dynamics of thefractional Lorenz systemrdquo Physical Review Letters vol 91 no3 pp 0341011ndash0341014 2003

[25] C P Li andG J Peng ldquoChaos in Chenrsquos systemwith a fractionalorderrdquo Chaos Solitons and Fractals vol 22 no 2 pp 443ndash4502004

[26] C G Li and G Chen ldquoChaos in the fractional order Chensystem and its controlrdquo Chaos Solitons and Fractals vol 22 no3 pp 549ndash554 2004

[27] T T Hartley C F Lorenzo and H K Qammer ldquoChaos in afractional order Chuarsquos systemrdquo IEEE Transactions on Circuitsand Systems I vol 42 no 8 pp 485ndash490 1995

[28] C G Li and G Chen ldquoChaos and hyperchaos in the fractional-order Rossler equationsrdquo Physica A vol 341 no 1ndash4 pp 55ndash612004

[29] M Shahiri R Ghaderi A Ranjbar N S H Hosseinnia andS Momani ldquoChaotic fractional-order Coullet system synchro-nization and control approachrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 3 pp 665ndash6742010

[30] A E Matouk ldquoChaos feedback control and synchronization ofa fractional-order modified Autonomous Van der Pol-Duffingcircuitrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 2 pp 975ndash986 2011

[31] X YWang andM JWang ldquoDynamic analysis of the fractional-order Liu system and its synchronizationrdquo Chaos vol 17 no 3Article ID 033106 2007

[32] V Daftardar-Gejji S Bhalekar and P Gade ldquoDynamics offractional-ordered Chen system with delayrdquo Journal of Physicsvol 79 pp 61ndash69 2012

[33] M Caputo Elasticita e Dissipazione Zanichelli Bologna Italy1969

[34] M Caputo ldquoMean fractional-order-derivatives differentialequations and filtersrdquo Annali dellrsquoUniversita di Ferrara vol 41no 1 pp 73ndash84 1995

[35] M Caputo ldquoDistributed order differential equations modellingdielectric induction and diffusionrdquo Fractional Calculus andApplied Analysis vol 4 pp 421ndash442 2001

[36] L Bagley and P J Torvik ldquoOn the existence of the order domainand the solution of distributed order equationsrdquo InternationalJournal of Applied Mathematics vol 1 pp 865ndash882 2000

[37] R L Bagley and P J Torvik ldquoOn the existence of the orderdomain and the solution of distributed order equationsrdquo Inter-national Journal of Applied Mathematics vol 2 pp 965ndash9872000

[38] K Diethelm andN J Ford ldquoNumerical analysis for distributed-order differential equationsrdquo Journal of Computational andApplied Mathematics vol 225 no 1 pp 96ndash104 2009

[39] H Saberi Najafi A Refahi Sheikhani and A Ansari ldquoStabilityanalysis of distributed order fractional differential equationsrdquoAbstract and Applied Analysis vol 2011 Article ID 175323 12pages 2011

[40] A Refahi Sheikhani H Saberi Nadjafi A Ansari and FMehrdoust ldquoAnalytic study on linear system of distributed-order fractional differntional equationsrdquo Le Matematiche vol67 p 313 2012

The Scientific World Journal 13

[41] Z Vukic and L Kuljaca Nonlinear Control Systems CRC PressNew York NY USA 2003

[42] L Dorciak Numerical Models for Simulation the Fractional-Order Control Systems UEF-04-94 The Academy of SciencesInstitute of Experimental Physic Kosiice SlovakRepublic 1994

[43] B M Vinagre Y Q Chen and I Petras ldquoTwo direct Tustindiscretization methods for fractional-order differentiatorintegratorrdquo Journal of the Franklin Institute vol 340 no 5 pp349ndash362 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 The Scientific World Journal

The idea of fractional derivative of distributed order isstated by Caputo [33] and later developed by Caputo himself[34 35] and Bagley and Torvik [36 37] Other researchersused this idea and interesting reviews appeared to describethe related mathematical models of partial fractional differ-ential equation of distributed order For example Diethelmand Ford [38] used a numerical technique along with itserror analysis to solve the distributed order differential equa-tion and analyze the physical phenomena and engineeringproblems see [38] and references therein Recently SaberiNajafi et al [39 40] studied stability analysis of distributedorder fractional differential equations with respect to thenonnegative density function The aim of the present workis twofold First we consider the stability of 119899-dimensionalnonlinear distributed order fractional differential systemwith respect to the nonnegative density function and thenwe study the stability of distributed order fractional Chensystem

This paper is organized as follows In the next sectionwe recall some basic definitions of the Caputo fractionalderivative operator systems with fractional derivatives ofdistributed order Section 3 contains the main definitionsand theorems for checking the stability analysis of dis-tributed order fractional system In Section 4 we presentthe distributed order fractional Chen system and afterwardsbased on the stability theorem of distributed order fractionalsystems the stability of the distributed order fractionalChen system is discussed Finally the numerical solution toillustrate the validity of the results is presented in Section 5

2 Elementary Definitions

In this section we consider the main definitions and proper-ties of fractional derivative operators of single and distributedorder

The differential and integral operator in fractional calcu-lus is denoted by

119886119863120572

119905 where 119886 and 119905 are the bounds of the

operation and 120572 is the fractional order which can be rationalirrational or even complex For simplicity and without lossof generality in the following we assume that 119886 = 0 and119863120572

119905=0119863120572

119905 The continuous integrodifferential operator is

defined as follows

119863120572

119905=

119889120572

119889119905120572 120572 gt 0

1 120572 = 0

int

119905

0

(119889120591)minus120572

120572 lt 0

(1)

There are many definitions of fractional derivatives of order120572 gt 0 [1 2] such as Grunwald-Letnikovrsquos definition (GL)Riemann-Liouvillersquos definition (RL) and Caputorsquos fractionalderivative The RL definition is given as

RL119863120572

119905119891 (119905) =

1

Γ (119899 minus 120572)

119889119899

119889119905119899int

119905

0

(119905 minus 120591)119899minus120572minus1

119891 (120591) 119889120591 (2)

where 119899 is the first integer which is not less than 120572 that is119899 minus 1 lt 120572 lt 119899 and Γ(sdot) is a Gamma function The Caputofractional derivative of 119891(119905) is defined as

119862

119863

120572

119905119891 (119905) =

1

Γ (119899 minus 120572)

int

119905

0

(119905 minus 120591)119899minus120572minus1

119891(119899)

(120591) 119889120591 (3)

Finally Grunwald-Letnikov definition is given by

GL119863

120572

119905119891 (119905) = lim

ℎrarr0

ℎminus120572

[(119905minus119886)ℎ]

sum

119894=0

(minus1)119894

(

120572

119894)119891 (119905 minus 119895ℎ) (4)

Fortunately the Laplace transform of the Caputo fractionalderivative satisfies

L 119888

119863

120572

119905119891 (119905) = 119904

120572

L 119891 (119905) minus

119898minus1

sum

119896=0

119891(119896)

(0+

) 119904120572minus1minus119896

(5)

where 119898 minus 1 lt 120572 le 119898 and 119904 is the Laplace variable TheLaplace transform of Caputo fractional derivative requiresthe knowledge of the initial values of the function and itsinteger derivatives of order 119896 = 1 2 119898 minus 1 When 120572 isin

(0 1] is given by

L 119888

119863

120572

119905119891 (119905) = 119904

120572

L 119891 (119905) minus 119891 (0+

) 119904120572minus1

(6)

When the fractional calculus operators act on 119891(119905) and weintegrate

119886119863120572

119905119891(119905) with respect to the order then distributed

order fractional differential equations can be obtained In thisbrief the following distributed order fractional differentialoperator notation is adopted

119888

119863

119887(120572)

119905119891 (119905) = int

119898

119898minus1

119887 (120572)119888

119863

120572

119905119891 (119905) 119889120572

119898 minus 1 le 120572 le 119898 119898 isin N

(7)

where the derivative 119888119863120572119905is taken to be a fractional derivative

of Caputo type of order 120572 with respect to the nonnegativedensity function 119887(120572)

The idea of distributed order is stated by Caputo [3334] Further the Laplace transform of the Caputo distributedorder satisfies

L 119888

119863

119887(120572)

119905119891 (119905) = int

119898

119898minus1

119887 (120572)

times [119904120572

119865 (119904) minus

119898minus1

sum

119896=0

119891(119896)

(0+

) 119904120572minus1minus119896

]119889120572

= 119861 (119904) 119865 (119904) minus

119898minus1

sum

119896=0

1

119904119896+1

119861 (119904) 119891(119896)

(0+

)

(8)where 119865(119904) is the Laplace transform of 119891(119905) and

119861 (119904) = int

119898

119898minus1

119887 (120572) 119904120572

119889120572 (9)

A distributed order fractional equations can be defined by thefollowing model

119888

119863

119887(120572)

119905119909 (119905) = 119891 (119905 119909 (119905))

119909 (0) = 1199090

(10)

where 119909(119905) isin R and 119888119863119887(120572)119905

119909(119905) = int

1

0

119887(120572)119888

119863

120572

119905119909(119905)119889120572

The Scientific World Journal 3

In [38] the following results about the existence anduniqueness of solutions for (10) are further presented

Theorem 1 Let the function 119887 be absolutely integrable on theinterval [0 1] and satisfy int1

0

119887(120572)119904120572

119889120572 = 0 for Re(119904) gt 0 and119891 isin L1[0infin) and 119909 is such that 119888119863120572

119905119909(119905) lt M for 119905 isin [0infin]

for every120572 isin [0 1] then initial value problem (10) has a uniquesolution

Furthermore the above definition in one dimension cannaturally be generalized to the case of multiple dimensionsthat is let 119883(119905) = (119909

1(119905) 1199092(119905) 119909

119899(119905))119879

isin R119899 and 119887(120572) =

(1198871(120572) 1198872(120572) 119887

119899(120572))119879 0 lt 120572 lt 1 The 119899-dimension

distributed order fractional system is described as follows

119888

119863

119887(120572)

119905119883(119905) = int

1

0

119887 (120572)119888

119863

120572

119905119883(119905) 119889120572 = 119865 (119905 119883 (119905)) (11)

where119888

119863

119887(120572)

119905119883 (119905)

= (119888

119863

1198871(120572)

1199051199091(119905) 119888

119863

1198872(120572)

1199051199092(119905)

119888

119863

119887119899(120572)

119905119909119899(119905))

119865 (119905 119883 (119905)) = (

1198911(119905 1199091(119905) 1199092(119905) 119909

119899(119905))

1198912(119905 1199091(119905) 1199092(119905) 119909

119899(119905))

119891119899(119905 1199091(119905) 1199092(119905) 119909

119899(119905))

)

(12)

The result ofTheorem 1 can be easily generalized to the initialvalue problem of (11)

3 Stability Analysis of Distributed OrderFractional Systems

In this section we generalize the main stability properties forsystems of distributed order fractionalThe linear distributedorder fractional systems are expressed as

119888

119863

119887(120572)

119905119883(119905) = 119860119883 (119905)

119883 (0) = 1198830

(13)

where 119883(119905) = (1199091(119905) 1199092(119905) 119909

119899(119905))119879

isin R119899 the matrix 119860 isin

R119899times119899 and 119887(120572) = (1198871(120572) 1198872(120572) 119887

119899(120572))119879 0 lt 120572 lt 1 Then

Saberi Najafi et al [39] have obtained the general solution ofthe distributed order fractional systems (13) which is writtenby

119883(119905)

= 119883 (0) +

1

120587

int

119905

0

int

infin

0

int

infin

0

119890minus119903119905+119860120591minus120588 cos(120587120574)

times sin (120588 sin (120587120574)) sin119860119883 (0) 119889119903119889120591119889119905

(14)

where 119861(119904) = 120588 cos(120587120574) + 119894120588 sin(120587120574) 120588 = |119861(119904)| 120574 = (1

120587) arg[119861(119904)] and 119903 = 119890119894120587

Now we will recall some theorems and definitions aboutlinear distributed order fractional equations and then we willshow this theorem for nonlinear distributed order fractionalequations as well

Theorem 2 (see [39]) The distributed order fractional systemof (13) is asymptotically stable if and only if all roots of119889119890119905(119861(119904)119868 minus 119860) = 0 have negative real parts

Definition 3 (see [39]) The value of det(119861(119904)119868 minus 119860) = 0 isthe characteristic function of the matrix 119860 with respect tothe distributed function 119861(119904) where 119861(119904) = int

1

0

119887(120572)119904120572

119889120572 isthe distributed function with respect to the density function119887(120572) ge 0

The inertia of a matrix is the triplet of the numbers ofeigenvalues of 119860 with positive negative and zero real partsAs pointed out in [39] authors have generalized the inertiaconcept for analyzing the stability of linear distributed orderfractional systems

Definition 4 The inertia of the system (13) is the triple

119868119899119861(119904)

(119860) = (120587119861(119904)

(119860) 120592119861(119904)

(119860) 120575119861(119904)

(119860)) (15)

where 120587119861(119904)

(119860) 120592119861(119904)

(119860) and 120575119861(119904)

(119860) are respectively thenumber of roots of det(119861(119904)119868minus119860) = 0with positive negativeand zero real parts where 119861(119904) = (119861

1(119904) 1198612(119904) 119861

119899(119904))119879 is

the distributed function with respect to the density function119887(120572) = (119887

1(120572) 1198872(120572) 119887

119899(120572))119879

Theorem 5 (see [39]) The linear distributed order fractionalsystem (13) is asymptotically stable if and only if any of thefollowing equivalent conditions holds

(1) 120587119899119861(119904)

(119860) = 120575119899119861(119904)

(119860) = 0(2) all roots 119904 of the characteristic function of119860with respect

to 119861(119904) = (1198611(119904) 1198612(119904) 119861

119899(119904))119879 satisfy |119886119903119892(119904)| gt

1205872

Next we will mainly discuss the stability of a nonlinearautonomous distributed order fractional system which canbe described by

119888

119863

119887(120572)

119905119883 (119905) = 119865 (119883 (119905)) (16)

with the initial value119883(0) = 1198830 where

119865 (119883 (119905)) = (

1198911(1199091(119905) 1199092(119905) 119909

119899(119905))

1198912(1199091(119905) 1199092(119905) 119909

119899(119905))

119891119899(1199091(119905) 1199092(119905) 119909

119899(119905))

) (17)

119883(119905) = (1199091(119905) 1199092(119905) 119909

119899(119905))119879

isin R119899 and 119887(120572) = (1198871(120572)

1198872(120572) 119887

119899(120572))119879 0 lt 120572 lt 1

Theorem 6 Let 119883 = (1199091 1199092 119909

119899)119879 be the equilibrium of

system (16) that is 119888119863119887(120572)119905

119883 = 119865(119883) = 0 and J = (120597119865

120597119883)|119883=

is the Jacobian matrix at the point 119883 then the point119883 is asymptotically stable if and only if all roots 119904 of the

4 The Scientific World Journal

minus50

0

50

minus40minus2002040

0

10

20

30

40

50

x(t)

z(t)

y(t)

(a)

minus30 minus20 minus10 0 10 20 30minus30

minus20

minus10

0

10

20

30

40

x(t)y(t)

(b)

minus30 minus20 minus10 0 10 20 30

0

5

10

15

20

25

30

35

40

45

50

x(t)

z(t)

(c)

minus30 minus20 minus10 0 10 20 30 40

0

5

10

15

20

25

30

35

40

45

50

z(t)

y(t)

(d)Figure 1 Chaotic attractor of the integer-order Chen system (21) with (119886 119887 119888) = (35 3 28)

characteristic function of J with respect to 119861(119904) = (1198611(119904)

1198612(119904) 119861

119899(119904))119879 satisfy |119886119903119892(119904)| gt 1205872

Proof Let 120577(119905) = 119883(119905)minus119883 where 120577(119905) = (1205771(119905) 1205772(119905) 120577

119899(119905))

is a small disturbance from a fixed point Therefore

119888

119863

119887(120572)

119905120577 (119905) =

119888

119863

119887(120572)

119905(119883 (119905) minus 119883) (18)

since 119888119863119887(120572)119905

(119883(119905)minus119883) =119888

119863

119887(120572)

119905119883(119905)minus

119888

119863

119887(120572)

119905119883 and 119888119863119887(120572)

119905119883 =

0 thus we have

119888

119863

119887(120572)

119905120577 (119905) =

119888

119863119887(120572)

119905119883 (119905) = 119865 (119883 (119905)) = 119865 (120577 (119905) + 119883)

= 119865 (119883) + J120577 (119905) + higher order terms asymp J120577 (119905) (19)

The Scientific World Journal 5

Table 1 Stability analysis of system (22) for various density functions

119887119894(120572) = 120575(120572 minus 119902

119894) 119894 = 1 2 3 119887

119894(120572) = 120575(120572 minus 119902

1119894) + 120575(120572 minus 119902

2119894) 119894 = 1 2 3

(119886 119887 119888) 119902 = (1199021 1199022 1199023) 119868

119899119861(119904)(J) (119886 119887 119888)

(11990211 11990212)

(11990221 11990222)

(11990231 11990232)

119868119899119861(119904)

(J)

(35 6 11) (095 09 09)

1199091997888rarr (0 2 0)

1199092= 1199093997888rarr (1 2 0)

(35 3 29)

(085 03)

(065 03)

(095 03)

1199091997888rarr (1 0 0)

1199092= 1199093997888rarr (2 0 0)

(30 3 28) (07 08 09)

1199091997888rarr (1 0 0)

1199092997888rarr (0 2 0)

(35 3 28)

(095 01)

(095 01)

(095 01)

1199091997888rarr (1 0 0)

1199092= 1199093997888rarr (20 18 0)

(35 3 28) (085 09 095)

1199091997888rarr (1 0 0)

1199092= 1199093997888rarr (2 0 0)

(50 3 20)

(025 085)

(025 095)

(025 075)

1199091997888rarr (0 4 0)

11990921199093997888rarr (1 0 0)

System (18) can be written as119888

119863

119887(120572)

119905120577 (119905) asymp J120577 (119905) (20)

with the initial value 120577(0) = 1198830minus 119883

The analytical procedure of linearization is based on thefact that if the matrix J has no purely imaginary eigenvaluesthen the trajectories of the nonlinear system in the neigh-borhood of the equilibrium point have the same form asthe trajectories of the linear system [41] Hence by applyingTheorem 2 the linear system (20) is asymptotically stable ifand only if all roots 119904 of the characteristic function of J withrespect to 119861(119904) = (119861

1(119904) 1198612(119904) 119861

119899(119904))119879 satisfy | arg(119904)| gt

1205872 which implies that the equilibrium 119883 of the nonlineardistributed order fractional system (16) is as asymptoticallystable

Remark 7 The nonlinear distributed order fractional system(16) in the point 119883 is asymptotically stable if and only if120587119899119861(119904)

(119869) = 120575119899119861(119904)

(119869) = 0

4 Distributed Order Fractional Chen System

The Chen system is described by the following nonlineardifferential equations on R3 [22 23]

(119905) = 119886 (119910 (119905) minus 119909 (119905))

119910 (119905) = (119888 minus 119886) 119909 (119905) minus 119909 (119905) 119911 (119905) + 119888119910 (119905)

(119905) = 119909 (119905) 119910 (119905) minus 119887119911 (119905)

(21)

where 119909 119910 and 119911 are the state variables and 119886 119887 and 119888

are three system parameters The above system has a chaoticattractor when 119886 = 35 119887 = 3 and 119888 = 28 as shown in Figure 1The corresponding distributed order fractional Chen system(21) can be written in the form119888

1198631198871(120572)

119905119909 (119905) = 119886 (119910 (119905) minus 119909 (119905))

119888

1198631198872(120572)

119905119910 (119905) = (119888 minus 119886) 119909 (119905) minus 119909 (119905) 119911 (119905) + 119888119910 (119905)

119888

1198631198873(120572)

119905119911 (119905) = 119909 (119905) 119910 (119905) minus 119887119911 (119905)

(22)

where 119887119894(120572) for 119894 = 1 2 3 denote the nonnegative density

function of order 120572 isin (0 1] As a generalization of nonlinearfractional order differential equation into nonlinear dis-tributed order fractional differential equation the linearizedform of the system (22) at the equilibrium points 119909

1=

(0 0 0) 1199092

= (radic119887(2119888 minus 119886) radic119887(2119888 minus 119886) 2119888 minus 119886) and 1199093

=

(minusradic119887(2119888 minus 119886) minusradic119887(2119888 minus 119886) 2119888 minus 119886) that is 1198881198631198871(120572)119905

119909(119905) =

119865(119909) = 0 can be written in the form

119888

119863

119887(120572)

119905119883 (119905) = J119883 (119905) (23)

where 119883(119905) = (119909(119905) 119910(119905) 119911(119905))119879 119887(120572) = (119887

1(120572) 1198872(120572) 1198873(120572))119879

0 lt 120572 le 1 and J = (120597119865120597119883)|119883=119909119894

for 119894 = 1 2 3 The Jacobianmatrix of distributed order fractional Chen system (22) at theequilibrium point119883lowast = (119909

lowast

119910lowast

119911lowast

) is given by

J = [

[

minus119886 119886 0

119888 minus 119886 minus 119911lowast

119888 minus119909lowast

119910lowast

119909lowast

minus119887

]

]

(24)

Remark 8 If 119887119894(120572) = 120575(120572 minus 119902

119894) where 0 lt 119902

119894le 1 for 119894 = 1 2 3

and 120575(120572) is the Dirac delta function then we have the follow-ing fractional incommensurate-order Chen system [25]

119888

1198631205721

119905119909 (119905) = 119886 (119910 (119905) minus 119909 (119905))

119888

1198631205722

119905119910 (119905) = (119888 minus 119886) 119909 (119905) minus 119909 (119905) 119911 (119905) + 119888119910 (119905)

119888

1198631205723

119905119911 (119905) = 119909 (119905) 119910 (119905) minus 119887119911 (119905)

(25)

Based upon Theorem 6 the stability of the distributedorder fractional Chen system can be reached with ease Foranalyzing the stability of the distributed order fractionalChen system we compute 119868

119899119861(119904)

(J) in the case that the density

6 The Scientific World Journal

02

015

01

005

0

02

02

0

0

minus02

minus02minus04

minus04minus06x(t)

z(t)

y(t)

(a)

0 500 1000 1500 2000

04

03

02

01

0

minus01

minus02

minus03

minus04

tx(t)

(b)

0 500 1000 1500 2000

t

06

04

02

0

minus02

minus04

minus06

y(t)

(c)Figure 2 The equilibrium point 119909

1of the distributed order fractional Chen system (22) with (119902

1 1199022 1199023) = (095 09 09) and (119886 119887 119888) =

(35 6 11) is asymptotically stable

function varies The results are shown in Table 1 for someparameters 119886 119887 and 119888

5 Numerical Methods

As pointed out in [38] distributed order fractional differen-tial equations may be regarded as a generalization of single-term fractional differential equations

119888

119863

120572

119905119909 (119905) = 119891 (119905 119909 (119905)) (26)

or multiterm fractional differential equations119899

sum

119894=1

120574119894

119888

119863

120572119894

119905119909 (119905) = 119891 (119905 119909 (119905)) 0 lt 120572

1lt 1205722lt sdot sdot sdot lt 120572

119899 (27)

Therefore in this section numerical method to solving of(26) and (27) is presented The approximate Grunwald-Letnikovrsquos definition is given below where the step size of ℎis assumed to be very small [2 42 43]

119888

119863

120572

119905119896

119910 (119905) asymp ℎminus120572

119896

sum

119894=0

119888(120572)

119894119910 (119905119896minus119894

) (28)

The Scientific World Journal 7

minus40

minus20

0

20

minus40

minus20

0

20

0

10

20

30

40

50

x(t)

z(t)

y(t)

(a)

0 500 1000 1500 2000

t

minus40

minus30

minus20

minus10

0

10

20

x(t)

(b)

minus40

minus30

minus20

minus10

0

10

20

0 500 1000 1500 2000

t

y(t)

(c)

0

5

10

15

20

25

30

35

40

45

50

0 500 1000 1500 2000

t

z(t)

(d)

Figure 3 The equilibrium point 1199092of the distributed order fractional Chen system (22) with (119902

1 1199022 1199023) = (07 08 09) and (119886 119887 119888) =

(30 3 28) is asymptotically stable

where 119905119896

= 119896ℎ (119896 = 0 1 ) and 119888(120572)

119894(119894 = 0 1 ) are

binomial coefficients which can be computed as [43]

119888(120572)

0= 1 119888

(120572)

119894= (1 minus

1 + 120572

119894

) 119888(120572)

119894minus1 (29)

Then general numerical solution of (26) and (27) can beexpressed as

ℎminus120572

119896

sum

119894=0

119888(120572)

119894119909 (119905119896minus119894

) = 119891 (119905 119909 (119905))

1205741ℎminus1205721

119896

sum

119894=0

119888(1205721)

119894119909 (119905119896minus119894

) + sdot sdot sdot + 120574119899ℎminus120572119899

119896

sum

119894=0

119888(120572119899)

119894119909 (119905119896minus119894

)

= 119891 (119905 119909 (119905))

(30)

where 119888(120572119895)119894

for 119895 = 1 119899 are binomial coefficients calculatedaccording to (29) Equations in (30) can be rewritten as thefollowing forms

8 The Scientific World Journal

z(t)

y(t)

x(t)

minus30 minus20 minus10 0 10 20

minus50

0

50

0

10

20

30

40

50

(a)

minus30

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

0 500 1000 1500 2000

tx(t)

(b)

0 500 1000 1500 2000

t

minus40

minus30

minus20

minus10

0

10

20

y(t)

(c)

0

5

10

15

20

25

30

35

40

45

0 500 1000 1500 2000

t

z(t)

(d)Figure 4 Chaotic attractor of the distributed order fractional Chen system (22) with (119902

1 1199022 1199023) = (085 09 095) and (119886 119887 119888) = (35 3 28)

119909 (119905119896) = 119891 (119909 (119905

119896minus1) 119905119896minus1

) ℎ120572

minus

119896

sum

119894=1

119888(120572)

119894119909 (119905119896minus119894

)

119909 (119905119896)

=

1

sum119899

119895=1119886119895ℎminus120572119895

times[

[

119891 (119909 (119905119896minus1

) 119905119896minus1

) minus

119899

sum

119895=1

120574119895ℎminus120572119895

119896

sum

119894=1

119888

(120572119895)

119894119909 (119905119896minus119894

)]

]

(31)

Based on the previous algorithmwe can obtain the numericalsolution of the fractional differential equations (26) andmultiterm fractional differential equations (27) where 119896 =

1 2 119873 for 119873 = 119879ℎ and where 119879 is the total time of thecalculation

To verify the efficiency of the obtained results in Table 1the numerical solution for the distributed order fractionalChen system has been computed In the following calcula-tions let 119879 = 10 ℎ = 0005 with the initial conditions(02 minus05 02)

The Scientific World Journal 9

x(t)

z(t)

y(t)

minus50

0

50

minus100minus50

050

0

10

20

30

40

50

60

(a)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

0 500 1000 1500 2000

tx(t)

(b)

minus80

minus60

minus40

minus20

0

20

40

0 500 1000 1500 2000

t

y(t)

(c)

0

10

20

30

40

50

60

z(t)

0 500 1000 1500 2000

t

(d)

Figure 5 Chaotic attractor of the distributed order fractional Chen system (22) with (11990211 11990212 11990221 11990222 11990231 11990232) = (085 03 085 03 085 03)

and (119886 119887 119888) = (35 3 29)

Figure 2 shows that the system (22) with parameters(119886 119887 119888) = (35 6 11) and (119902

1 1199022 1199023) = (095 09 09) is

asymptotically stable in the equilibrium 1199091 Figure 3 dem-

onstrates that the system (22) with parameters (119886 119887 119888) =

(30 3 28) and (1199021 1199022 1199023) = (07 08 09) is asymptotically

stable in the equilibrium 1199092 From Figure 4 we can see that

for the distributed order Chen system (21) with parameters(119886 119887 119888) = (35 3 28) and (119902

1 1199022 1199023) = (085 09 095)

is chaotic Figure 5 demonstrates that the system (22)with parameters (119886 119887 119888) = (35 3 29) and (119902

11 11990212 11990221

11990222 11990231 11990232) = (085 03 085 03 085 03) has chaotic

attractor Figure 6 shows the chaotic attractor for the

10 The Scientific World Journal

minus50

0

50

minus40minus2002040

0

10

20

30

40

50

x(t)

z(t)

y(t)

(a)

minus30

minus20

minus10

0

10

20

30

0 500 1000 1500 2000

tx(t)

(b)

minus40

minus30

minus20

minus10

10

20

30

0

0 500 1000 1500 2000

t

y(t)

(c)

0 500 1000 1500 2000

t

z(t)

0

5

10

15

20

25

30

35

40

45

(d)

Figure 6 Chaotic attractor of the distributed order fractional Chen system (22) with (11990211 11990212 11990221 11990222 11990231 11990232) = (095 01 095 01 095 01)

and (119886 119887 119888) = (35 3 28)

distributed order Chen system (22) for parameters(119886 119887 119888) = (35 3 28) and (119902

11 11990212 11990221 11990222 11990231 11990232) =

(095 01 095 01 095 01) From Figure 7 we cansee that the system (22) is asymptotically stable in theequilibrium 119909

1with parameters (119886 119887 119888) = (50 3 20)

and (11990211 11990212 11990221 11990222 11990231 11990232) = (025 085 025 095

025 075)

6 Conclusion

In this paper we introduced the nonlinear distributed orderfractional differential equations with respect to a nonnegativedensity function hence the asymptotical stability for suchsystems has been investigated In addition we presented thedistributed order fractional Chen system and then in two

The Scientific World Journal 11

02

015

01

005

0

0

minus02

minus02

minus04

minus04

minus06

minus08

0

02

x(t)

z(t)

y(t)

(a)

minus03

minus04

02

03

01

minus02

minus01

0

0 500 1000 1500 2000

t

x(t)

(b)

02

01

0 500 1000 1500 2000

t

minus025

minus015

minus005

0

005

015

025

minus01

minus02

y(t)

(c)

minus008

minus006

minus004

minus002

0

002

004

006

008

01

minus01

0 500 1000 1500 2000

t

z(t)

(d)

Figure 7 The equilibrium point 1199091of the distributed order fractional Chen system (22) with (119902

11 11990212 11990221 11990222 11990231 11990232) = (025 085 025

095 025 075) and (119886 119887 119888) = (50 3 20) is asymptotically stable

special cases the stability for the distributed order fractionalChen is discussed Numerical solutions were coincident withresults of Table 1 described in Section 4 All numerical resultsare obtained using MATLAB 78

References

[1] A A Kilbas H M Srivastava and J J Trujillo Theory andApplication of Fractional Differential Equations Elsevier NewYork NY USA 2006

[2] K B Oldham and J SpanierThe Fractional Calculus AcadmicPress New York NY USA 1974

[3] E Reyes-Melo J Martinez-Vega C Guerrero-Salazar andU Ortiz-Mendez ldquoApplication of fractional calculus to themodeling of dielectric relaxation phenomena in polymericmaterialsrdquo Journal of Applied Polymer Science vol 98 no 2 pp923ndash935 2005

[4] R Schumer D A Benson M M Meerschaert and SW Wheatcraft ldquoEulerian derivation of the fractional

12 The Scientific World Journal

advection-dispersion equationrdquo Journal of ContaminantHydrology vol 48 no 1-2 pp 69ndash88 2001

[5] B I Henry and S L Wearne ldquoExistence of turing instabilitiesin a two-species fractional reaction-diffusion systemrdquo SIAMJournal on Applied Mathematics vol 62 no 3 pp 870ndash8872002

[6] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportvol 339 no 1 pp 1ndash77 2000

[7] S Picozzi andB JWest ldquoFractional Langevinmodel ofmemoryin financialmarketsrdquo Physical Review E vol 66 no 4 Article ID046118 pp 0461181ndash04611812 2002

[8] A Ansari A Refahi Sheikhani and S Kordrostami ldquoOn thegenerating function 119890119909119905+119910120593(119905) and its fractional calculusrdquo CentralEuropean Journal of Physics 2013

[9] A Ansari A Refahi Sheikhani and H Saberi Najafi ldquoSolutionto system of partial fractional differential equations using thefractional exponential operatorsrdquoMathematical Methods in theApplied Sciences vol 35 no 1 pp 119ndash123 2012

[10] D Matignon ldquoStability results for fractional differential equa-tions with applications to control processingrdquo in Proceedingsof the Computational Engineering in Systems and ApplicationMulticonference (IMACS IEEE-SMC rsquo96) vol 2 pp 963ndash968Lille France 1996

[11] M S Tavazoei and M Haeri ldquoA note on the stability offractional order systemsrdquoMathematics and Computers in Simu-lation vol 79 no 5 pp 1566ndash1576 2009

[12] V Daftardar-Gejji and A Babakhani ldquoAnalysis of a systemof fractional differential equationsrdquo Journal of MathematicalAnalysis and Applications vol 293 no 2 pp 511ndash522 2004

[13] K Diethelm and N J Ford ldquoAnalysis of fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 265 no 2 pp 229ndash248 2002

[14] J Sabatier M Moze and C Farges ldquoLMI stability conditionsfor fractional order systemsrdquo Computers and Mathematics withApplications vol 59 no 5 pp 1594ndash1609 2010

[15] Y Li Y Chen and I Podlubny ldquoMittag-Leffler stability offractional order nonlinear dynamic systemsrdquo Automatica vol45 no 8 pp 1965ndash1969 2009

[16] Y Li Y Chen and I Podlubny ldquoStability of fractional-ordernonlinear dynamic systems lyapunov directmethod and gener-alizedMittag-Leffler stabilityrdquoComputers andMathematics withApplications vol 59 no 5 pp 1810ndash1821 2010

[17] M Lakshmanan and K Murali Chaos in Nonlinear OscillatorsControlling and Synchronization World Scientific Singapore1996

[18] S KHan C Kurrer andY Kuramoto ldquoDephasing and burstingin coupled neural oscillatorsrdquo Physical Review Letters vol 75no 17 pp 3190ndash3193 1995

[19] B Blasius A Huppert and L Stone ldquoComplex dynamics andphase synchronization in spatially extended ecological systemsrdquoNature vol 399 no 6734 pp 354ndash359 1999

[20] C Sparrow The Lorenz Equations Bifurcations Chaos andStrange Attractors Springer New York NY USA 1982

[21] K T Alligood T D Sauer and J A Yorke Chaos An Intro-duction to Dynamical Systems Springer New York NY USA2008

[22] G Chen and T Ueta ldquoYet another chaotic attractorrdquo Interna-tional Journal of Bifurcation and Chaos vol 9 no 7 pp 1465ndash1466 1999

[23] T Ueta and G Chen ldquoBifurcation analysis of Chenrsquos equationrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 10 no 8 pp 1917ndash1931 2000

[24] I Grigorenko and E Grigorenko ldquoChaotic dynamics of thefractional Lorenz systemrdquo Physical Review Letters vol 91 no3 pp 0341011ndash0341014 2003

[25] C P Li andG J Peng ldquoChaos in Chenrsquos systemwith a fractionalorderrdquo Chaos Solitons and Fractals vol 22 no 2 pp 443ndash4502004

[26] C G Li and G Chen ldquoChaos in the fractional order Chensystem and its controlrdquo Chaos Solitons and Fractals vol 22 no3 pp 549ndash554 2004

[27] T T Hartley C F Lorenzo and H K Qammer ldquoChaos in afractional order Chuarsquos systemrdquo IEEE Transactions on Circuitsand Systems I vol 42 no 8 pp 485ndash490 1995

[28] C G Li and G Chen ldquoChaos and hyperchaos in the fractional-order Rossler equationsrdquo Physica A vol 341 no 1ndash4 pp 55ndash612004

[29] M Shahiri R Ghaderi A Ranjbar N S H Hosseinnia andS Momani ldquoChaotic fractional-order Coullet system synchro-nization and control approachrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 3 pp 665ndash6742010

[30] A E Matouk ldquoChaos feedback control and synchronization ofa fractional-order modified Autonomous Van der Pol-Duffingcircuitrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 2 pp 975ndash986 2011

[31] X YWang andM JWang ldquoDynamic analysis of the fractional-order Liu system and its synchronizationrdquo Chaos vol 17 no 3Article ID 033106 2007

[32] V Daftardar-Gejji S Bhalekar and P Gade ldquoDynamics offractional-ordered Chen system with delayrdquo Journal of Physicsvol 79 pp 61ndash69 2012

[33] M Caputo Elasticita e Dissipazione Zanichelli Bologna Italy1969

[34] M Caputo ldquoMean fractional-order-derivatives differentialequations and filtersrdquo Annali dellrsquoUniversita di Ferrara vol 41no 1 pp 73ndash84 1995

[35] M Caputo ldquoDistributed order differential equations modellingdielectric induction and diffusionrdquo Fractional Calculus andApplied Analysis vol 4 pp 421ndash442 2001

[36] L Bagley and P J Torvik ldquoOn the existence of the order domainand the solution of distributed order equationsrdquo InternationalJournal of Applied Mathematics vol 1 pp 865ndash882 2000

[37] R L Bagley and P J Torvik ldquoOn the existence of the orderdomain and the solution of distributed order equationsrdquo Inter-national Journal of Applied Mathematics vol 2 pp 965ndash9872000

[38] K Diethelm andN J Ford ldquoNumerical analysis for distributed-order differential equationsrdquo Journal of Computational andApplied Mathematics vol 225 no 1 pp 96ndash104 2009

[39] H Saberi Najafi A Refahi Sheikhani and A Ansari ldquoStabilityanalysis of distributed order fractional differential equationsrdquoAbstract and Applied Analysis vol 2011 Article ID 175323 12pages 2011

[40] A Refahi Sheikhani H Saberi Nadjafi A Ansari and FMehrdoust ldquoAnalytic study on linear system of distributed-order fractional differntional equationsrdquo Le Matematiche vol67 p 313 2012

The Scientific World Journal 13

[41] Z Vukic and L Kuljaca Nonlinear Control Systems CRC PressNew York NY USA 2003

[42] L Dorciak Numerical Models for Simulation the Fractional-Order Control Systems UEF-04-94 The Academy of SciencesInstitute of Experimental Physic Kosiice SlovakRepublic 1994

[43] B M Vinagre Y Q Chen and I Petras ldquoTwo direct Tustindiscretization methods for fractional-order differentiatorintegratorrdquo Journal of the Franklin Institute vol 340 no 5 pp349ndash362 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Journal of

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 3

In [38] the following results about the existence anduniqueness of solutions for (10) are further presented

Theorem 1 Let the function 119887 be absolutely integrable on theinterval [0 1] and satisfy int1

0

119887(120572)119904120572

119889120572 = 0 for Re(119904) gt 0 and119891 isin L1[0infin) and 119909 is such that 119888119863120572

119905119909(119905) lt M for 119905 isin [0infin]

for every120572 isin [0 1] then initial value problem (10) has a uniquesolution

Furthermore the above definition in one dimension cannaturally be generalized to the case of multiple dimensionsthat is let 119883(119905) = (119909

1(119905) 1199092(119905) 119909

119899(119905))119879

isin R119899 and 119887(120572) =

(1198871(120572) 1198872(120572) 119887

119899(120572))119879 0 lt 120572 lt 1 The 119899-dimension

distributed order fractional system is described as follows

119888

119863

119887(120572)

119905119883(119905) = int

1

0

119887 (120572)119888

119863

120572

119905119883(119905) 119889120572 = 119865 (119905 119883 (119905)) (11)

where119888

119863

119887(120572)

119905119883 (119905)

= (119888

119863

1198871(120572)

1199051199091(119905) 119888

119863

1198872(120572)

1199051199092(119905)

119888

119863

119887119899(120572)

119905119909119899(119905))

119865 (119905 119883 (119905)) = (

1198911(119905 1199091(119905) 1199092(119905) 119909

119899(119905))

1198912(119905 1199091(119905) 1199092(119905) 119909

119899(119905))

119891119899(119905 1199091(119905) 1199092(119905) 119909

119899(119905))

)

(12)

The result ofTheorem 1 can be easily generalized to the initialvalue problem of (11)

3 Stability Analysis of Distributed OrderFractional Systems

In this section we generalize the main stability properties forsystems of distributed order fractionalThe linear distributedorder fractional systems are expressed as

119888

119863

119887(120572)

119905119883(119905) = 119860119883 (119905)

119883 (0) = 1198830

(13)

where 119883(119905) = (1199091(119905) 1199092(119905) 119909

119899(119905))119879

isin R119899 the matrix 119860 isin

R119899times119899 and 119887(120572) = (1198871(120572) 1198872(120572) 119887

119899(120572))119879 0 lt 120572 lt 1 Then

Saberi Najafi et al [39] have obtained the general solution ofthe distributed order fractional systems (13) which is writtenby

119883(119905)

= 119883 (0) +

1

120587

int

119905

0

int

infin

0

int

infin

0

119890minus119903119905+119860120591minus120588 cos(120587120574)

times sin (120588 sin (120587120574)) sin119860119883 (0) 119889119903119889120591119889119905

(14)

where 119861(119904) = 120588 cos(120587120574) + 119894120588 sin(120587120574) 120588 = |119861(119904)| 120574 = (1

120587) arg[119861(119904)] and 119903 = 119890119894120587

Now we will recall some theorems and definitions aboutlinear distributed order fractional equations and then we willshow this theorem for nonlinear distributed order fractionalequations as well

Theorem 2 (see [39]) The distributed order fractional systemof (13) is asymptotically stable if and only if all roots of119889119890119905(119861(119904)119868 minus 119860) = 0 have negative real parts

Definition 3 (see [39]) The value of det(119861(119904)119868 minus 119860) = 0 isthe characteristic function of the matrix 119860 with respect tothe distributed function 119861(119904) where 119861(119904) = int

1

0

119887(120572)119904120572

119889120572 isthe distributed function with respect to the density function119887(120572) ge 0

The inertia of a matrix is the triplet of the numbers ofeigenvalues of 119860 with positive negative and zero real partsAs pointed out in [39] authors have generalized the inertiaconcept for analyzing the stability of linear distributed orderfractional systems

Definition 4 The inertia of the system (13) is the triple

119868119899119861(119904)

(119860) = (120587119861(119904)

(119860) 120592119861(119904)

(119860) 120575119861(119904)

(119860)) (15)

where 120587119861(119904)

(119860) 120592119861(119904)

(119860) and 120575119861(119904)

(119860) are respectively thenumber of roots of det(119861(119904)119868minus119860) = 0with positive negativeand zero real parts where 119861(119904) = (119861

1(119904) 1198612(119904) 119861

119899(119904))119879 is

the distributed function with respect to the density function119887(120572) = (119887

1(120572) 1198872(120572) 119887

119899(120572))119879

Theorem 5 (see [39]) The linear distributed order fractionalsystem (13) is asymptotically stable if and only if any of thefollowing equivalent conditions holds

(1) 120587119899119861(119904)

(119860) = 120575119899119861(119904)

(119860) = 0(2) all roots 119904 of the characteristic function of119860with respect

to 119861(119904) = (1198611(119904) 1198612(119904) 119861

119899(119904))119879 satisfy |119886119903119892(119904)| gt

1205872

Next we will mainly discuss the stability of a nonlinearautonomous distributed order fractional system which canbe described by

119888

119863

119887(120572)

119905119883 (119905) = 119865 (119883 (119905)) (16)

with the initial value119883(0) = 1198830 where

119865 (119883 (119905)) = (

1198911(1199091(119905) 1199092(119905) 119909

119899(119905))

1198912(1199091(119905) 1199092(119905) 119909

119899(119905))

119891119899(1199091(119905) 1199092(119905) 119909

119899(119905))

) (17)

119883(119905) = (1199091(119905) 1199092(119905) 119909

119899(119905))119879

isin R119899 and 119887(120572) = (1198871(120572)

1198872(120572) 119887

119899(120572))119879 0 lt 120572 lt 1

Theorem 6 Let 119883 = (1199091 1199092 119909

119899)119879 be the equilibrium of

system (16) that is 119888119863119887(120572)119905

119883 = 119865(119883) = 0 and J = (120597119865

120597119883)|119883=

is the Jacobian matrix at the point 119883 then the point119883 is asymptotically stable if and only if all roots 119904 of the

4 The Scientific World Journal

minus50

0

50

minus40minus2002040

0

10

20

30

40

50

x(t)

z(t)

y(t)

(a)

minus30 minus20 minus10 0 10 20 30minus30

minus20

minus10

0

10

20

30

40

x(t)y(t)

(b)

minus30 minus20 minus10 0 10 20 30

0

5

10

15

20

25

30

35

40

45

50

x(t)

z(t)

(c)

minus30 minus20 minus10 0 10 20 30 40

0

5

10

15

20

25

30

35

40

45

50

z(t)

y(t)

(d)Figure 1 Chaotic attractor of the integer-order Chen system (21) with (119886 119887 119888) = (35 3 28)

characteristic function of J with respect to 119861(119904) = (1198611(119904)

1198612(119904) 119861

119899(119904))119879 satisfy |119886119903119892(119904)| gt 1205872

Proof Let 120577(119905) = 119883(119905)minus119883 where 120577(119905) = (1205771(119905) 1205772(119905) 120577

119899(119905))

is a small disturbance from a fixed point Therefore

119888

119863

119887(120572)

119905120577 (119905) =

119888

119863

119887(120572)

119905(119883 (119905) minus 119883) (18)

since 119888119863119887(120572)119905

(119883(119905)minus119883) =119888

119863

119887(120572)

119905119883(119905)minus

119888

119863

119887(120572)

119905119883 and 119888119863119887(120572)

119905119883 =

0 thus we have

119888

119863

119887(120572)

119905120577 (119905) =

119888

119863119887(120572)

119905119883 (119905) = 119865 (119883 (119905)) = 119865 (120577 (119905) + 119883)

= 119865 (119883) + J120577 (119905) + higher order terms asymp J120577 (119905) (19)

The Scientific World Journal 5

Table 1 Stability analysis of system (22) for various density functions

119887119894(120572) = 120575(120572 minus 119902

119894) 119894 = 1 2 3 119887

119894(120572) = 120575(120572 minus 119902

1119894) + 120575(120572 minus 119902

2119894) 119894 = 1 2 3

(119886 119887 119888) 119902 = (1199021 1199022 1199023) 119868

119899119861(119904)(J) (119886 119887 119888)

(11990211 11990212)

(11990221 11990222)

(11990231 11990232)

119868119899119861(119904)

(J)

(35 6 11) (095 09 09)

1199091997888rarr (0 2 0)

1199092= 1199093997888rarr (1 2 0)

(35 3 29)

(085 03)

(065 03)

(095 03)

1199091997888rarr (1 0 0)

1199092= 1199093997888rarr (2 0 0)

(30 3 28) (07 08 09)

1199091997888rarr (1 0 0)

1199092997888rarr (0 2 0)

(35 3 28)

(095 01)

(095 01)

(095 01)

1199091997888rarr (1 0 0)

1199092= 1199093997888rarr (20 18 0)

(35 3 28) (085 09 095)

1199091997888rarr (1 0 0)

1199092= 1199093997888rarr (2 0 0)

(50 3 20)

(025 085)

(025 095)

(025 075)

1199091997888rarr (0 4 0)

11990921199093997888rarr (1 0 0)

System (18) can be written as119888

119863

119887(120572)

119905120577 (119905) asymp J120577 (119905) (20)

with the initial value 120577(0) = 1198830minus 119883

The analytical procedure of linearization is based on thefact that if the matrix J has no purely imaginary eigenvaluesthen the trajectories of the nonlinear system in the neigh-borhood of the equilibrium point have the same form asthe trajectories of the linear system [41] Hence by applyingTheorem 2 the linear system (20) is asymptotically stable ifand only if all roots 119904 of the characteristic function of J withrespect to 119861(119904) = (119861

1(119904) 1198612(119904) 119861

119899(119904))119879 satisfy | arg(119904)| gt

1205872 which implies that the equilibrium 119883 of the nonlineardistributed order fractional system (16) is as asymptoticallystable

Remark 7 The nonlinear distributed order fractional system(16) in the point 119883 is asymptotically stable if and only if120587119899119861(119904)

(119869) = 120575119899119861(119904)

(119869) = 0

4 Distributed Order Fractional Chen System

The Chen system is described by the following nonlineardifferential equations on R3 [22 23]

(119905) = 119886 (119910 (119905) minus 119909 (119905))

119910 (119905) = (119888 minus 119886) 119909 (119905) minus 119909 (119905) 119911 (119905) + 119888119910 (119905)

(119905) = 119909 (119905) 119910 (119905) minus 119887119911 (119905)

(21)

where 119909 119910 and 119911 are the state variables and 119886 119887 and 119888

are three system parameters The above system has a chaoticattractor when 119886 = 35 119887 = 3 and 119888 = 28 as shown in Figure 1The corresponding distributed order fractional Chen system(21) can be written in the form119888

1198631198871(120572)

119905119909 (119905) = 119886 (119910 (119905) minus 119909 (119905))

119888

1198631198872(120572)

119905119910 (119905) = (119888 minus 119886) 119909 (119905) minus 119909 (119905) 119911 (119905) + 119888119910 (119905)

119888

1198631198873(120572)

119905119911 (119905) = 119909 (119905) 119910 (119905) minus 119887119911 (119905)

(22)

where 119887119894(120572) for 119894 = 1 2 3 denote the nonnegative density

function of order 120572 isin (0 1] As a generalization of nonlinearfractional order differential equation into nonlinear dis-tributed order fractional differential equation the linearizedform of the system (22) at the equilibrium points 119909

1=

(0 0 0) 1199092

= (radic119887(2119888 minus 119886) radic119887(2119888 minus 119886) 2119888 minus 119886) and 1199093

=

(minusradic119887(2119888 minus 119886) minusradic119887(2119888 minus 119886) 2119888 minus 119886) that is 1198881198631198871(120572)119905

119909(119905) =

119865(119909) = 0 can be written in the form

119888

119863

119887(120572)

119905119883 (119905) = J119883 (119905) (23)

where 119883(119905) = (119909(119905) 119910(119905) 119911(119905))119879 119887(120572) = (119887

1(120572) 1198872(120572) 1198873(120572))119879

0 lt 120572 le 1 and J = (120597119865120597119883)|119883=119909119894

for 119894 = 1 2 3 The Jacobianmatrix of distributed order fractional Chen system (22) at theequilibrium point119883lowast = (119909

lowast

119910lowast

119911lowast

) is given by

J = [

[

minus119886 119886 0

119888 minus 119886 minus 119911lowast

119888 minus119909lowast

119910lowast

119909lowast

minus119887

]

]

(24)

Remark 8 If 119887119894(120572) = 120575(120572 minus 119902

119894) where 0 lt 119902

119894le 1 for 119894 = 1 2 3

and 120575(120572) is the Dirac delta function then we have the follow-ing fractional incommensurate-order Chen system [25]

119888

1198631205721

119905119909 (119905) = 119886 (119910 (119905) minus 119909 (119905))

119888

1198631205722

119905119910 (119905) = (119888 minus 119886) 119909 (119905) minus 119909 (119905) 119911 (119905) + 119888119910 (119905)

119888

1198631205723

119905119911 (119905) = 119909 (119905) 119910 (119905) minus 119887119911 (119905)

(25)

Based upon Theorem 6 the stability of the distributedorder fractional Chen system can be reached with ease Foranalyzing the stability of the distributed order fractionalChen system we compute 119868

119899119861(119904)

(J) in the case that the density

6 The Scientific World Journal

02

015

01

005

0

02

02

0

0

minus02

minus02minus04

minus04minus06x(t)

z(t)

y(t)

(a)

0 500 1000 1500 2000

04

03

02

01

0

minus01

minus02

minus03

minus04

tx(t)

(b)

0 500 1000 1500 2000

t

06

04

02

0

minus02

minus04

minus06

y(t)

(c)Figure 2 The equilibrium point 119909

1of the distributed order fractional Chen system (22) with (119902

1 1199022 1199023) = (095 09 09) and (119886 119887 119888) =

(35 6 11) is asymptotically stable

function varies The results are shown in Table 1 for someparameters 119886 119887 and 119888

5 Numerical Methods

As pointed out in [38] distributed order fractional differen-tial equations may be regarded as a generalization of single-term fractional differential equations

119888

119863

120572

119905119909 (119905) = 119891 (119905 119909 (119905)) (26)

or multiterm fractional differential equations119899

sum

119894=1

120574119894

119888

119863

120572119894

119905119909 (119905) = 119891 (119905 119909 (119905)) 0 lt 120572

1lt 1205722lt sdot sdot sdot lt 120572

119899 (27)

Therefore in this section numerical method to solving of(26) and (27) is presented The approximate Grunwald-Letnikovrsquos definition is given below where the step size of ℎis assumed to be very small [2 42 43]

119888

119863

120572

119905119896

119910 (119905) asymp ℎminus120572

119896

sum

119894=0

119888(120572)

119894119910 (119905119896minus119894

) (28)

The Scientific World Journal 7

minus40

minus20

0

20

minus40

minus20

0

20

0

10

20

30

40

50

x(t)

z(t)

y(t)

(a)

0 500 1000 1500 2000

t

minus40

minus30

minus20

minus10

0

10

20

x(t)

(b)

minus40

minus30

minus20

minus10

0

10

20

0 500 1000 1500 2000

t

y(t)

(c)

0

5

10

15

20

25

30

35

40

45

50

0 500 1000 1500 2000

t

z(t)

(d)

Figure 3 The equilibrium point 1199092of the distributed order fractional Chen system (22) with (119902

1 1199022 1199023) = (07 08 09) and (119886 119887 119888) =

(30 3 28) is asymptotically stable

where 119905119896

= 119896ℎ (119896 = 0 1 ) and 119888(120572)

119894(119894 = 0 1 ) are

binomial coefficients which can be computed as [43]

119888(120572)

0= 1 119888

(120572)

119894= (1 minus

1 + 120572

119894

) 119888(120572)

119894minus1 (29)

Then general numerical solution of (26) and (27) can beexpressed as

ℎminus120572

119896

sum

119894=0

119888(120572)

119894119909 (119905119896minus119894

) = 119891 (119905 119909 (119905))

1205741ℎminus1205721

119896

sum

119894=0

119888(1205721)

119894119909 (119905119896minus119894

) + sdot sdot sdot + 120574119899ℎminus120572119899

119896

sum

119894=0

119888(120572119899)

119894119909 (119905119896minus119894

)

= 119891 (119905 119909 (119905))

(30)

where 119888(120572119895)119894

for 119895 = 1 119899 are binomial coefficients calculatedaccording to (29) Equations in (30) can be rewritten as thefollowing forms

8 The Scientific World Journal

z(t)

y(t)

x(t)

minus30 minus20 minus10 0 10 20

minus50

0

50

0

10

20

30

40

50

(a)

minus30

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

0 500 1000 1500 2000

tx(t)

(b)

0 500 1000 1500 2000

t

minus40

minus30

minus20

minus10

0

10

20

y(t)

(c)

0

5

10

15

20

25

30

35

40

45

0 500 1000 1500 2000

t

z(t)

(d)Figure 4 Chaotic attractor of the distributed order fractional Chen system (22) with (119902

1 1199022 1199023) = (085 09 095) and (119886 119887 119888) = (35 3 28)

119909 (119905119896) = 119891 (119909 (119905

119896minus1) 119905119896minus1

) ℎ120572

minus

119896

sum

119894=1

119888(120572)

119894119909 (119905119896minus119894

)

119909 (119905119896)

=

1

sum119899

119895=1119886119895ℎminus120572119895

times[

[

119891 (119909 (119905119896minus1

) 119905119896minus1

) minus

119899

sum

119895=1

120574119895ℎminus120572119895

119896

sum

119894=1

119888

(120572119895)

119894119909 (119905119896minus119894

)]

]

(31)

Based on the previous algorithmwe can obtain the numericalsolution of the fractional differential equations (26) andmultiterm fractional differential equations (27) where 119896 =

1 2 119873 for 119873 = 119879ℎ and where 119879 is the total time of thecalculation

To verify the efficiency of the obtained results in Table 1the numerical solution for the distributed order fractionalChen system has been computed In the following calcula-tions let 119879 = 10 ℎ = 0005 with the initial conditions(02 minus05 02)

The Scientific World Journal 9

x(t)

z(t)

y(t)

minus50

0

50

minus100minus50

050

0

10

20

30

40

50

60

(a)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

0 500 1000 1500 2000

tx(t)

(b)

minus80

minus60

minus40

minus20

0

20

40

0 500 1000 1500 2000

t

y(t)

(c)

0

10

20

30

40

50

60

z(t)

0 500 1000 1500 2000

t

(d)

Figure 5 Chaotic attractor of the distributed order fractional Chen system (22) with (11990211 11990212 11990221 11990222 11990231 11990232) = (085 03 085 03 085 03)

and (119886 119887 119888) = (35 3 29)

Figure 2 shows that the system (22) with parameters(119886 119887 119888) = (35 6 11) and (119902

1 1199022 1199023) = (095 09 09) is

asymptotically stable in the equilibrium 1199091 Figure 3 dem-

onstrates that the system (22) with parameters (119886 119887 119888) =

(30 3 28) and (1199021 1199022 1199023) = (07 08 09) is asymptotically

stable in the equilibrium 1199092 From Figure 4 we can see that

for the distributed order Chen system (21) with parameters(119886 119887 119888) = (35 3 28) and (119902

1 1199022 1199023) = (085 09 095)

is chaotic Figure 5 demonstrates that the system (22)with parameters (119886 119887 119888) = (35 3 29) and (119902

11 11990212 11990221

11990222 11990231 11990232) = (085 03 085 03 085 03) has chaotic

attractor Figure 6 shows the chaotic attractor for the

10 The Scientific World Journal

minus50

0

50

minus40minus2002040

0

10

20

30

40

50

x(t)

z(t)

y(t)

(a)

minus30

minus20

minus10

0

10

20

30

0 500 1000 1500 2000

tx(t)

(b)

minus40

minus30

minus20

minus10

10

20

30

0

0 500 1000 1500 2000

t

y(t)

(c)

0 500 1000 1500 2000

t

z(t)

0

5

10

15

20

25

30

35

40

45

(d)

Figure 6 Chaotic attractor of the distributed order fractional Chen system (22) with (11990211 11990212 11990221 11990222 11990231 11990232) = (095 01 095 01 095 01)

and (119886 119887 119888) = (35 3 28)

distributed order Chen system (22) for parameters(119886 119887 119888) = (35 3 28) and (119902

11 11990212 11990221 11990222 11990231 11990232) =

(095 01 095 01 095 01) From Figure 7 we cansee that the system (22) is asymptotically stable in theequilibrium 119909

1with parameters (119886 119887 119888) = (50 3 20)

and (11990211 11990212 11990221 11990222 11990231 11990232) = (025 085 025 095

025 075)

6 Conclusion

In this paper we introduced the nonlinear distributed orderfractional differential equations with respect to a nonnegativedensity function hence the asymptotical stability for suchsystems has been investigated In addition we presented thedistributed order fractional Chen system and then in two

The Scientific World Journal 11

02

015

01

005

0

0

minus02

minus02

minus04

minus04

minus06

minus08

0

02

x(t)

z(t)

y(t)

(a)

minus03

minus04

02

03

01

minus02

minus01

0

0 500 1000 1500 2000

t

x(t)

(b)

02

01

0 500 1000 1500 2000

t

minus025

minus015

minus005

0

005

015

025

minus01

minus02

y(t)

(c)

minus008

minus006

minus004

minus002

0

002

004

006

008

01

minus01

0 500 1000 1500 2000

t

z(t)

(d)

Figure 7 The equilibrium point 1199091of the distributed order fractional Chen system (22) with (119902

11 11990212 11990221 11990222 11990231 11990232) = (025 085 025

095 025 075) and (119886 119887 119888) = (50 3 20) is asymptotically stable

special cases the stability for the distributed order fractionalChen is discussed Numerical solutions were coincident withresults of Table 1 described in Section 4 All numerical resultsare obtained using MATLAB 78

References

[1] A A Kilbas H M Srivastava and J J Trujillo Theory andApplication of Fractional Differential Equations Elsevier NewYork NY USA 2006

[2] K B Oldham and J SpanierThe Fractional Calculus AcadmicPress New York NY USA 1974

[3] E Reyes-Melo J Martinez-Vega C Guerrero-Salazar andU Ortiz-Mendez ldquoApplication of fractional calculus to themodeling of dielectric relaxation phenomena in polymericmaterialsrdquo Journal of Applied Polymer Science vol 98 no 2 pp923ndash935 2005

[4] R Schumer D A Benson M M Meerschaert and SW Wheatcraft ldquoEulerian derivation of the fractional

12 The Scientific World Journal

advection-dispersion equationrdquo Journal of ContaminantHydrology vol 48 no 1-2 pp 69ndash88 2001

[5] B I Henry and S L Wearne ldquoExistence of turing instabilitiesin a two-species fractional reaction-diffusion systemrdquo SIAMJournal on Applied Mathematics vol 62 no 3 pp 870ndash8872002

[6] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportvol 339 no 1 pp 1ndash77 2000

[7] S Picozzi andB JWest ldquoFractional Langevinmodel ofmemoryin financialmarketsrdquo Physical Review E vol 66 no 4 Article ID046118 pp 0461181ndash04611812 2002

[8] A Ansari A Refahi Sheikhani and S Kordrostami ldquoOn thegenerating function 119890119909119905+119910120593(119905) and its fractional calculusrdquo CentralEuropean Journal of Physics 2013

[9] A Ansari A Refahi Sheikhani and H Saberi Najafi ldquoSolutionto system of partial fractional differential equations using thefractional exponential operatorsrdquoMathematical Methods in theApplied Sciences vol 35 no 1 pp 119ndash123 2012

[10] D Matignon ldquoStability results for fractional differential equa-tions with applications to control processingrdquo in Proceedingsof the Computational Engineering in Systems and ApplicationMulticonference (IMACS IEEE-SMC rsquo96) vol 2 pp 963ndash968Lille France 1996

[11] M S Tavazoei and M Haeri ldquoA note on the stability offractional order systemsrdquoMathematics and Computers in Simu-lation vol 79 no 5 pp 1566ndash1576 2009

[12] V Daftardar-Gejji and A Babakhani ldquoAnalysis of a systemof fractional differential equationsrdquo Journal of MathematicalAnalysis and Applications vol 293 no 2 pp 511ndash522 2004

[13] K Diethelm and N J Ford ldquoAnalysis of fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 265 no 2 pp 229ndash248 2002

[14] J Sabatier M Moze and C Farges ldquoLMI stability conditionsfor fractional order systemsrdquo Computers and Mathematics withApplications vol 59 no 5 pp 1594ndash1609 2010

[15] Y Li Y Chen and I Podlubny ldquoMittag-Leffler stability offractional order nonlinear dynamic systemsrdquo Automatica vol45 no 8 pp 1965ndash1969 2009

[16] Y Li Y Chen and I Podlubny ldquoStability of fractional-ordernonlinear dynamic systems lyapunov directmethod and gener-alizedMittag-Leffler stabilityrdquoComputers andMathematics withApplications vol 59 no 5 pp 1810ndash1821 2010

[17] M Lakshmanan and K Murali Chaos in Nonlinear OscillatorsControlling and Synchronization World Scientific Singapore1996

[18] S KHan C Kurrer andY Kuramoto ldquoDephasing and burstingin coupled neural oscillatorsrdquo Physical Review Letters vol 75no 17 pp 3190ndash3193 1995

[19] B Blasius A Huppert and L Stone ldquoComplex dynamics andphase synchronization in spatially extended ecological systemsrdquoNature vol 399 no 6734 pp 354ndash359 1999

[20] C Sparrow The Lorenz Equations Bifurcations Chaos andStrange Attractors Springer New York NY USA 1982

[21] K T Alligood T D Sauer and J A Yorke Chaos An Intro-duction to Dynamical Systems Springer New York NY USA2008

[22] G Chen and T Ueta ldquoYet another chaotic attractorrdquo Interna-tional Journal of Bifurcation and Chaos vol 9 no 7 pp 1465ndash1466 1999

[23] T Ueta and G Chen ldquoBifurcation analysis of Chenrsquos equationrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 10 no 8 pp 1917ndash1931 2000

[24] I Grigorenko and E Grigorenko ldquoChaotic dynamics of thefractional Lorenz systemrdquo Physical Review Letters vol 91 no3 pp 0341011ndash0341014 2003

[25] C P Li andG J Peng ldquoChaos in Chenrsquos systemwith a fractionalorderrdquo Chaos Solitons and Fractals vol 22 no 2 pp 443ndash4502004

[26] C G Li and G Chen ldquoChaos in the fractional order Chensystem and its controlrdquo Chaos Solitons and Fractals vol 22 no3 pp 549ndash554 2004

[27] T T Hartley C F Lorenzo and H K Qammer ldquoChaos in afractional order Chuarsquos systemrdquo IEEE Transactions on Circuitsand Systems I vol 42 no 8 pp 485ndash490 1995

[28] C G Li and G Chen ldquoChaos and hyperchaos in the fractional-order Rossler equationsrdquo Physica A vol 341 no 1ndash4 pp 55ndash612004

[29] M Shahiri R Ghaderi A Ranjbar N S H Hosseinnia andS Momani ldquoChaotic fractional-order Coullet system synchro-nization and control approachrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 3 pp 665ndash6742010

[30] A E Matouk ldquoChaos feedback control and synchronization ofa fractional-order modified Autonomous Van der Pol-Duffingcircuitrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 2 pp 975ndash986 2011

[31] X YWang andM JWang ldquoDynamic analysis of the fractional-order Liu system and its synchronizationrdquo Chaos vol 17 no 3Article ID 033106 2007

[32] V Daftardar-Gejji S Bhalekar and P Gade ldquoDynamics offractional-ordered Chen system with delayrdquo Journal of Physicsvol 79 pp 61ndash69 2012

[33] M Caputo Elasticita e Dissipazione Zanichelli Bologna Italy1969

[34] M Caputo ldquoMean fractional-order-derivatives differentialequations and filtersrdquo Annali dellrsquoUniversita di Ferrara vol 41no 1 pp 73ndash84 1995

[35] M Caputo ldquoDistributed order differential equations modellingdielectric induction and diffusionrdquo Fractional Calculus andApplied Analysis vol 4 pp 421ndash442 2001

[36] L Bagley and P J Torvik ldquoOn the existence of the order domainand the solution of distributed order equationsrdquo InternationalJournal of Applied Mathematics vol 1 pp 865ndash882 2000

[37] R L Bagley and P J Torvik ldquoOn the existence of the orderdomain and the solution of distributed order equationsrdquo Inter-national Journal of Applied Mathematics vol 2 pp 965ndash9872000

[38] K Diethelm andN J Ford ldquoNumerical analysis for distributed-order differential equationsrdquo Journal of Computational andApplied Mathematics vol 225 no 1 pp 96ndash104 2009

[39] H Saberi Najafi A Refahi Sheikhani and A Ansari ldquoStabilityanalysis of distributed order fractional differential equationsrdquoAbstract and Applied Analysis vol 2011 Article ID 175323 12pages 2011

[40] A Refahi Sheikhani H Saberi Nadjafi A Ansari and FMehrdoust ldquoAnalytic study on linear system of distributed-order fractional differntional equationsrdquo Le Matematiche vol67 p 313 2012

The Scientific World Journal 13

[41] Z Vukic and L Kuljaca Nonlinear Control Systems CRC PressNew York NY USA 2003

[42] L Dorciak Numerical Models for Simulation the Fractional-Order Control Systems UEF-04-94 The Academy of SciencesInstitute of Experimental Physic Kosiice SlovakRepublic 1994

[43] B M Vinagre Y Q Chen and I Petras ldquoTwo direct Tustindiscretization methods for fractional-order differentiatorintegratorrdquo Journal of the Franklin Institute vol 340 no 5 pp349ndash362 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 The Scientific World Journal

minus50

0

50

minus40minus2002040

0

10

20

30

40

50

x(t)

z(t)

y(t)

(a)

minus30 minus20 minus10 0 10 20 30minus30

minus20

minus10

0

10

20

30

40

x(t)y(t)

(b)

minus30 minus20 minus10 0 10 20 30

0

5

10

15

20

25

30

35

40

45

50

x(t)

z(t)

(c)

minus30 minus20 minus10 0 10 20 30 40

0

5

10

15

20

25

30

35

40

45

50

z(t)

y(t)

(d)Figure 1 Chaotic attractor of the integer-order Chen system (21) with (119886 119887 119888) = (35 3 28)

characteristic function of J with respect to 119861(119904) = (1198611(119904)

1198612(119904) 119861

119899(119904))119879 satisfy |119886119903119892(119904)| gt 1205872

Proof Let 120577(119905) = 119883(119905)minus119883 where 120577(119905) = (1205771(119905) 1205772(119905) 120577

119899(119905))

is a small disturbance from a fixed point Therefore

119888

119863

119887(120572)

119905120577 (119905) =

119888

119863

119887(120572)

119905(119883 (119905) minus 119883) (18)

since 119888119863119887(120572)119905

(119883(119905)minus119883) =119888

119863

119887(120572)

119905119883(119905)minus

119888

119863

119887(120572)

119905119883 and 119888119863119887(120572)

119905119883 =

0 thus we have

119888

119863

119887(120572)

119905120577 (119905) =

119888

119863119887(120572)

119905119883 (119905) = 119865 (119883 (119905)) = 119865 (120577 (119905) + 119883)

= 119865 (119883) + J120577 (119905) + higher order terms asymp J120577 (119905) (19)

The Scientific World Journal 5

Table 1 Stability analysis of system (22) for various density functions

119887119894(120572) = 120575(120572 minus 119902

119894) 119894 = 1 2 3 119887

119894(120572) = 120575(120572 minus 119902

1119894) + 120575(120572 minus 119902

2119894) 119894 = 1 2 3

(119886 119887 119888) 119902 = (1199021 1199022 1199023) 119868

119899119861(119904)(J) (119886 119887 119888)

(11990211 11990212)

(11990221 11990222)

(11990231 11990232)

119868119899119861(119904)

(J)

(35 6 11) (095 09 09)

1199091997888rarr (0 2 0)

1199092= 1199093997888rarr (1 2 0)

(35 3 29)

(085 03)

(065 03)

(095 03)

1199091997888rarr (1 0 0)

1199092= 1199093997888rarr (2 0 0)

(30 3 28) (07 08 09)

1199091997888rarr (1 0 0)

1199092997888rarr (0 2 0)

(35 3 28)

(095 01)

(095 01)

(095 01)

1199091997888rarr (1 0 0)

1199092= 1199093997888rarr (20 18 0)

(35 3 28) (085 09 095)

1199091997888rarr (1 0 0)

1199092= 1199093997888rarr (2 0 0)

(50 3 20)

(025 085)

(025 095)

(025 075)

1199091997888rarr (0 4 0)

11990921199093997888rarr (1 0 0)

System (18) can be written as119888

119863

119887(120572)

119905120577 (119905) asymp J120577 (119905) (20)

with the initial value 120577(0) = 1198830minus 119883

The analytical procedure of linearization is based on thefact that if the matrix J has no purely imaginary eigenvaluesthen the trajectories of the nonlinear system in the neigh-borhood of the equilibrium point have the same form asthe trajectories of the linear system [41] Hence by applyingTheorem 2 the linear system (20) is asymptotically stable ifand only if all roots 119904 of the characteristic function of J withrespect to 119861(119904) = (119861

1(119904) 1198612(119904) 119861

119899(119904))119879 satisfy | arg(119904)| gt

1205872 which implies that the equilibrium 119883 of the nonlineardistributed order fractional system (16) is as asymptoticallystable

Remark 7 The nonlinear distributed order fractional system(16) in the point 119883 is asymptotically stable if and only if120587119899119861(119904)

(119869) = 120575119899119861(119904)

(119869) = 0

4 Distributed Order Fractional Chen System

The Chen system is described by the following nonlineardifferential equations on R3 [22 23]

(119905) = 119886 (119910 (119905) minus 119909 (119905))

119910 (119905) = (119888 minus 119886) 119909 (119905) minus 119909 (119905) 119911 (119905) + 119888119910 (119905)

(119905) = 119909 (119905) 119910 (119905) minus 119887119911 (119905)

(21)

where 119909 119910 and 119911 are the state variables and 119886 119887 and 119888

are three system parameters The above system has a chaoticattractor when 119886 = 35 119887 = 3 and 119888 = 28 as shown in Figure 1The corresponding distributed order fractional Chen system(21) can be written in the form119888

1198631198871(120572)

119905119909 (119905) = 119886 (119910 (119905) minus 119909 (119905))

119888

1198631198872(120572)

119905119910 (119905) = (119888 minus 119886) 119909 (119905) minus 119909 (119905) 119911 (119905) + 119888119910 (119905)

119888

1198631198873(120572)

119905119911 (119905) = 119909 (119905) 119910 (119905) minus 119887119911 (119905)

(22)

where 119887119894(120572) for 119894 = 1 2 3 denote the nonnegative density

function of order 120572 isin (0 1] As a generalization of nonlinearfractional order differential equation into nonlinear dis-tributed order fractional differential equation the linearizedform of the system (22) at the equilibrium points 119909

1=

(0 0 0) 1199092

= (radic119887(2119888 minus 119886) radic119887(2119888 minus 119886) 2119888 minus 119886) and 1199093

=

(minusradic119887(2119888 minus 119886) minusradic119887(2119888 minus 119886) 2119888 minus 119886) that is 1198881198631198871(120572)119905

119909(119905) =

119865(119909) = 0 can be written in the form

119888

119863

119887(120572)

119905119883 (119905) = J119883 (119905) (23)

where 119883(119905) = (119909(119905) 119910(119905) 119911(119905))119879 119887(120572) = (119887

1(120572) 1198872(120572) 1198873(120572))119879

0 lt 120572 le 1 and J = (120597119865120597119883)|119883=119909119894

for 119894 = 1 2 3 The Jacobianmatrix of distributed order fractional Chen system (22) at theequilibrium point119883lowast = (119909

lowast

119910lowast

119911lowast

) is given by

J = [

[

minus119886 119886 0

119888 minus 119886 minus 119911lowast

119888 minus119909lowast

119910lowast

119909lowast

minus119887

]

]

(24)

Remark 8 If 119887119894(120572) = 120575(120572 minus 119902

119894) where 0 lt 119902

119894le 1 for 119894 = 1 2 3

and 120575(120572) is the Dirac delta function then we have the follow-ing fractional incommensurate-order Chen system [25]

119888

1198631205721

119905119909 (119905) = 119886 (119910 (119905) minus 119909 (119905))

119888

1198631205722

119905119910 (119905) = (119888 minus 119886) 119909 (119905) minus 119909 (119905) 119911 (119905) + 119888119910 (119905)

119888

1198631205723

119905119911 (119905) = 119909 (119905) 119910 (119905) minus 119887119911 (119905)

(25)

Based upon Theorem 6 the stability of the distributedorder fractional Chen system can be reached with ease Foranalyzing the stability of the distributed order fractionalChen system we compute 119868

119899119861(119904)

(J) in the case that the density

6 The Scientific World Journal

02

015

01

005

0

02

02

0

0

minus02

minus02minus04

minus04minus06x(t)

z(t)

y(t)

(a)

0 500 1000 1500 2000

04

03

02

01

0

minus01

minus02

minus03

minus04

tx(t)

(b)

0 500 1000 1500 2000

t

06

04

02

0

minus02

minus04

minus06

y(t)

(c)Figure 2 The equilibrium point 119909

1of the distributed order fractional Chen system (22) with (119902

1 1199022 1199023) = (095 09 09) and (119886 119887 119888) =

(35 6 11) is asymptotically stable

function varies The results are shown in Table 1 for someparameters 119886 119887 and 119888

5 Numerical Methods

As pointed out in [38] distributed order fractional differen-tial equations may be regarded as a generalization of single-term fractional differential equations

119888

119863

120572

119905119909 (119905) = 119891 (119905 119909 (119905)) (26)

or multiterm fractional differential equations119899

sum

119894=1

120574119894

119888

119863

120572119894

119905119909 (119905) = 119891 (119905 119909 (119905)) 0 lt 120572

1lt 1205722lt sdot sdot sdot lt 120572

119899 (27)

Therefore in this section numerical method to solving of(26) and (27) is presented The approximate Grunwald-Letnikovrsquos definition is given below where the step size of ℎis assumed to be very small [2 42 43]

119888

119863

120572

119905119896

119910 (119905) asymp ℎminus120572

119896

sum

119894=0

119888(120572)

119894119910 (119905119896minus119894

) (28)

The Scientific World Journal 7

minus40

minus20

0

20

minus40

minus20

0

20

0

10

20

30

40

50

x(t)

z(t)

y(t)

(a)

0 500 1000 1500 2000

t

minus40

minus30

minus20

minus10

0

10

20

x(t)

(b)

minus40

minus30

minus20

minus10

0

10

20

0 500 1000 1500 2000

t

y(t)

(c)

0

5

10

15

20

25

30

35

40

45

50

0 500 1000 1500 2000

t

z(t)

(d)

Figure 3 The equilibrium point 1199092of the distributed order fractional Chen system (22) with (119902

1 1199022 1199023) = (07 08 09) and (119886 119887 119888) =

(30 3 28) is asymptotically stable

where 119905119896

= 119896ℎ (119896 = 0 1 ) and 119888(120572)

119894(119894 = 0 1 ) are

binomial coefficients which can be computed as [43]

119888(120572)

0= 1 119888

(120572)

119894= (1 minus

1 + 120572

119894

) 119888(120572)

119894minus1 (29)

Then general numerical solution of (26) and (27) can beexpressed as

ℎminus120572

119896

sum

119894=0

119888(120572)

119894119909 (119905119896minus119894

) = 119891 (119905 119909 (119905))

1205741ℎminus1205721

119896

sum

119894=0

119888(1205721)

119894119909 (119905119896minus119894

) + sdot sdot sdot + 120574119899ℎminus120572119899

119896

sum

119894=0

119888(120572119899)

119894119909 (119905119896minus119894

)

= 119891 (119905 119909 (119905))

(30)

where 119888(120572119895)119894

for 119895 = 1 119899 are binomial coefficients calculatedaccording to (29) Equations in (30) can be rewritten as thefollowing forms

8 The Scientific World Journal

z(t)

y(t)

x(t)

minus30 minus20 minus10 0 10 20

minus50

0

50

0

10

20

30

40

50

(a)

minus30

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

0 500 1000 1500 2000

tx(t)

(b)

0 500 1000 1500 2000

t

minus40

minus30

minus20

minus10

0

10

20

y(t)

(c)

0

5

10

15

20

25

30

35

40

45

0 500 1000 1500 2000

t

z(t)

(d)Figure 4 Chaotic attractor of the distributed order fractional Chen system (22) with (119902

1 1199022 1199023) = (085 09 095) and (119886 119887 119888) = (35 3 28)

119909 (119905119896) = 119891 (119909 (119905

119896minus1) 119905119896minus1

) ℎ120572

minus

119896

sum

119894=1

119888(120572)

119894119909 (119905119896minus119894

)

119909 (119905119896)

=

1

sum119899

119895=1119886119895ℎminus120572119895

times[

[

119891 (119909 (119905119896minus1

) 119905119896minus1

) minus

119899

sum

119895=1

120574119895ℎminus120572119895

119896

sum

119894=1

119888

(120572119895)

119894119909 (119905119896minus119894

)]

]

(31)

Based on the previous algorithmwe can obtain the numericalsolution of the fractional differential equations (26) andmultiterm fractional differential equations (27) where 119896 =

1 2 119873 for 119873 = 119879ℎ and where 119879 is the total time of thecalculation

To verify the efficiency of the obtained results in Table 1the numerical solution for the distributed order fractionalChen system has been computed In the following calcula-tions let 119879 = 10 ℎ = 0005 with the initial conditions(02 minus05 02)

The Scientific World Journal 9

x(t)

z(t)

y(t)

minus50

0

50

minus100minus50

050

0

10

20

30

40

50

60

(a)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

0 500 1000 1500 2000

tx(t)

(b)

minus80

minus60

minus40

minus20

0

20

40

0 500 1000 1500 2000

t

y(t)

(c)

0

10

20

30

40

50

60

z(t)

0 500 1000 1500 2000

t

(d)

Figure 5 Chaotic attractor of the distributed order fractional Chen system (22) with (11990211 11990212 11990221 11990222 11990231 11990232) = (085 03 085 03 085 03)

and (119886 119887 119888) = (35 3 29)

Figure 2 shows that the system (22) with parameters(119886 119887 119888) = (35 6 11) and (119902

1 1199022 1199023) = (095 09 09) is

asymptotically stable in the equilibrium 1199091 Figure 3 dem-

onstrates that the system (22) with parameters (119886 119887 119888) =

(30 3 28) and (1199021 1199022 1199023) = (07 08 09) is asymptotically

stable in the equilibrium 1199092 From Figure 4 we can see that

for the distributed order Chen system (21) with parameters(119886 119887 119888) = (35 3 28) and (119902

1 1199022 1199023) = (085 09 095)

is chaotic Figure 5 demonstrates that the system (22)with parameters (119886 119887 119888) = (35 3 29) and (119902

11 11990212 11990221

11990222 11990231 11990232) = (085 03 085 03 085 03) has chaotic

attractor Figure 6 shows the chaotic attractor for the

10 The Scientific World Journal

minus50

0

50

minus40minus2002040

0

10

20

30

40

50

x(t)

z(t)

y(t)

(a)

minus30

minus20

minus10

0

10

20

30

0 500 1000 1500 2000

tx(t)

(b)

minus40

minus30

minus20

minus10

10

20

30

0

0 500 1000 1500 2000

t

y(t)

(c)

0 500 1000 1500 2000

t

z(t)

0

5

10

15

20

25

30

35

40

45

(d)

Figure 6 Chaotic attractor of the distributed order fractional Chen system (22) with (11990211 11990212 11990221 11990222 11990231 11990232) = (095 01 095 01 095 01)

and (119886 119887 119888) = (35 3 28)

distributed order Chen system (22) for parameters(119886 119887 119888) = (35 3 28) and (119902

11 11990212 11990221 11990222 11990231 11990232) =

(095 01 095 01 095 01) From Figure 7 we cansee that the system (22) is asymptotically stable in theequilibrium 119909

1with parameters (119886 119887 119888) = (50 3 20)

and (11990211 11990212 11990221 11990222 11990231 11990232) = (025 085 025 095

025 075)

6 Conclusion

In this paper we introduced the nonlinear distributed orderfractional differential equations with respect to a nonnegativedensity function hence the asymptotical stability for suchsystems has been investigated In addition we presented thedistributed order fractional Chen system and then in two

The Scientific World Journal 11

02

015

01

005

0

0

minus02

minus02

minus04

minus04

minus06

minus08

0

02

x(t)

z(t)

y(t)

(a)

minus03

minus04

02

03

01

minus02

minus01

0

0 500 1000 1500 2000

t

x(t)

(b)

02

01

0 500 1000 1500 2000

t

minus025

minus015

minus005

0

005

015

025

minus01

minus02

y(t)

(c)

minus008

minus006

minus004

minus002

0

002

004

006

008

01

minus01

0 500 1000 1500 2000

t

z(t)

(d)

Figure 7 The equilibrium point 1199091of the distributed order fractional Chen system (22) with (119902

11 11990212 11990221 11990222 11990231 11990232) = (025 085 025

095 025 075) and (119886 119887 119888) = (50 3 20) is asymptotically stable

special cases the stability for the distributed order fractionalChen is discussed Numerical solutions were coincident withresults of Table 1 described in Section 4 All numerical resultsare obtained using MATLAB 78

References

[1] A A Kilbas H M Srivastava and J J Trujillo Theory andApplication of Fractional Differential Equations Elsevier NewYork NY USA 2006

[2] K B Oldham and J SpanierThe Fractional Calculus AcadmicPress New York NY USA 1974

[3] E Reyes-Melo J Martinez-Vega C Guerrero-Salazar andU Ortiz-Mendez ldquoApplication of fractional calculus to themodeling of dielectric relaxation phenomena in polymericmaterialsrdquo Journal of Applied Polymer Science vol 98 no 2 pp923ndash935 2005

[4] R Schumer D A Benson M M Meerschaert and SW Wheatcraft ldquoEulerian derivation of the fractional

12 The Scientific World Journal

advection-dispersion equationrdquo Journal of ContaminantHydrology vol 48 no 1-2 pp 69ndash88 2001

[5] B I Henry and S L Wearne ldquoExistence of turing instabilitiesin a two-species fractional reaction-diffusion systemrdquo SIAMJournal on Applied Mathematics vol 62 no 3 pp 870ndash8872002

[6] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportvol 339 no 1 pp 1ndash77 2000

[7] S Picozzi andB JWest ldquoFractional Langevinmodel ofmemoryin financialmarketsrdquo Physical Review E vol 66 no 4 Article ID046118 pp 0461181ndash04611812 2002

[8] A Ansari A Refahi Sheikhani and S Kordrostami ldquoOn thegenerating function 119890119909119905+119910120593(119905) and its fractional calculusrdquo CentralEuropean Journal of Physics 2013

[9] A Ansari A Refahi Sheikhani and H Saberi Najafi ldquoSolutionto system of partial fractional differential equations using thefractional exponential operatorsrdquoMathematical Methods in theApplied Sciences vol 35 no 1 pp 119ndash123 2012

[10] D Matignon ldquoStability results for fractional differential equa-tions with applications to control processingrdquo in Proceedingsof the Computational Engineering in Systems and ApplicationMulticonference (IMACS IEEE-SMC rsquo96) vol 2 pp 963ndash968Lille France 1996

[11] M S Tavazoei and M Haeri ldquoA note on the stability offractional order systemsrdquoMathematics and Computers in Simu-lation vol 79 no 5 pp 1566ndash1576 2009

[12] V Daftardar-Gejji and A Babakhani ldquoAnalysis of a systemof fractional differential equationsrdquo Journal of MathematicalAnalysis and Applications vol 293 no 2 pp 511ndash522 2004

[13] K Diethelm and N J Ford ldquoAnalysis of fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 265 no 2 pp 229ndash248 2002

[14] J Sabatier M Moze and C Farges ldquoLMI stability conditionsfor fractional order systemsrdquo Computers and Mathematics withApplications vol 59 no 5 pp 1594ndash1609 2010

[15] Y Li Y Chen and I Podlubny ldquoMittag-Leffler stability offractional order nonlinear dynamic systemsrdquo Automatica vol45 no 8 pp 1965ndash1969 2009

[16] Y Li Y Chen and I Podlubny ldquoStability of fractional-ordernonlinear dynamic systems lyapunov directmethod and gener-alizedMittag-Leffler stabilityrdquoComputers andMathematics withApplications vol 59 no 5 pp 1810ndash1821 2010

[17] M Lakshmanan and K Murali Chaos in Nonlinear OscillatorsControlling and Synchronization World Scientific Singapore1996

[18] S KHan C Kurrer andY Kuramoto ldquoDephasing and burstingin coupled neural oscillatorsrdquo Physical Review Letters vol 75no 17 pp 3190ndash3193 1995

[19] B Blasius A Huppert and L Stone ldquoComplex dynamics andphase synchronization in spatially extended ecological systemsrdquoNature vol 399 no 6734 pp 354ndash359 1999

[20] C Sparrow The Lorenz Equations Bifurcations Chaos andStrange Attractors Springer New York NY USA 1982

[21] K T Alligood T D Sauer and J A Yorke Chaos An Intro-duction to Dynamical Systems Springer New York NY USA2008

[22] G Chen and T Ueta ldquoYet another chaotic attractorrdquo Interna-tional Journal of Bifurcation and Chaos vol 9 no 7 pp 1465ndash1466 1999

[23] T Ueta and G Chen ldquoBifurcation analysis of Chenrsquos equationrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 10 no 8 pp 1917ndash1931 2000

[24] I Grigorenko and E Grigorenko ldquoChaotic dynamics of thefractional Lorenz systemrdquo Physical Review Letters vol 91 no3 pp 0341011ndash0341014 2003

[25] C P Li andG J Peng ldquoChaos in Chenrsquos systemwith a fractionalorderrdquo Chaos Solitons and Fractals vol 22 no 2 pp 443ndash4502004

[26] C G Li and G Chen ldquoChaos in the fractional order Chensystem and its controlrdquo Chaos Solitons and Fractals vol 22 no3 pp 549ndash554 2004

[27] T T Hartley C F Lorenzo and H K Qammer ldquoChaos in afractional order Chuarsquos systemrdquo IEEE Transactions on Circuitsand Systems I vol 42 no 8 pp 485ndash490 1995

[28] C G Li and G Chen ldquoChaos and hyperchaos in the fractional-order Rossler equationsrdquo Physica A vol 341 no 1ndash4 pp 55ndash612004

[29] M Shahiri R Ghaderi A Ranjbar N S H Hosseinnia andS Momani ldquoChaotic fractional-order Coullet system synchro-nization and control approachrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 3 pp 665ndash6742010

[30] A E Matouk ldquoChaos feedback control and synchronization ofa fractional-order modified Autonomous Van der Pol-Duffingcircuitrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 2 pp 975ndash986 2011

[31] X YWang andM JWang ldquoDynamic analysis of the fractional-order Liu system and its synchronizationrdquo Chaos vol 17 no 3Article ID 033106 2007

[32] V Daftardar-Gejji S Bhalekar and P Gade ldquoDynamics offractional-ordered Chen system with delayrdquo Journal of Physicsvol 79 pp 61ndash69 2012

[33] M Caputo Elasticita e Dissipazione Zanichelli Bologna Italy1969

[34] M Caputo ldquoMean fractional-order-derivatives differentialequations and filtersrdquo Annali dellrsquoUniversita di Ferrara vol 41no 1 pp 73ndash84 1995

[35] M Caputo ldquoDistributed order differential equations modellingdielectric induction and diffusionrdquo Fractional Calculus andApplied Analysis vol 4 pp 421ndash442 2001

[36] L Bagley and P J Torvik ldquoOn the existence of the order domainand the solution of distributed order equationsrdquo InternationalJournal of Applied Mathematics vol 1 pp 865ndash882 2000

[37] R L Bagley and P J Torvik ldquoOn the existence of the orderdomain and the solution of distributed order equationsrdquo Inter-national Journal of Applied Mathematics vol 2 pp 965ndash9872000

[38] K Diethelm andN J Ford ldquoNumerical analysis for distributed-order differential equationsrdquo Journal of Computational andApplied Mathematics vol 225 no 1 pp 96ndash104 2009

[39] H Saberi Najafi A Refahi Sheikhani and A Ansari ldquoStabilityanalysis of distributed order fractional differential equationsrdquoAbstract and Applied Analysis vol 2011 Article ID 175323 12pages 2011

[40] A Refahi Sheikhani H Saberi Nadjafi A Ansari and FMehrdoust ldquoAnalytic study on linear system of distributed-order fractional differntional equationsrdquo Le Matematiche vol67 p 313 2012

The Scientific World Journal 13

[41] Z Vukic and L Kuljaca Nonlinear Control Systems CRC PressNew York NY USA 2003

[42] L Dorciak Numerical Models for Simulation the Fractional-Order Control Systems UEF-04-94 The Academy of SciencesInstitute of Experimental Physic Kosiice SlovakRepublic 1994

[43] B M Vinagre Y Q Chen and I Petras ldquoTwo direct Tustindiscretization methods for fractional-order differentiatorintegratorrdquo Journal of the Franklin Institute vol 340 no 5 pp349ndash362 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 5

Table 1 Stability analysis of system (22) for various density functions

119887119894(120572) = 120575(120572 minus 119902

119894) 119894 = 1 2 3 119887

119894(120572) = 120575(120572 minus 119902

1119894) + 120575(120572 minus 119902

2119894) 119894 = 1 2 3

(119886 119887 119888) 119902 = (1199021 1199022 1199023) 119868

119899119861(119904)(J) (119886 119887 119888)

(11990211 11990212)

(11990221 11990222)

(11990231 11990232)

119868119899119861(119904)

(J)

(35 6 11) (095 09 09)

1199091997888rarr (0 2 0)

1199092= 1199093997888rarr (1 2 0)

(35 3 29)

(085 03)

(065 03)

(095 03)

1199091997888rarr (1 0 0)

1199092= 1199093997888rarr (2 0 0)

(30 3 28) (07 08 09)

1199091997888rarr (1 0 0)

1199092997888rarr (0 2 0)

(35 3 28)

(095 01)

(095 01)

(095 01)

1199091997888rarr (1 0 0)

1199092= 1199093997888rarr (20 18 0)

(35 3 28) (085 09 095)

1199091997888rarr (1 0 0)

1199092= 1199093997888rarr (2 0 0)

(50 3 20)

(025 085)

(025 095)

(025 075)

1199091997888rarr (0 4 0)

11990921199093997888rarr (1 0 0)

System (18) can be written as119888

119863

119887(120572)

119905120577 (119905) asymp J120577 (119905) (20)

with the initial value 120577(0) = 1198830minus 119883

The analytical procedure of linearization is based on thefact that if the matrix J has no purely imaginary eigenvaluesthen the trajectories of the nonlinear system in the neigh-borhood of the equilibrium point have the same form asthe trajectories of the linear system [41] Hence by applyingTheorem 2 the linear system (20) is asymptotically stable ifand only if all roots 119904 of the characteristic function of J withrespect to 119861(119904) = (119861

1(119904) 1198612(119904) 119861

119899(119904))119879 satisfy | arg(119904)| gt

1205872 which implies that the equilibrium 119883 of the nonlineardistributed order fractional system (16) is as asymptoticallystable

Remark 7 The nonlinear distributed order fractional system(16) in the point 119883 is asymptotically stable if and only if120587119899119861(119904)

(119869) = 120575119899119861(119904)

(119869) = 0

4 Distributed Order Fractional Chen System

The Chen system is described by the following nonlineardifferential equations on R3 [22 23]

(119905) = 119886 (119910 (119905) minus 119909 (119905))

119910 (119905) = (119888 minus 119886) 119909 (119905) minus 119909 (119905) 119911 (119905) + 119888119910 (119905)

(119905) = 119909 (119905) 119910 (119905) minus 119887119911 (119905)

(21)

where 119909 119910 and 119911 are the state variables and 119886 119887 and 119888

are three system parameters The above system has a chaoticattractor when 119886 = 35 119887 = 3 and 119888 = 28 as shown in Figure 1The corresponding distributed order fractional Chen system(21) can be written in the form119888

1198631198871(120572)

119905119909 (119905) = 119886 (119910 (119905) minus 119909 (119905))

119888

1198631198872(120572)

119905119910 (119905) = (119888 minus 119886) 119909 (119905) minus 119909 (119905) 119911 (119905) + 119888119910 (119905)

119888

1198631198873(120572)

119905119911 (119905) = 119909 (119905) 119910 (119905) minus 119887119911 (119905)

(22)

where 119887119894(120572) for 119894 = 1 2 3 denote the nonnegative density

function of order 120572 isin (0 1] As a generalization of nonlinearfractional order differential equation into nonlinear dis-tributed order fractional differential equation the linearizedform of the system (22) at the equilibrium points 119909

1=

(0 0 0) 1199092

= (radic119887(2119888 minus 119886) radic119887(2119888 minus 119886) 2119888 minus 119886) and 1199093

=

(minusradic119887(2119888 minus 119886) minusradic119887(2119888 minus 119886) 2119888 minus 119886) that is 1198881198631198871(120572)119905

119909(119905) =

119865(119909) = 0 can be written in the form

119888

119863

119887(120572)

119905119883 (119905) = J119883 (119905) (23)

where 119883(119905) = (119909(119905) 119910(119905) 119911(119905))119879 119887(120572) = (119887

1(120572) 1198872(120572) 1198873(120572))119879

0 lt 120572 le 1 and J = (120597119865120597119883)|119883=119909119894

for 119894 = 1 2 3 The Jacobianmatrix of distributed order fractional Chen system (22) at theequilibrium point119883lowast = (119909

lowast

119910lowast

119911lowast

) is given by

J = [

[

minus119886 119886 0

119888 minus 119886 minus 119911lowast

119888 minus119909lowast

119910lowast

119909lowast

minus119887

]

]

(24)

Remark 8 If 119887119894(120572) = 120575(120572 minus 119902

119894) where 0 lt 119902

119894le 1 for 119894 = 1 2 3

and 120575(120572) is the Dirac delta function then we have the follow-ing fractional incommensurate-order Chen system [25]

119888

1198631205721

119905119909 (119905) = 119886 (119910 (119905) minus 119909 (119905))

119888

1198631205722

119905119910 (119905) = (119888 minus 119886) 119909 (119905) minus 119909 (119905) 119911 (119905) + 119888119910 (119905)

119888

1198631205723

119905119911 (119905) = 119909 (119905) 119910 (119905) minus 119887119911 (119905)

(25)

Based upon Theorem 6 the stability of the distributedorder fractional Chen system can be reached with ease Foranalyzing the stability of the distributed order fractionalChen system we compute 119868

119899119861(119904)

(J) in the case that the density

6 The Scientific World Journal

02

015

01

005

0

02

02

0

0

minus02

minus02minus04

minus04minus06x(t)

z(t)

y(t)

(a)

0 500 1000 1500 2000

04

03

02

01

0

minus01

minus02

minus03

minus04

tx(t)

(b)

0 500 1000 1500 2000

t

06

04

02

0

minus02

minus04

minus06

y(t)

(c)Figure 2 The equilibrium point 119909

1of the distributed order fractional Chen system (22) with (119902

1 1199022 1199023) = (095 09 09) and (119886 119887 119888) =

(35 6 11) is asymptotically stable

function varies The results are shown in Table 1 for someparameters 119886 119887 and 119888

5 Numerical Methods

As pointed out in [38] distributed order fractional differen-tial equations may be regarded as a generalization of single-term fractional differential equations

119888

119863

120572

119905119909 (119905) = 119891 (119905 119909 (119905)) (26)

or multiterm fractional differential equations119899

sum

119894=1

120574119894

119888

119863

120572119894

119905119909 (119905) = 119891 (119905 119909 (119905)) 0 lt 120572

1lt 1205722lt sdot sdot sdot lt 120572

119899 (27)

Therefore in this section numerical method to solving of(26) and (27) is presented The approximate Grunwald-Letnikovrsquos definition is given below where the step size of ℎis assumed to be very small [2 42 43]

119888

119863

120572

119905119896

119910 (119905) asymp ℎminus120572

119896

sum

119894=0

119888(120572)

119894119910 (119905119896minus119894

) (28)

The Scientific World Journal 7

minus40

minus20

0

20

minus40

minus20

0

20

0

10

20

30

40

50

x(t)

z(t)

y(t)

(a)

0 500 1000 1500 2000

t

minus40

minus30

minus20

minus10

0

10

20

x(t)

(b)

minus40

minus30

minus20

minus10

0

10

20

0 500 1000 1500 2000

t

y(t)

(c)

0

5

10

15

20

25

30

35

40

45

50

0 500 1000 1500 2000

t

z(t)

(d)

Figure 3 The equilibrium point 1199092of the distributed order fractional Chen system (22) with (119902

1 1199022 1199023) = (07 08 09) and (119886 119887 119888) =

(30 3 28) is asymptotically stable

where 119905119896

= 119896ℎ (119896 = 0 1 ) and 119888(120572)

119894(119894 = 0 1 ) are

binomial coefficients which can be computed as [43]

119888(120572)

0= 1 119888

(120572)

119894= (1 minus

1 + 120572

119894

) 119888(120572)

119894minus1 (29)

Then general numerical solution of (26) and (27) can beexpressed as

ℎminus120572

119896

sum

119894=0

119888(120572)

119894119909 (119905119896minus119894

) = 119891 (119905 119909 (119905))

1205741ℎminus1205721

119896

sum

119894=0

119888(1205721)

119894119909 (119905119896minus119894

) + sdot sdot sdot + 120574119899ℎminus120572119899

119896

sum

119894=0

119888(120572119899)

119894119909 (119905119896minus119894

)

= 119891 (119905 119909 (119905))

(30)

where 119888(120572119895)119894

for 119895 = 1 119899 are binomial coefficients calculatedaccording to (29) Equations in (30) can be rewritten as thefollowing forms

8 The Scientific World Journal

z(t)

y(t)

x(t)

minus30 minus20 minus10 0 10 20

minus50

0

50

0

10

20

30

40

50

(a)

minus30

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

0 500 1000 1500 2000

tx(t)

(b)

0 500 1000 1500 2000

t

minus40

minus30

minus20

minus10

0

10

20

y(t)

(c)

0

5

10

15

20

25

30

35

40

45

0 500 1000 1500 2000

t

z(t)

(d)Figure 4 Chaotic attractor of the distributed order fractional Chen system (22) with (119902

1 1199022 1199023) = (085 09 095) and (119886 119887 119888) = (35 3 28)

119909 (119905119896) = 119891 (119909 (119905

119896minus1) 119905119896minus1

) ℎ120572

minus

119896

sum

119894=1

119888(120572)

119894119909 (119905119896minus119894

)

119909 (119905119896)

=

1

sum119899

119895=1119886119895ℎminus120572119895

times[

[

119891 (119909 (119905119896minus1

) 119905119896minus1

) minus

119899

sum

119895=1

120574119895ℎminus120572119895

119896

sum

119894=1

119888

(120572119895)

119894119909 (119905119896minus119894

)]

]

(31)

Based on the previous algorithmwe can obtain the numericalsolution of the fractional differential equations (26) andmultiterm fractional differential equations (27) where 119896 =

1 2 119873 for 119873 = 119879ℎ and where 119879 is the total time of thecalculation

To verify the efficiency of the obtained results in Table 1the numerical solution for the distributed order fractionalChen system has been computed In the following calcula-tions let 119879 = 10 ℎ = 0005 with the initial conditions(02 minus05 02)

The Scientific World Journal 9

x(t)

z(t)

y(t)

minus50

0

50

minus100minus50

050

0

10

20

30

40

50

60

(a)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

0 500 1000 1500 2000

tx(t)

(b)

minus80

minus60

minus40

minus20

0

20

40

0 500 1000 1500 2000

t

y(t)

(c)

0

10

20

30

40

50

60

z(t)

0 500 1000 1500 2000

t

(d)

Figure 5 Chaotic attractor of the distributed order fractional Chen system (22) with (11990211 11990212 11990221 11990222 11990231 11990232) = (085 03 085 03 085 03)

and (119886 119887 119888) = (35 3 29)

Figure 2 shows that the system (22) with parameters(119886 119887 119888) = (35 6 11) and (119902

1 1199022 1199023) = (095 09 09) is

asymptotically stable in the equilibrium 1199091 Figure 3 dem-

onstrates that the system (22) with parameters (119886 119887 119888) =

(30 3 28) and (1199021 1199022 1199023) = (07 08 09) is asymptotically

stable in the equilibrium 1199092 From Figure 4 we can see that

for the distributed order Chen system (21) with parameters(119886 119887 119888) = (35 3 28) and (119902

1 1199022 1199023) = (085 09 095)

is chaotic Figure 5 demonstrates that the system (22)with parameters (119886 119887 119888) = (35 3 29) and (119902

11 11990212 11990221

11990222 11990231 11990232) = (085 03 085 03 085 03) has chaotic

attractor Figure 6 shows the chaotic attractor for the

10 The Scientific World Journal

minus50

0

50

minus40minus2002040

0

10

20

30

40

50

x(t)

z(t)

y(t)

(a)

minus30

minus20

minus10

0

10

20

30

0 500 1000 1500 2000

tx(t)

(b)

minus40

minus30

minus20

minus10

10

20

30

0

0 500 1000 1500 2000

t

y(t)

(c)

0 500 1000 1500 2000

t

z(t)

0

5

10

15

20

25

30

35

40

45

(d)

Figure 6 Chaotic attractor of the distributed order fractional Chen system (22) with (11990211 11990212 11990221 11990222 11990231 11990232) = (095 01 095 01 095 01)

and (119886 119887 119888) = (35 3 28)

distributed order Chen system (22) for parameters(119886 119887 119888) = (35 3 28) and (119902

11 11990212 11990221 11990222 11990231 11990232) =

(095 01 095 01 095 01) From Figure 7 we cansee that the system (22) is asymptotically stable in theequilibrium 119909

1with parameters (119886 119887 119888) = (50 3 20)

and (11990211 11990212 11990221 11990222 11990231 11990232) = (025 085 025 095

025 075)

6 Conclusion

In this paper we introduced the nonlinear distributed orderfractional differential equations with respect to a nonnegativedensity function hence the asymptotical stability for suchsystems has been investigated In addition we presented thedistributed order fractional Chen system and then in two

The Scientific World Journal 11

02

015

01

005

0

0

minus02

minus02

minus04

minus04

minus06

minus08

0

02

x(t)

z(t)

y(t)

(a)

minus03

minus04

02

03

01

minus02

minus01

0

0 500 1000 1500 2000

t

x(t)

(b)

02

01

0 500 1000 1500 2000

t

minus025

minus015

minus005

0

005

015

025

minus01

minus02

y(t)

(c)

minus008

minus006

minus004

minus002

0

002

004

006

008

01

minus01

0 500 1000 1500 2000

t

z(t)

(d)

Figure 7 The equilibrium point 1199091of the distributed order fractional Chen system (22) with (119902

11 11990212 11990221 11990222 11990231 11990232) = (025 085 025

095 025 075) and (119886 119887 119888) = (50 3 20) is asymptotically stable

special cases the stability for the distributed order fractionalChen is discussed Numerical solutions were coincident withresults of Table 1 described in Section 4 All numerical resultsare obtained using MATLAB 78

References

[1] A A Kilbas H M Srivastava and J J Trujillo Theory andApplication of Fractional Differential Equations Elsevier NewYork NY USA 2006

[2] K B Oldham and J SpanierThe Fractional Calculus AcadmicPress New York NY USA 1974

[3] E Reyes-Melo J Martinez-Vega C Guerrero-Salazar andU Ortiz-Mendez ldquoApplication of fractional calculus to themodeling of dielectric relaxation phenomena in polymericmaterialsrdquo Journal of Applied Polymer Science vol 98 no 2 pp923ndash935 2005

[4] R Schumer D A Benson M M Meerschaert and SW Wheatcraft ldquoEulerian derivation of the fractional

12 The Scientific World Journal

advection-dispersion equationrdquo Journal of ContaminantHydrology vol 48 no 1-2 pp 69ndash88 2001

[5] B I Henry and S L Wearne ldquoExistence of turing instabilitiesin a two-species fractional reaction-diffusion systemrdquo SIAMJournal on Applied Mathematics vol 62 no 3 pp 870ndash8872002

[6] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportvol 339 no 1 pp 1ndash77 2000

[7] S Picozzi andB JWest ldquoFractional Langevinmodel ofmemoryin financialmarketsrdquo Physical Review E vol 66 no 4 Article ID046118 pp 0461181ndash04611812 2002

[8] A Ansari A Refahi Sheikhani and S Kordrostami ldquoOn thegenerating function 119890119909119905+119910120593(119905) and its fractional calculusrdquo CentralEuropean Journal of Physics 2013

[9] A Ansari A Refahi Sheikhani and H Saberi Najafi ldquoSolutionto system of partial fractional differential equations using thefractional exponential operatorsrdquoMathematical Methods in theApplied Sciences vol 35 no 1 pp 119ndash123 2012

[10] D Matignon ldquoStability results for fractional differential equa-tions with applications to control processingrdquo in Proceedingsof the Computational Engineering in Systems and ApplicationMulticonference (IMACS IEEE-SMC rsquo96) vol 2 pp 963ndash968Lille France 1996

[11] M S Tavazoei and M Haeri ldquoA note on the stability offractional order systemsrdquoMathematics and Computers in Simu-lation vol 79 no 5 pp 1566ndash1576 2009

[12] V Daftardar-Gejji and A Babakhani ldquoAnalysis of a systemof fractional differential equationsrdquo Journal of MathematicalAnalysis and Applications vol 293 no 2 pp 511ndash522 2004

[13] K Diethelm and N J Ford ldquoAnalysis of fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 265 no 2 pp 229ndash248 2002

[14] J Sabatier M Moze and C Farges ldquoLMI stability conditionsfor fractional order systemsrdquo Computers and Mathematics withApplications vol 59 no 5 pp 1594ndash1609 2010

[15] Y Li Y Chen and I Podlubny ldquoMittag-Leffler stability offractional order nonlinear dynamic systemsrdquo Automatica vol45 no 8 pp 1965ndash1969 2009

[16] Y Li Y Chen and I Podlubny ldquoStability of fractional-ordernonlinear dynamic systems lyapunov directmethod and gener-alizedMittag-Leffler stabilityrdquoComputers andMathematics withApplications vol 59 no 5 pp 1810ndash1821 2010

[17] M Lakshmanan and K Murali Chaos in Nonlinear OscillatorsControlling and Synchronization World Scientific Singapore1996

[18] S KHan C Kurrer andY Kuramoto ldquoDephasing and burstingin coupled neural oscillatorsrdquo Physical Review Letters vol 75no 17 pp 3190ndash3193 1995

[19] B Blasius A Huppert and L Stone ldquoComplex dynamics andphase synchronization in spatially extended ecological systemsrdquoNature vol 399 no 6734 pp 354ndash359 1999

[20] C Sparrow The Lorenz Equations Bifurcations Chaos andStrange Attractors Springer New York NY USA 1982

[21] K T Alligood T D Sauer and J A Yorke Chaos An Intro-duction to Dynamical Systems Springer New York NY USA2008

[22] G Chen and T Ueta ldquoYet another chaotic attractorrdquo Interna-tional Journal of Bifurcation and Chaos vol 9 no 7 pp 1465ndash1466 1999

[23] T Ueta and G Chen ldquoBifurcation analysis of Chenrsquos equationrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 10 no 8 pp 1917ndash1931 2000

[24] I Grigorenko and E Grigorenko ldquoChaotic dynamics of thefractional Lorenz systemrdquo Physical Review Letters vol 91 no3 pp 0341011ndash0341014 2003

[25] C P Li andG J Peng ldquoChaos in Chenrsquos systemwith a fractionalorderrdquo Chaos Solitons and Fractals vol 22 no 2 pp 443ndash4502004

[26] C G Li and G Chen ldquoChaos in the fractional order Chensystem and its controlrdquo Chaos Solitons and Fractals vol 22 no3 pp 549ndash554 2004

[27] T T Hartley C F Lorenzo and H K Qammer ldquoChaos in afractional order Chuarsquos systemrdquo IEEE Transactions on Circuitsand Systems I vol 42 no 8 pp 485ndash490 1995

[28] C G Li and G Chen ldquoChaos and hyperchaos in the fractional-order Rossler equationsrdquo Physica A vol 341 no 1ndash4 pp 55ndash612004

[29] M Shahiri R Ghaderi A Ranjbar N S H Hosseinnia andS Momani ldquoChaotic fractional-order Coullet system synchro-nization and control approachrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 3 pp 665ndash6742010

[30] A E Matouk ldquoChaos feedback control and synchronization ofa fractional-order modified Autonomous Van der Pol-Duffingcircuitrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 2 pp 975ndash986 2011

[31] X YWang andM JWang ldquoDynamic analysis of the fractional-order Liu system and its synchronizationrdquo Chaos vol 17 no 3Article ID 033106 2007

[32] V Daftardar-Gejji S Bhalekar and P Gade ldquoDynamics offractional-ordered Chen system with delayrdquo Journal of Physicsvol 79 pp 61ndash69 2012

[33] M Caputo Elasticita e Dissipazione Zanichelli Bologna Italy1969

[34] M Caputo ldquoMean fractional-order-derivatives differentialequations and filtersrdquo Annali dellrsquoUniversita di Ferrara vol 41no 1 pp 73ndash84 1995

[35] M Caputo ldquoDistributed order differential equations modellingdielectric induction and diffusionrdquo Fractional Calculus andApplied Analysis vol 4 pp 421ndash442 2001

[36] L Bagley and P J Torvik ldquoOn the existence of the order domainand the solution of distributed order equationsrdquo InternationalJournal of Applied Mathematics vol 1 pp 865ndash882 2000

[37] R L Bagley and P J Torvik ldquoOn the existence of the orderdomain and the solution of distributed order equationsrdquo Inter-national Journal of Applied Mathematics vol 2 pp 965ndash9872000

[38] K Diethelm andN J Ford ldquoNumerical analysis for distributed-order differential equationsrdquo Journal of Computational andApplied Mathematics vol 225 no 1 pp 96ndash104 2009

[39] H Saberi Najafi A Refahi Sheikhani and A Ansari ldquoStabilityanalysis of distributed order fractional differential equationsrdquoAbstract and Applied Analysis vol 2011 Article ID 175323 12pages 2011

[40] A Refahi Sheikhani H Saberi Nadjafi A Ansari and FMehrdoust ldquoAnalytic study on linear system of distributed-order fractional differntional equationsrdquo Le Matematiche vol67 p 313 2012

The Scientific World Journal 13

[41] Z Vukic and L Kuljaca Nonlinear Control Systems CRC PressNew York NY USA 2003

[42] L Dorciak Numerical Models for Simulation the Fractional-Order Control Systems UEF-04-94 The Academy of SciencesInstitute of Experimental Physic Kosiice SlovakRepublic 1994

[43] B M Vinagre Y Q Chen and I Petras ldquoTwo direct Tustindiscretization methods for fractional-order differentiatorintegratorrdquo Journal of the Franklin Institute vol 340 no 5 pp349ndash362 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 The Scientific World Journal

02

015

01

005

0

02

02

0

0

minus02

minus02minus04

minus04minus06x(t)

z(t)

y(t)

(a)

0 500 1000 1500 2000

04

03

02

01

0

minus01

minus02

minus03

minus04

tx(t)

(b)

0 500 1000 1500 2000

t

06

04

02

0

minus02

minus04

minus06

y(t)

(c)Figure 2 The equilibrium point 119909

1of the distributed order fractional Chen system (22) with (119902

1 1199022 1199023) = (095 09 09) and (119886 119887 119888) =

(35 6 11) is asymptotically stable

function varies The results are shown in Table 1 for someparameters 119886 119887 and 119888

5 Numerical Methods

As pointed out in [38] distributed order fractional differen-tial equations may be regarded as a generalization of single-term fractional differential equations

119888

119863

120572

119905119909 (119905) = 119891 (119905 119909 (119905)) (26)

or multiterm fractional differential equations119899

sum

119894=1

120574119894

119888

119863

120572119894

119905119909 (119905) = 119891 (119905 119909 (119905)) 0 lt 120572

1lt 1205722lt sdot sdot sdot lt 120572

119899 (27)

Therefore in this section numerical method to solving of(26) and (27) is presented The approximate Grunwald-Letnikovrsquos definition is given below where the step size of ℎis assumed to be very small [2 42 43]

119888

119863

120572

119905119896

119910 (119905) asymp ℎminus120572

119896

sum

119894=0

119888(120572)

119894119910 (119905119896minus119894

) (28)

The Scientific World Journal 7

minus40

minus20

0

20

minus40

minus20

0

20

0

10

20

30

40

50

x(t)

z(t)

y(t)

(a)

0 500 1000 1500 2000

t

minus40

minus30

minus20

minus10

0

10

20

x(t)

(b)

minus40

minus30

minus20

minus10

0

10

20

0 500 1000 1500 2000

t

y(t)

(c)

0

5

10

15

20

25

30

35

40

45

50

0 500 1000 1500 2000

t

z(t)

(d)

Figure 3 The equilibrium point 1199092of the distributed order fractional Chen system (22) with (119902

1 1199022 1199023) = (07 08 09) and (119886 119887 119888) =

(30 3 28) is asymptotically stable

where 119905119896

= 119896ℎ (119896 = 0 1 ) and 119888(120572)

119894(119894 = 0 1 ) are

binomial coefficients which can be computed as [43]

119888(120572)

0= 1 119888

(120572)

119894= (1 minus

1 + 120572

119894

) 119888(120572)

119894minus1 (29)

Then general numerical solution of (26) and (27) can beexpressed as

ℎminus120572

119896

sum

119894=0

119888(120572)

119894119909 (119905119896minus119894

) = 119891 (119905 119909 (119905))

1205741ℎminus1205721

119896

sum

119894=0

119888(1205721)

119894119909 (119905119896minus119894

) + sdot sdot sdot + 120574119899ℎminus120572119899

119896

sum

119894=0

119888(120572119899)

119894119909 (119905119896minus119894

)

= 119891 (119905 119909 (119905))

(30)

where 119888(120572119895)119894

for 119895 = 1 119899 are binomial coefficients calculatedaccording to (29) Equations in (30) can be rewritten as thefollowing forms

8 The Scientific World Journal

z(t)

y(t)

x(t)

minus30 minus20 minus10 0 10 20

minus50

0

50

0

10

20

30

40

50

(a)

minus30

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

0 500 1000 1500 2000

tx(t)

(b)

0 500 1000 1500 2000

t

minus40

minus30

minus20

minus10

0

10

20

y(t)

(c)

0

5

10

15

20

25

30

35

40

45

0 500 1000 1500 2000

t

z(t)

(d)Figure 4 Chaotic attractor of the distributed order fractional Chen system (22) with (119902

1 1199022 1199023) = (085 09 095) and (119886 119887 119888) = (35 3 28)

119909 (119905119896) = 119891 (119909 (119905

119896minus1) 119905119896minus1

) ℎ120572

minus

119896

sum

119894=1

119888(120572)

119894119909 (119905119896minus119894

)

119909 (119905119896)

=

1

sum119899

119895=1119886119895ℎminus120572119895

times[

[

119891 (119909 (119905119896minus1

) 119905119896minus1

) minus

119899

sum

119895=1

120574119895ℎminus120572119895

119896

sum

119894=1

119888

(120572119895)

119894119909 (119905119896minus119894

)]

]

(31)

Based on the previous algorithmwe can obtain the numericalsolution of the fractional differential equations (26) andmultiterm fractional differential equations (27) where 119896 =

1 2 119873 for 119873 = 119879ℎ and where 119879 is the total time of thecalculation

To verify the efficiency of the obtained results in Table 1the numerical solution for the distributed order fractionalChen system has been computed In the following calcula-tions let 119879 = 10 ℎ = 0005 with the initial conditions(02 minus05 02)

The Scientific World Journal 9

x(t)

z(t)

y(t)

minus50

0

50

minus100minus50

050

0

10

20

30

40

50

60

(a)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

0 500 1000 1500 2000

tx(t)

(b)

minus80

minus60

minus40

minus20

0

20

40

0 500 1000 1500 2000

t

y(t)

(c)

0

10

20

30

40

50

60

z(t)

0 500 1000 1500 2000

t

(d)

Figure 5 Chaotic attractor of the distributed order fractional Chen system (22) with (11990211 11990212 11990221 11990222 11990231 11990232) = (085 03 085 03 085 03)

and (119886 119887 119888) = (35 3 29)

Figure 2 shows that the system (22) with parameters(119886 119887 119888) = (35 6 11) and (119902

1 1199022 1199023) = (095 09 09) is

asymptotically stable in the equilibrium 1199091 Figure 3 dem-

onstrates that the system (22) with parameters (119886 119887 119888) =

(30 3 28) and (1199021 1199022 1199023) = (07 08 09) is asymptotically

stable in the equilibrium 1199092 From Figure 4 we can see that

for the distributed order Chen system (21) with parameters(119886 119887 119888) = (35 3 28) and (119902

1 1199022 1199023) = (085 09 095)

is chaotic Figure 5 demonstrates that the system (22)with parameters (119886 119887 119888) = (35 3 29) and (119902

11 11990212 11990221

11990222 11990231 11990232) = (085 03 085 03 085 03) has chaotic

attractor Figure 6 shows the chaotic attractor for the

10 The Scientific World Journal

minus50

0

50

minus40minus2002040

0

10

20

30

40

50

x(t)

z(t)

y(t)

(a)

minus30

minus20

minus10

0

10

20

30

0 500 1000 1500 2000

tx(t)

(b)

minus40

minus30

minus20

minus10

10

20

30

0

0 500 1000 1500 2000

t

y(t)

(c)

0 500 1000 1500 2000

t

z(t)

0

5

10

15

20

25

30

35

40

45

(d)

Figure 6 Chaotic attractor of the distributed order fractional Chen system (22) with (11990211 11990212 11990221 11990222 11990231 11990232) = (095 01 095 01 095 01)

and (119886 119887 119888) = (35 3 28)

distributed order Chen system (22) for parameters(119886 119887 119888) = (35 3 28) and (119902

11 11990212 11990221 11990222 11990231 11990232) =

(095 01 095 01 095 01) From Figure 7 we cansee that the system (22) is asymptotically stable in theequilibrium 119909

1with parameters (119886 119887 119888) = (50 3 20)

and (11990211 11990212 11990221 11990222 11990231 11990232) = (025 085 025 095

025 075)

6 Conclusion

In this paper we introduced the nonlinear distributed orderfractional differential equations with respect to a nonnegativedensity function hence the asymptotical stability for suchsystems has been investigated In addition we presented thedistributed order fractional Chen system and then in two

The Scientific World Journal 11

02

015

01

005

0

0

minus02

minus02

minus04

minus04

minus06

minus08

0

02

x(t)

z(t)

y(t)

(a)

minus03

minus04

02

03

01

minus02

minus01

0

0 500 1000 1500 2000

t

x(t)

(b)

02

01

0 500 1000 1500 2000

t

minus025

minus015

minus005

0

005

015

025

minus01

minus02

y(t)

(c)

minus008

minus006

minus004

minus002

0

002

004

006

008

01

minus01

0 500 1000 1500 2000

t

z(t)

(d)

Figure 7 The equilibrium point 1199091of the distributed order fractional Chen system (22) with (119902

11 11990212 11990221 11990222 11990231 11990232) = (025 085 025

095 025 075) and (119886 119887 119888) = (50 3 20) is asymptotically stable

special cases the stability for the distributed order fractionalChen is discussed Numerical solutions were coincident withresults of Table 1 described in Section 4 All numerical resultsare obtained using MATLAB 78

References

[1] A A Kilbas H M Srivastava and J J Trujillo Theory andApplication of Fractional Differential Equations Elsevier NewYork NY USA 2006

[2] K B Oldham and J SpanierThe Fractional Calculus AcadmicPress New York NY USA 1974

[3] E Reyes-Melo J Martinez-Vega C Guerrero-Salazar andU Ortiz-Mendez ldquoApplication of fractional calculus to themodeling of dielectric relaxation phenomena in polymericmaterialsrdquo Journal of Applied Polymer Science vol 98 no 2 pp923ndash935 2005

[4] R Schumer D A Benson M M Meerschaert and SW Wheatcraft ldquoEulerian derivation of the fractional

12 The Scientific World Journal

advection-dispersion equationrdquo Journal of ContaminantHydrology vol 48 no 1-2 pp 69ndash88 2001

[5] B I Henry and S L Wearne ldquoExistence of turing instabilitiesin a two-species fractional reaction-diffusion systemrdquo SIAMJournal on Applied Mathematics vol 62 no 3 pp 870ndash8872002

[6] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportvol 339 no 1 pp 1ndash77 2000

[7] S Picozzi andB JWest ldquoFractional Langevinmodel ofmemoryin financialmarketsrdquo Physical Review E vol 66 no 4 Article ID046118 pp 0461181ndash04611812 2002

[8] A Ansari A Refahi Sheikhani and S Kordrostami ldquoOn thegenerating function 119890119909119905+119910120593(119905) and its fractional calculusrdquo CentralEuropean Journal of Physics 2013

[9] A Ansari A Refahi Sheikhani and H Saberi Najafi ldquoSolutionto system of partial fractional differential equations using thefractional exponential operatorsrdquoMathematical Methods in theApplied Sciences vol 35 no 1 pp 119ndash123 2012

[10] D Matignon ldquoStability results for fractional differential equa-tions with applications to control processingrdquo in Proceedingsof the Computational Engineering in Systems and ApplicationMulticonference (IMACS IEEE-SMC rsquo96) vol 2 pp 963ndash968Lille France 1996

[11] M S Tavazoei and M Haeri ldquoA note on the stability offractional order systemsrdquoMathematics and Computers in Simu-lation vol 79 no 5 pp 1566ndash1576 2009

[12] V Daftardar-Gejji and A Babakhani ldquoAnalysis of a systemof fractional differential equationsrdquo Journal of MathematicalAnalysis and Applications vol 293 no 2 pp 511ndash522 2004

[13] K Diethelm and N J Ford ldquoAnalysis of fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 265 no 2 pp 229ndash248 2002

[14] J Sabatier M Moze and C Farges ldquoLMI stability conditionsfor fractional order systemsrdquo Computers and Mathematics withApplications vol 59 no 5 pp 1594ndash1609 2010

[15] Y Li Y Chen and I Podlubny ldquoMittag-Leffler stability offractional order nonlinear dynamic systemsrdquo Automatica vol45 no 8 pp 1965ndash1969 2009

[16] Y Li Y Chen and I Podlubny ldquoStability of fractional-ordernonlinear dynamic systems lyapunov directmethod and gener-alizedMittag-Leffler stabilityrdquoComputers andMathematics withApplications vol 59 no 5 pp 1810ndash1821 2010

[17] M Lakshmanan and K Murali Chaos in Nonlinear OscillatorsControlling and Synchronization World Scientific Singapore1996

[18] S KHan C Kurrer andY Kuramoto ldquoDephasing and burstingin coupled neural oscillatorsrdquo Physical Review Letters vol 75no 17 pp 3190ndash3193 1995

[19] B Blasius A Huppert and L Stone ldquoComplex dynamics andphase synchronization in spatially extended ecological systemsrdquoNature vol 399 no 6734 pp 354ndash359 1999

[20] C Sparrow The Lorenz Equations Bifurcations Chaos andStrange Attractors Springer New York NY USA 1982

[21] K T Alligood T D Sauer and J A Yorke Chaos An Intro-duction to Dynamical Systems Springer New York NY USA2008

[22] G Chen and T Ueta ldquoYet another chaotic attractorrdquo Interna-tional Journal of Bifurcation and Chaos vol 9 no 7 pp 1465ndash1466 1999

[23] T Ueta and G Chen ldquoBifurcation analysis of Chenrsquos equationrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 10 no 8 pp 1917ndash1931 2000

[24] I Grigorenko and E Grigorenko ldquoChaotic dynamics of thefractional Lorenz systemrdquo Physical Review Letters vol 91 no3 pp 0341011ndash0341014 2003

[25] C P Li andG J Peng ldquoChaos in Chenrsquos systemwith a fractionalorderrdquo Chaos Solitons and Fractals vol 22 no 2 pp 443ndash4502004

[26] C G Li and G Chen ldquoChaos in the fractional order Chensystem and its controlrdquo Chaos Solitons and Fractals vol 22 no3 pp 549ndash554 2004

[27] T T Hartley C F Lorenzo and H K Qammer ldquoChaos in afractional order Chuarsquos systemrdquo IEEE Transactions on Circuitsand Systems I vol 42 no 8 pp 485ndash490 1995

[28] C G Li and G Chen ldquoChaos and hyperchaos in the fractional-order Rossler equationsrdquo Physica A vol 341 no 1ndash4 pp 55ndash612004

[29] M Shahiri R Ghaderi A Ranjbar N S H Hosseinnia andS Momani ldquoChaotic fractional-order Coullet system synchro-nization and control approachrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 3 pp 665ndash6742010

[30] A E Matouk ldquoChaos feedback control and synchronization ofa fractional-order modified Autonomous Van der Pol-Duffingcircuitrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 2 pp 975ndash986 2011

[31] X YWang andM JWang ldquoDynamic analysis of the fractional-order Liu system and its synchronizationrdquo Chaos vol 17 no 3Article ID 033106 2007

[32] V Daftardar-Gejji S Bhalekar and P Gade ldquoDynamics offractional-ordered Chen system with delayrdquo Journal of Physicsvol 79 pp 61ndash69 2012

[33] M Caputo Elasticita e Dissipazione Zanichelli Bologna Italy1969

[34] M Caputo ldquoMean fractional-order-derivatives differentialequations and filtersrdquo Annali dellrsquoUniversita di Ferrara vol 41no 1 pp 73ndash84 1995

[35] M Caputo ldquoDistributed order differential equations modellingdielectric induction and diffusionrdquo Fractional Calculus andApplied Analysis vol 4 pp 421ndash442 2001

[36] L Bagley and P J Torvik ldquoOn the existence of the order domainand the solution of distributed order equationsrdquo InternationalJournal of Applied Mathematics vol 1 pp 865ndash882 2000

[37] R L Bagley and P J Torvik ldquoOn the existence of the orderdomain and the solution of distributed order equationsrdquo Inter-national Journal of Applied Mathematics vol 2 pp 965ndash9872000

[38] K Diethelm andN J Ford ldquoNumerical analysis for distributed-order differential equationsrdquo Journal of Computational andApplied Mathematics vol 225 no 1 pp 96ndash104 2009

[39] H Saberi Najafi A Refahi Sheikhani and A Ansari ldquoStabilityanalysis of distributed order fractional differential equationsrdquoAbstract and Applied Analysis vol 2011 Article ID 175323 12pages 2011

[40] A Refahi Sheikhani H Saberi Nadjafi A Ansari and FMehrdoust ldquoAnalytic study on linear system of distributed-order fractional differntional equationsrdquo Le Matematiche vol67 p 313 2012

The Scientific World Journal 13

[41] Z Vukic and L Kuljaca Nonlinear Control Systems CRC PressNew York NY USA 2003

[42] L Dorciak Numerical Models for Simulation the Fractional-Order Control Systems UEF-04-94 The Academy of SciencesInstitute of Experimental Physic Kosiice SlovakRepublic 1994

[43] B M Vinagre Y Q Chen and I Petras ldquoTwo direct Tustindiscretization methods for fractional-order differentiatorintegratorrdquo Journal of the Franklin Institute vol 340 no 5 pp349ndash362 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 7

minus40

minus20

0

20

minus40

minus20

0

20

0

10

20

30

40

50

x(t)

z(t)

y(t)

(a)

0 500 1000 1500 2000

t

minus40

minus30

minus20

minus10

0

10

20

x(t)

(b)

minus40

minus30

minus20

minus10

0

10

20

0 500 1000 1500 2000

t

y(t)

(c)

0

5

10

15

20

25

30

35

40

45

50

0 500 1000 1500 2000

t

z(t)

(d)

Figure 3 The equilibrium point 1199092of the distributed order fractional Chen system (22) with (119902

1 1199022 1199023) = (07 08 09) and (119886 119887 119888) =

(30 3 28) is asymptotically stable

where 119905119896

= 119896ℎ (119896 = 0 1 ) and 119888(120572)

119894(119894 = 0 1 ) are

binomial coefficients which can be computed as [43]

119888(120572)

0= 1 119888

(120572)

119894= (1 minus

1 + 120572

119894

) 119888(120572)

119894minus1 (29)

Then general numerical solution of (26) and (27) can beexpressed as

ℎminus120572

119896

sum

119894=0

119888(120572)

119894119909 (119905119896minus119894

) = 119891 (119905 119909 (119905))

1205741ℎminus1205721

119896

sum

119894=0

119888(1205721)

119894119909 (119905119896minus119894

) + sdot sdot sdot + 120574119899ℎminus120572119899

119896

sum

119894=0

119888(120572119899)

119894119909 (119905119896minus119894

)

= 119891 (119905 119909 (119905))

(30)

where 119888(120572119895)119894

for 119895 = 1 119899 are binomial coefficients calculatedaccording to (29) Equations in (30) can be rewritten as thefollowing forms

8 The Scientific World Journal

z(t)

y(t)

x(t)

minus30 minus20 minus10 0 10 20

minus50

0

50

0

10

20

30

40

50

(a)

minus30

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

0 500 1000 1500 2000

tx(t)

(b)

0 500 1000 1500 2000

t

minus40

minus30

minus20

minus10

0

10

20

y(t)

(c)

0

5

10

15

20

25

30

35

40

45

0 500 1000 1500 2000

t

z(t)

(d)Figure 4 Chaotic attractor of the distributed order fractional Chen system (22) with (119902

1 1199022 1199023) = (085 09 095) and (119886 119887 119888) = (35 3 28)

119909 (119905119896) = 119891 (119909 (119905

119896minus1) 119905119896minus1

) ℎ120572

minus

119896

sum

119894=1

119888(120572)

119894119909 (119905119896minus119894

)

119909 (119905119896)

=

1

sum119899

119895=1119886119895ℎminus120572119895

times[

[

119891 (119909 (119905119896minus1

) 119905119896minus1

) minus

119899

sum

119895=1

120574119895ℎminus120572119895

119896

sum

119894=1

119888

(120572119895)

119894119909 (119905119896minus119894

)]

]

(31)

Based on the previous algorithmwe can obtain the numericalsolution of the fractional differential equations (26) andmultiterm fractional differential equations (27) where 119896 =

1 2 119873 for 119873 = 119879ℎ and where 119879 is the total time of thecalculation

To verify the efficiency of the obtained results in Table 1the numerical solution for the distributed order fractionalChen system has been computed In the following calcula-tions let 119879 = 10 ℎ = 0005 with the initial conditions(02 minus05 02)

The Scientific World Journal 9

x(t)

z(t)

y(t)

minus50

0

50

minus100minus50

050

0

10

20

30

40

50

60

(a)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

0 500 1000 1500 2000

tx(t)

(b)

minus80

minus60

minus40

minus20

0

20

40

0 500 1000 1500 2000

t

y(t)

(c)

0

10

20

30

40

50

60

z(t)

0 500 1000 1500 2000

t

(d)

Figure 5 Chaotic attractor of the distributed order fractional Chen system (22) with (11990211 11990212 11990221 11990222 11990231 11990232) = (085 03 085 03 085 03)

and (119886 119887 119888) = (35 3 29)

Figure 2 shows that the system (22) with parameters(119886 119887 119888) = (35 6 11) and (119902

1 1199022 1199023) = (095 09 09) is

asymptotically stable in the equilibrium 1199091 Figure 3 dem-

onstrates that the system (22) with parameters (119886 119887 119888) =

(30 3 28) and (1199021 1199022 1199023) = (07 08 09) is asymptotically

stable in the equilibrium 1199092 From Figure 4 we can see that

for the distributed order Chen system (21) with parameters(119886 119887 119888) = (35 3 28) and (119902

1 1199022 1199023) = (085 09 095)

is chaotic Figure 5 demonstrates that the system (22)with parameters (119886 119887 119888) = (35 3 29) and (119902

11 11990212 11990221

11990222 11990231 11990232) = (085 03 085 03 085 03) has chaotic

attractor Figure 6 shows the chaotic attractor for the

10 The Scientific World Journal

minus50

0

50

minus40minus2002040

0

10

20

30

40

50

x(t)

z(t)

y(t)

(a)

minus30

minus20

minus10

0

10

20

30

0 500 1000 1500 2000

tx(t)

(b)

minus40

minus30

minus20

minus10

10

20

30

0

0 500 1000 1500 2000

t

y(t)

(c)

0 500 1000 1500 2000

t

z(t)

0

5

10

15

20

25

30

35

40

45

(d)

Figure 6 Chaotic attractor of the distributed order fractional Chen system (22) with (11990211 11990212 11990221 11990222 11990231 11990232) = (095 01 095 01 095 01)

and (119886 119887 119888) = (35 3 28)

distributed order Chen system (22) for parameters(119886 119887 119888) = (35 3 28) and (119902

11 11990212 11990221 11990222 11990231 11990232) =

(095 01 095 01 095 01) From Figure 7 we cansee that the system (22) is asymptotically stable in theequilibrium 119909

1with parameters (119886 119887 119888) = (50 3 20)

and (11990211 11990212 11990221 11990222 11990231 11990232) = (025 085 025 095

025 075)

6 Conclusion

In this paper we introduced the nonlinear distributed orderfractional differential equations with respect to a nonnegativedensity function hence the asymptotical stability for suchsystems has been investigated In addition we presented thedistributed order fractional Chen system and then in two

The Scientific World Journal 11

02

015

01

005

0

0

minus02

minus02

minus04

minus04

minus06

minus08

0

02

x(t)

z(t)

y(t)

(a)

minus03

minus04

02

03

01

minus02

minus01

0

0 500 1000 1500 2000

t

x(t)

(b)

02

01

0 500 1000 1500 2000

t

minus025

minus015

minus005

0

005

015

025

minus01

minus02

y(t)

(c)

minus008

minus006

minus004

minus002

0

002

004

006

008

01

minus01

0 500 1000 1500 2000

t

z(t)

(d)

Figure 7 The equilibrium point 1199091of the distributed order fractional Chen system (22) with (119902

11 11990212 11990221 11990222 11990231 11990232) = (025 085 025

095 025 075) and (119886 119887 119888) = (50 3 20) is asymptotically stable

special cases the stability for the distributed order fractionalChen is discussed Numerical solutions were coincident withresults of Table 1 described in Section 4 All numerical resultsare obtained using MATLAB 78

References

[1] A A Kilbas H M Srivastava and J J Trujillo Theory andApplication of Fractional Differential Equations Elsevier NewYork NY USA 2006

[2] K B Oldham and J SpanierThe Fractional Calculus AcadmicPress New York NY USA 1974

[3] E Reyes-Melo J Martinez-Vega C Guerrero-Salazar andU Ortiz-Mendez ldquoApplication of fractional calculus to themodeling of dielectric relaxation phenomena in polymericmaterialsrdquo Journal of Applied Polymer Science vol 98 no 2 pp923ndash935 2005

[4] R Schumer D A Benson M M Meerschaert and SW Wheatcraft ldquoEulerian derivation of the fractional

12 The Scientific World Journal

advection-dispersion equationrdquo Journal of ContaminantHydrology vol 48 no 1-2 pp 69ndash88 2001

[5] B I Henry and S L Wearne ldquoExistence of turing instabilitiesin a two-species fractional reaction-diffusion systemrdquo SIAMJournal on Applied Mathematics vol 62 no 3 pp 870ndash8872002

[6] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportvol 339 no 1 pp 1ndash77 2000

[7] S Picozzi andB JWest ldquoFractional Langevinmodel ofmemoryin financialmarketsrdquo Physical Review E vol 66 no 4 Article ID046118 pp 0461181ndash04611812 2002

[8] A Ansari A Refahi Sheikhani and S Kordrostami ldquoOn thegenerating function 119890119909119905+119910120593(119905) and its fractional calculusrdquo CentralEuropean Journal of Physics 2013

[9] A Ansari A Refahi Sheikhani and H Saberi Najafi ldquoSolutionto system of partial fractional differential equations using thefractional exponential operatorsrdquoMathematical Methods in theApplied Sciences vol 35 no 1 pp 119ndash123 2012

[10] D Matignon ldquoStability results for fractional differential equa-tions with applications to control processingrdquo in Proceedingsof the Computational Engineering in Systems and ApplicationMulticonference (IMACS IEEE-SMC rsquo96) vol 2 pp 963ndash968Lille France 1996

[11] M S Tavazoei and M Haeri ldquoA note on the stability offractional order systemsrdquoMathematics and Computers in Simu-lation vol 79 no 5 pp 1566ndash1576 2009

[12] V Daftardar-Gejji and A Babakhani ldquoAnalysis of a systemof fractional differential equationsrdquo Journal of MathematicalAnalysis and Applications vol 293 no 2 pp 511ndash522 2004

[13] K Diethelm and N J Ford ldquoAnalysis of fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 265 no 2 pp 229ndash248 2002

[14] J Sabatier M Moze and C Farges ldquoLMI stability conditionsfor fractional order systemsrdquo Computers and Mathematics withApplications vol 59 no 5 pp 1594ndash1609 2010

[15] Y Li Y Chen and I Podlubny ldquoMittag-Leffler stability offractional order nonlinear dynamic systemsrdquo Automatica vol45 no 8 pp 1965ndash1969 2009

[16] Y Li Y Chen and I Podlubny ldquoStability of fractional-ordernonlinear dynamic systems lyapunov directmethod and gener-alizedMittag-Leffler stabilityrdquoComputers andMathematics withApplications vol 59 no 5 pp 1810ndash1821 2010

[17] M Lakshmanan and K Murali Chaos in Nonlinear OscillatorsControlling and Synchronization World Scientific Singapore1996

[18] S KHan C Kurrer andY Kuramoto ldquoDephasing and burstingin coupled neural oscillatorsrdquo Physical Review Letters vol 75no 17 pp 3190ndash3193 1995

[19] B Blasius A Huppert and L Stone ldquoComplex dynamics andphase synchronization in spatially extended ecological systemsrdquoNature vol 399 no 6734 pp 354ndash359 1999

[20] C Sparrow The Lorenz Equations Bifurcations Chaos andStrange Attractors Springer New York NY USA 1982

[21] K T Alligood T D Sauer and J A Yorke Chaos An Intro-duction to Dynamical Systems Springer New York NY USA2008

[22] G Chen and T Ueta ldquoYet another chaotic attractorrdquo Interna-tional Journal of Bifurcation and Chaos vol 9 no 7 pp 1465ndash1466 1999

[23] T Ueta and G Chen ldquoBifurcation analysis of Chenrsquos equationrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 10 no 8 pp 1917ndash1931 2000

[24] I Grigorenko and E Grigorenko ldquoChaotic dynamics of thefractional Lorenz systemrdquo Physical Review Letters vol 91 no3 pp 0341011ndash0341014 2003

[25] C P Li andG J Peng ldquoChaos in Chenrsquos systemwith a fractionalorderrdquo Chaos Solitons and Fractals vol 22 no 2 pp 443ndash4502004

[26] C G Li and G Chen ldquoChaos in the fractional order Chensystem and its controlrdquo Chaos Solitons and Fractals vol 22 no3 pp 549ndash554 2004

[27] T T Hartley C F Lorenzo and H K Qammer ldquoChaos in afractional order Chuarsquos systemrdquo IEEE Transactions on Circuitsand Systems I vol 42 no 8 pp 485ndash490 1995

[28] C G Li and G Chen ldquoChaos and hyperchaos in the fractional-order Rossler equationsrdquo Physica A vol 341 no 1ndash4 pp 55ndash612004

[29] M Shahiri R Ghaderi A Ranjbar N S H Hosseinnia andS Momani ldquoChaotic fractional-order Coullet system synchro-nization and control approachrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 3 pp 665ndash6742010

[30] A E Matouk ldquoChaos feedback control and synchronization ofa fractional-order modified Autonomous Van der Pol-Duffingcircuitrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 2 pp 975ndash986 2011

[31] X YWang andM JWang ldquoDynamic analysis of the fractional-order Liu system and its synchronizationrdquo Chaos vol 17 no 3Article ID 033106 2007

[32] V Daftardar-Gejji S Bhalekar and P Gade ldquoDynamics offractional-ordered Chen system with delayrdquo Journal of Physicsvol 79 pp 61ndash69 2012

[33] M Caputo Elasticita e Dissipazione Zanichelli Bologna Italy1969

[34] M Caputo ldquoMean fractional-order-derivatives differentialequations and filtersrdquo Annali dellrsquoUniversita di Ferrara vol 41no 1 pp 73ndash84 1995

[35] M Caputo ldquoDistributed order differential equations modellingdielectric induction and diffusionrdquo Fractional Calculus andApplied Analysis vol 4 pp 421ndash442 2001

[36] L Bagley and P J Torvik ldquoOn the existence of the order domainand the solution of distributed order equationsrdquo InternationalJournal of Applied Mathematics vol 1 pp 865ndash882 2000

[37] R L Bagley and P J Torvik ldquoOn the existence of the orderdomain and the solution of distributed order equationsrdquo Inter-national Journal of Applied Mathematics vol 2 pp 965ndash9872000

[38] K Diethelm andN J Ford ldquoNumerical analysis for distributed-order differential equationsrdquo Journal of Computational andApplied Mathematics vol 225 no 1 pp 96ndash104 2009

[39] H Saberi Najafi A Refahi Sheikhani and A Ansari ldquoStabilityanalysis of distributed order fractional differential equationsrdquoAbstract and Applied Analysis vol 2011 Article ID 175323 12pages 2011

[40] A Refahi Sheikhani H Saberi Nadjafi A Ansari and FMehrdoust ldquoAnalytic study on linear system of distributed-order fractional differntional equationsrdquo Le Matematiche vol67 p 313 2012

The Scientific World Journal 13

[41] Z Vukic and L Kuljaca Nonlinear Control Systems CRC PressNew York NY USA 2003

[42] L Dorciak Numerical Models for Simulation the Fractional-Order Control Systems UEF-04-94 The Academy of SciencesInstitute of Experimental Physic Kosiice SlovakRepublic 1994

[43] B M Vinagre Y Q Chen and I Petras ldquoTwo direct Tustindiscretization methods for fractional-order differentiatorintegratorrdquo Journal of the Franklin Institute vol 340 no 5 pp349ndash362 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 The Scientific World Journal

z(t)

y(t)

x(t)

minus30 minus20 minus10 0 10 20

minus50

0

50

0

10

20

30

40

50

(a)

minus30

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

0 500 1000 1500 2000

tx(t)

(b)

0 500 1000 1500 2000

t

minus40

minus30

minus20

minus10

0

10

20

y(t)

(c)

0

5

10

15

20

25

30

35

40

45

0 500 1000 1500 2000

t

z(t)

(d)Figure 4 Chaotic attractor of the distributed order fractional Chen system (22) with (119902

1 1199022 1199023) = (085 09 095) and (119886 119887 119888) = (35 3 28)

119909 (119905119896) = 119891 (119909 (119905

119896minus1) 119905119896minus1

) ℎ120572

minus

119896

sum

119894=1

119888(120572)

119894119909 (119905119896minus119894

)

119909 (119905119896)

=

1

sum119899

119895=1119886119895ℎminus120572119895

times[

[

119891 (119909 (119905119896minus1

) 119905119896minus1

) minus

119899

sum

119895=1

120574119895ℎminus120572119895

119896

sum

119894=1

119888

(120572119895)

119894119909 (119905119896minus119894

)]

]

(31)

Based on the previous algorithmwe can obtain the numericalsolution of the fractional differential equations (26) andmultiterm fractional differential equations (27) where 119896 =

1 2 119873 for 119873 = 119879ℎ and where 119879 is the total time of thecalculation

To verify the efficiency of the obtained results in Table 1the numerical solution for the distributed order fractionalChen system has been computed In the following calcula-tions let 119879 = 10 ℎ = 0005 with the initial conditions(02 minus05 02)

The Scientific World Journal 9

x(t)

z(t)

y(t)

minus50

0

50

minus100minus50

050

0

10

20

30

40

50

60

(a)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

0 500 1000 1500 2000

tx(t)

(b)

minus80

minus60

minus40

minus20

0

20

40

0 500 1000 1500 2000

t

y(t)

(c)

0

10

20

30

40

50

60

z(t)

0 500 1000 1500 2000

t

(d)

Figure 5 Chaotic attractor of the distributed order fractional Chen system (22) with (11990211 11990212 11990221 11990222 11990231 11990232) = (085 03 085 03 085 03)

and (119886 119887 119888) = (35 3 29)

Figure 2 shows that the system (22) with parameters(119886 119887 119888) = (35 6 11) and (119902

1 1199022 1199023) = (095 09 09) is

asymptotically stable in the equilibrium 1199091 Figure 3 dem-

onstrates that the system (22) with parameters (119886 119887 119888) =

(30 3 28) and (1199021 1199022 1199023) = (07 08 09) is asymptotically

stable in the equilibrium 1199092 From Figure 4 we can see that

for the distributed order Chen system (21) with parameters(119886 119887 119888) = (35 3 28) and (119902

1 1199022 1199023) = (085 09 095)

is chaotic Figure 5 demonstrates that the system (22)with parameters (119886 119887 119888) = (35 3 29) and (119902

11 11990212 11990221

11990222 11990231 11990232) = (085 03 085 03 085 03) has chaotic

attractor Figure 6 shows the chaotic attractor for the

10 The Scientific World Journal

minus50

0

50

minus40minus2002040

0

10

20

30

40

50

x(t)

z(t)

y(t)

(a)

minus30

minus20

minus10

0

10

20

30

0 500 1000 1500 2000

tx(t)

(b)

minus40

minus30

minus20

minus10

10

20

30

0

0 500 1000 1500 2000

t

y(t)

(c)

0 500 1000 1500 2000

t

z(t)

0

5

10

15

20

25

30

35

40

45

(d)

Figure 6 Chaotic attractor of the distributed order fractional Chen system (22) with (11990211 11990212 11990221 11990222 11990231 11990232) = (095 01 095 01 095 01)

and (119886 119887 119888) = (35 3 28)

distributed order Chen system (22) for parameters(119886 119887 119888) = (35 3 28) and (119902

11 11990212 11990221 11990222 11990231 11990232) =

(095 01 095 01 095 01) From Figure 7 we cansee that the system (22) is asymptotically stable in theequilibrium 119909

1with parameters (119886 119887 119888) = (50 3 20)

and (11990211 11990212 11990221 11990222 11990231 11990232) = (025 085 025 095

025 075)

6 Conclusion

In this paper we introduced the nonlinear distributed orderfractional differential equations with respect to a nonnegativedensity function hence the asymptotical stability for suchsystems has been investigated In addition we presented thedistributed order fractional Chen system and then in two

The Scientific World Journal 11

02

015

01

005

0

0

minus02

minus02

minus04

minus04

minus06

minus08

0

02

x(t)

z(t)

y(t)

(a)

minus03

minus04

02

03

01

minus02

minus01

0

0 500 1000 1500 2000

t

x(t)

(b)

02

01

0 500 1000 1500 2000

t

minus025

minus015

minus005

0

005

015

025

minus01

minus02

y(t)

(c)

minus008

minus006

minus004

minus002

0

002

004

006

008

01

minus01

0 500 1000 1500 2000

t

z(t)

(d)

Figure 7 The equilibrium point 1199091of the distributed order fractional Chen system (22) with (119902

11 11990212 11990221 11990222 11990231 11990232) = (025 085 025

095 025 075) and (119886 119887 119888) = (50 3 20) is asymptotically stable

special cases the stability for the distributed order fractionalChen is discussed Numerical solutions were coincident withresults of Table 1 described in Section 4 All numerical resultsare obtained using MATLAB 78

References

[1] A A Kilbas H M Srivastava and J J Trujillo Theory andApplication of Fractional Differential Equations Elsevier NewYork NY USA 2006

[2] K B Oldham and J SpanierThe Fractional Calculus AcadmicPress New York NY USA 1974

[3] E Reyes-Melo J Martinez-Vega C Guerrero-Salazar andU Ortiz-Mendez ldquoApplication of fractional calculus to themodeling of dielectric relaxation phenomena in polymericmaterialsrdquo Journal of Applied Polymer Science vol 98 no 2 pp923ndash935 2005

[4] R Schumer D A Benson M M Meerschaert and SW Wheatcraft ldquoEulerian derivation of the fractional

12 The Scientific World Journal

advection-dispersion equationrdquo Journal of ContaminantHydrology vol 48 no 1-2 pp 69ndash88 2001

[5] B I Henry and S L Wearne ldquoExistence of turing instabilitiesin a two-species fractional reaction-diffusion systemrdquo SIAMJournal on Applied Mathematics vol 62 no 3 pp 870ndash8872002

[6] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportvol 339 no 1 pp 1ndash77 2000

[7] S Picozzi andB JWest ldquoFractional Langevinmodel ofmemoryin financialmarketsrdquo Physical Review E vol 66 no 4 Article ID046118 pp 0461181ndash04611812 2002

[8] A Ansari A Refahi Sheikhani and S Kordrostami ldquoOn thegenerating function 119890119909119905+119910120593(119905) and its fractional calculusrdquo CentralEuropean Journal of Physics 2013

[9] A Ansari A Refahi Sheikhani and H Saberi Najafi ldquoSolutionto system of partial fractional differential equations using thefractional exponential operatorsrdquoMathematical Methods in theApplied Sciences vol 35 no 1 pp 119ndash123 2012

[10] D Matignon ldquoStability results for fractional differential equa-tions with applications to control processingrdquo in Proceedingsof the Computational Engineering in Systems and ApplicationMulticonference (IMACS IEEE-SMC rsquo96) vol 2 pp 963ndash968Lille France 1996

[11] M S Tavazoei and M Haeri ldquoA note on the stability offractional order systemsrdquoMathematics and Computers in Simu-lation vol 79 no 5 pp 1566ndash1576 2009

[12] V Daftardar-Gejji and A Babakhani ldquoAnalysis of a systemof fractional differential equationsrdquo Journal of MathematicalAnalysis and Applications vol 293 no 2 pp 511ndash522 2004

[13] K Diethelm and N J Ford ldquoAnalysis of fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 265 no 2 pp 229ndash248 2002

[14] J Sabatier M Moze and C Farges ldquoLMI stability conditionsfor fractional order systemsrdquo Computers and Mathematics withApplications vol 59 no 5 pp 1594ndash1609 2010

[15] Y Li Y Chen and I Podlubny ldquoMittag-Leffler stability offractional order nonlinear dynamic systemsrdquo Automatica vol45 no 8 pp 1965ndash1969 2009

[16] Y Li Y Chen and I Podlubny ldquoStability of fractional-ordernonlinear dynamic systems lyapunov directmethod and gener-alizedMittag-Leffler stabilityrdquoComputers andMathematics withApplications vol 59 no 5 pp 1810ndash1821 2010

[17] M Lakshmanan and K Murali Chaos in Nonlinear OscillatorsControlling and Synchronization World Scientific Singapore1996

[18] S KHan C Kurrer andY Kuramoto ldquoDephasing and burstingin coupled neural oscillatorsrdquo Physical Review Letters vol 75no 17 pp 3190ndash3193 1995

[19] B Blasius A Huppert and L Stone ldquoComplex dynamics andphase synchronization in spatially extended ecological systemsrdquoNature vol 399 no 6734 pp 354ndash359 1999

[20] C Sparrow The Lorenz Equations Bifurcations Chaos andStrange Attractors Springer New York NY USA 1982

[21] K T Alligood T D Sauer and J A Yorke Chaos An Intro-duction to Dynamical Systems Springer New York NY USA2008

[22] G Chen and T Ueta ldquoYet another chaotic attractorrdquo Interna-tional Journal of Bifurcation and Chaos vol 9 no 7 pp 1465ndash1466 1999

[23] T Ueta and G Chen ldquoBifurcation analysis of Chenrsquos equationrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 10 no 8 pp 1917ndash1931 2000

[24] I Grigorenko and E Grigorenko ldquoChaotic dynamics of thefractional Lorenz systemrdquo Physical Review Letters vol 91 no3 pp 0341011ndash0341014 2003

[25] C P Li andG J Peng ldquoChaos in Chenrsquos systemwith a fractionalorderrdquo Chaos Solitons and Fractals vol 22 no 2 pp 443ndash4502004

[26] C G Li and G Chen ldquoChaos in the fractional order Chensystem and its controlrdquo Chaos Solitons and Fractals vol 22 no3 pp 549ndash554 2004

[27] T T Hartley C F Lorenzo and H K Qammer ldquoChaos in afractional order Chuarsquos systemrdquo IEEE Transactions on Circuitsand Systems I vol 42 no 8 pp 485ndash490 1995

[28] C G Li and G Chen ldquoChaos and hyperchaos in the fractional-order Rossler equationsrdquo Physica A vol 341 no 1ndash4 pp 55ndash612004

[29] M Shahiri R Ghaderi A Ranjbar N S H Hosseinnia andS Momani ldquoChaotic fractional-order Coullet system synchro-nization and control approachrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 3 pp 665ndash6742010

[30] A E Matouk ldquoChaos feedback control and synchronization ofa fractional-order modified Autonomous Van der Pol-Duffingcircuitrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 2 pp 975ndash986 2011

[31] X YWang andM JWang ldquoDynamic analysis of the fractional-order Liu system and its synchronizationrdquo Chaos vol 17 no 3Article ID 033106 2007

[32] V Daftardar-Gejji S Bhalekar and P Gade ldquoDynamics offractional-ordered Chen system with delayrdquo Journal of Physicsvol 79 pp 61ndash69 2012

[33] M Caputo Elasticita e Dissipazione Zanichelli Bologna Italy1969

[34] M Caputo ldquoMean fractional-order-derivatives differentialequations and filtersrdquo Annali dellrsquoUniversita di Ferrara vol 41no 1 pp 73ndash84 1995

[35] M Caputo ldquoDistributed order differential equations modellingdielectric induction and diffusionrdquo Fractional Calculus andApplied Analysis vol 4 pp 421ndash442 2001

[36] L Bagley and P J Torvik ldquoOn the existence of the order domainand the solution of distributed order equationsrdquo InternationalJournal of Applied Mathematics vol 1 pp 865ndash882 2000

[37] R L Bagley and P J Torvik ldquoOn the existence of the orderdomain and the solution of distributed order equationsrdquo Inter-national Journal of Applied Mathematics vol 2 pp 965ndash9872000

[38] K Diethelm andN J Ford ldquoNumerical analysis for distributed-order differential equationsrdquo Journal of Computational andApplied Mathematics vol 225 no 1 pp 96ndash104 2009

[39] H Saberi Najafi A Refahi Sheikhani and A Ansari ldquoStabilityanalysis of distributed order fractional differential equationsrdquoAbstract and Applied Analysis vol 2011 Article ID 175323 12pages 2011

[40] A Refahi Sheikhani H Saberi Nadjafi A Ansari and FMehrdoust ldquoAnalytic study on linear system of distributed-order fractional differntional equationsrdquo Le Matematiche vol67 p 313 2012

The Scientific World Journal 13

[41] Z Vukic and L Kuljaca Nonlinear Control Systems CRC PressNew York NY USA 2003

[42] L Dorciak Numerical Models for Simulation the Fractional-Order Control Systems UEF-04-94 The Academy of SciencesInstitute of Experimental Physic Kosiice SlovakRepublic 1994

[43] B M Vinagre Y Q Chen and I Petras ldquoTwo direct Tustindiscretization methods for fractional-order differentiatorintegratorrdquo Journal of the Franklin Institute vol 340 no 5 pp349ndash362 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 9

x(t)

z(t)

y(t)

minus50

0

50

minus100minus50

050

0

10

20

30

40

50

60

(a)

minus50

minus40

minus30

minus20

minus10

0

10

20

30

0 500 1000 1500 2000

tx(t)

(b)

minus80

minus60

minus40

minus20

0

20

40

0 500 1000 1500 2000

t

y(t)

(c)

0

10

20

30

40

50

60

z(t)

0 500 1000 1500 2000

t

(d)

Figure 5 Chaotic attractor of the distributed order fractional Chen system (22) with (11990211 11990212 11990221 11990222 11990231 11990232) = (085 03 085 03 085 03)

and (119886 119887 119888) = (35 3 29)

Figure 2 shows that the system (22) with parameters(119886 119887 119888) = (35 6 11) and (119902

1 1199022 1199023) = (095 09 09) is

asymptotically stable in the equilibrium 1199091 Figure 3 dem-

onstrates that the system (22) with parameters (119886 119887 119888) =

(30 3 28) and (1199021 1199022 1199023) = (07 08 09) is asymptotically

stable in the equilibrium 1199092 From Figure 4 we can see that

for the distributed order Chen system (21) with parameters(119886 119887 119888) = (35 3 28) and (119902

1 1199022 1199023) = (085 09 095)

is chaotic Figure 5 demonstrates that the system (22)with parameters (119886 119887 119888) = (35 3 29) and (119902

11 11990212 11990221

11990222 11990231 11990232) = (085 03 085 03 085 03) has chaotic

attractor Figure 6 shows the chaotic attractor for the

10 The Scientific World Journal

minus50

0

50

minus40minus2002040

0

10

20

30

40

50

x(t)

z(t)

y(t)

(a)

minus30

minus20

minus10

0

10

20

30

0 500 1000 1500 2000

tx(t)

(b)

minus40

minus30

minus20

minus10

10

20

30

0

0 500 1000 1500 2000

t

y(t)

(c)

0 500 1000 1500 2000

t

z(t)

0

5

10

15

20

25

30

35

40

45

(d)

Figure 6 Chaotic attractor of the distributed order fractional Chen system (22) with (11990211 11990212 11990221 11990222 11990231 11990232) = (095 01 095 01 095 01)

and (119886 119887 119888) = (35 3 28)

distributed order Chen system (22) for parameters(119886 119887 119888) = (35 3 28) and (119902

11 11990212 11990221 11990222 11990231 11990232) =

(095 01 095 01 095 01) From Figure 7 we cansee that the system (22) is asymptotically stable in theequilibrium 119909

1with parameters (119886 119887 119888) = (50 3 20)

and (11990211 11990212 11990221 11990222 11990231 11990232) = (025 085 025 095

025 075)

6 Conclusion

In this paper we introduced the nonlinear distributed orderfractional differential equations with respect to a nonnegativedensity function hence the asymptotical stability for suchsystems has been investigated In addition we presented thedistributed order fractional Chen system and then in two

The Scientific World Journal 11

02

015

01

005

0

0

minus02

minus02

minus04

minus04

minus06

minus08

0

02

x(t)

z(t)

y(t)

(a)

minus03

minus04

02

03

01

minus02

minus01

0

0 500 1000 1500 2000

t

x(t)

(b)

02

01

0 500 1000 1500 2000

t

minus025

minus015

minus005

0

005

015

025

minus01

minus02

y(t)

(c)

minus008

minus006

minus004

minus002

0

002

004

006

008

01

minus01

0 500 1000 1500 2000

t

z(t)

(d)

Figure 7 The equilibrium point 1199091of the distributed order fractional Chen system (22) with (119902

11 11990212 11990221 11990222 11990231 11990232) = (025 085 025

095 025 075) and (119886 119887 119888) = (50 3 20) is asymptotically stable

special cases the stability for the distributed order fractionalChen is discussed Numerical solutions were coincident withresults of Table 1 described in Section 4 All numerical resultsare obtained using MATLAB 78

References

[1] A A Kilbas H M Srivastava and J J Trujillo Theory andApplication of Fractional Differential Equations Elsevier NewYork NY USA 2006

[2] K B Oldham and J SpanierThe Fractional Calculus AcadmicPress New York NY USA 1974

[3] E Reyes-Melo J Martinez-Vega C Guerrero-Salazar andU Ortiz-Mendez ldquoApplication of fractional calculus to themodeling of dielectric relaxation phenomena in polymericmaterialsrdquo Journal of Applied Polymer Science vol 98 no 2 pp923ndash935 2005

[4] R Schumer D A Benson M M Meerschaert and SW Wheatcraft ldquoEulerian derivation of the fractional

12 The Scientific World Journal

advection-dispersion equationrdquo Journal of ContaminantHydrology vol 48 no 1-2 pp 69ndash88 2001

[5] B I Henry and S L Wearne ldquoExistence of turing instabilitiesin a two-species fractional reaction-diffusion systemrdquo SIAMJournal on Applied Mathematics vol 62 no 3 pp 870ndash8872002

[6] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportvol 339 no 1 pp 1ndash77 2000

[7] S Picozzi andB JWest ldquoFractional Langevinmodel ofmemoryin financialmarketsrdquo Physical Review E vol 66 no 4 Article ID046118 pp 0461181ndash04611812 2002

[8] A Ansari A Refahi Sheikhani and S Kordrostami ldquoOn thegenerating function 119890119909119905+119910120593(119905) and its fractional calculusrdquo CentralEuropean Journal of Physics 2013

[9] A Ansari A Refahi Sheikhani and H Saberi Najafi ldquoSolutionto system of partial fractional differential equations using thefractional exponential operatorsrdquoMathematical Methods in theApplied Sciences vol 35 no 1 pp 119ndash123 2012

[10] D Matignon ldquoStability results for fractional differential equa-tions with applications to control processingrdquo in Proceedingsof the Computational Engineering in Systems and ApplicationMulticonference (IMACS IEEE-SMC rsquo96) vol 2 pp 963ndash968Lille France 1996

[11] M S Tavazoei and M Haeri ldquoA note on the stability offractional order systemsrdquoMathematics and Computers in Simu-lation vol 79 no 5 pp 1566ndash1576 2009

[12] V Daftardar-Gejji and A Babakhani ldquoAnalysis of a systemof fractional differential equationsrdquo Journal of MathematicalAnalysis and Applications vol 293 no 2 pp 511ndash522 2004

[13] K Diethelm and N J Ford ldquoAnalysis of fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 265 no 2 pp 229ndash248 2002

[14] J Sabatier M Moze and C Farges ldquoLMI stability conditionsfor fractional order systemsrdquo Computers and Mathematics withApplications vol 59 no 5 pp 1594ndash1609 2010

[15] Y Li Y Chen and I Podlubny ldquoMittag-Leffler stability offractional order nonlinear dynamic systemsrdquo Automatica vol45 no 8 pp 1965ndash1969 2009

[16] Y Li Y Chen and I Podlubny ldquoStability of fractional-ordernonlinear dynamic systems lyapunov directmethod and gener-alizedMittag-Leffler stabilityrdquoComputers andMathematics withApplications vol 59 no 5 pp 1810ndash1821 2010

[17] M Lakshmanan and K Murali Chaos in Nonlinear OscillatorsControlling and Synchronization World Scientific Singapore1996

[18] S KHan C Kurrer andY Kuramoto ldquoDephasing and burstingin coupled neural oscillatorsrdquo Physical Review Letters vol 75no 17 pp 3190ndash3193 1995

[19] B Blasius A Huppert and L Stone ldquoComplex dynamics andphase synchronization in spatially extended ecological systemsrdquoNature vol 399 no 6734 pp 354ndash359 1999

[20] C Sparrow The Lorenz Equations Bifurcations Chaos andStrange Attractors Springer New York NY USA 1982

[21] K T Alligood T D Sauer and J A Yorke Chaos An Intro-duction to Dynamical Systems Springer New York NY USA2008

[22] G Chen and T Ueta ldquoYet another chaotic attractorrdquo Interna-tional Journal of Bifurcation and Chaos vol 9 no 7 pp 1465ndash1466 1999

[23] T Ueta and G Chen ldquoBifurcation analysis of Chenrsquos equationrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 10 no 8 pp 1917ndash1931 2000

[24] I Grigorenko and E Grigorenko ldquoChaotic dynamics of thefractional Lorenz systemrdquo Physical Review Letters vol 91 no3 pp 0341011ndash0341014 2003

[25] C P Li andG J Peng ldquoChaos in Chenrsquos systemwith a fractionalorderrdquo Chaos Solitons and Fractals vol 22 no 2 pp 443ndash4502004

[26] C G Li and G Chen ldquoChaos in the fractional order Chensystem and its controlrdquo Chaos Solitons and Fractals vol 22 no3 pp 549ndash554 2004

[27] T T Hartley C F Lorenzo and H K Qammer ldquoChaos in afractional order Chuarsquos systemrdquo IEEE Transactions on Circuitsand Systems I vol 42 no 8 pp 485ndash490 1995

[28] C G Li and G Chen ldquoChaos and hyperchaos in the fractional-order Rossler equationsrdquo Physica A vol 341 no 1ndash4 pp 55ndash612004

[29] M Shahiri R Ghaderi A Ranjbar N S H Hosseinnia andS Momani ldquoChaotic fractional-order Coullet system synchro-nization and control approachrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 3 pp 665ndash6742010

[30] A E Matouk ldquoChaos feedback control and synchronization ofa fractional-order modified Autonomous Van der Pol-Duffingcircuitrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 2 pp 975ndash986 2011

[31] X YWang andM JWang ldquoDynamic analysis of the fractional-order Liu system and its synchronizationrdquo Chaos vol 17 no 3Article ID 033106 2007

[32] V Daftardar-Gejji S Bhalekar and P Gade ldquoDynamics offractional-ordered Chen system with delayrdquo Journal of Physicsvol 79 pp 61ndash69 2012

[33] M Caputo Elasticita e Dissipazione Zanichelli Bologna Italy1969

[34] M Caputo ldquoMean fractional-order-derivatives differentialequations and filtersrdquo Annali dellrsquoUniversita di Ferrara vol 41no 1 pp 73ndash84 1995

[35] M Caputo ldquoDistributed order differential equations modellingdielectric induction and diffusionrdquo Fractional Calculus andApplied Analysis vol 4 pp 421ndash442 2001

[36] L Bagley and P J Torvik ldquoOn the existence of the order domainand the solution of distributed order equationsrdquo InternationalJournal of Applied Mathematics vol 1 pp 865ndash882 2000

[37] R L Bagley and P J Torvik ldquoOn the existence of the orderdomain and the solution of distributed order equationsrdquo Inter-national Journal of Applied Mathematics vol 2 pp 965ndash9872000

[38] K Diethelm andN J Ford ldquoNumerical analysis for distributed-order differential equationsrdquo Journal of Computational andApplied Mathematics vol 225 no 1 pp 96ndash104 2009

[39] H Saberi Najafi A Refahi Sheikhani and A Ansari ldquoStabilityanalysis of distributed order fractional differential equationsrdquoAbstract and Applied Analysis vol 2011 Article ID 175323 12pages 2011

[40] A Refahi Sheikhani H Saberi Nadjafi A Ansari and FMehrdoust ldquoAnalytic study on linear system of distributed-order fractional differntional equationsrdquo Le Matematiche vol67 p 313 2012

The Scientific World Journal 13

[41] Z Vukic and L Kuljaca Nonlinear Control Systems CRC PressNew York NY USA 2003

[42] L Dorciak Numerical Models for Simulation the Fractional-Order Control Systems UEF-04-94 The Academy of SciencesInstitute of Experimental Physic Kosiice SlovakRepublic 1994

[43] B M Vinagre Y Q Chen and I Petras ldquoTwo direct Tustindiscretization methods for fractional-order differentiatorintegratorrdquo Journal of the Franklin Institute vol 340 no 5 pp349ndash362 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 The Scientific World Journal

minus50

0

50

minus40minus2002040

0

10

20

30

40

50

x(t)

z(t)

y(t)

(a)

minus30

minus20

minus10

0

10

20

30

0 500 1000 1500 2000

tx(t)

(b)

minus40

minus30

minus20

minus10

10

20

30

0

0 500 1000 1500 2000

t

y(t)

(c)

0 500 1000 1500 2000

t

z(t)

0

5

10

15

20

25

30

35

40

45

(d)

Figure 6 Chaotic attractor of the distributed order fractional Chen system (22) with (11990211 11990212 11990221 11990222 11990231 11990232) = (095 01 095 01 095 01)

and (119886 119887 119888) = (35 3 28)

distributed order Chen system (22) for parameters(119886 119887 119888) = (35 3 28) and (119902

11 11990212 11990221 11990222 11990231 11990232) =

(095 01 095 01 095 01) From Figure 7 we cansee that the system (22) is asymptotically stable in theequilibrium 119909

1with parameters (119886 119887 119888) = (50 3 20)

and (11990211 11990212 11990221 11990222 11990231 11990232) = (025 085 025 095

025 075)

6 Conclusion

In this paper we introduced the nonlinear distributed orderfractional differential equations with respect to a nonnegativedensity function hence the asymptotical stability for suchsystems has been investigated In addition we presented thedistributed order fractional Chen system and then in two

The Scientific World Journal 11

02

015

01

005

0

0

minus02

minus02

minus04

minus04

minus06

minus08

0

02

x(t)

z(t)

y(t)

(a)

minus03

minus04

02

03

01

minus02

minus01

0

0 500 1000 1500 2000

t

x(t)

(b)

02

01

0 500 1000 1500 2000

t

minus025

minus015

minus005

0

005

015

025

minus01

minus02

y(t)

(c)

minus008

minus006

minus004

minus002

0

002

004

006

008

01

minus01

0 500 1000 1500 2000

t

z(t)

(d)

Figure 7 The equilibrium point 1199091of the distributed order fractional Chen system (22) with (119902

11 11990212 11990221 11990222 11990231 11990232) = (025 085 025

095 025 075) and (119886 119887 119888) = (50 3 20) is asymptotically stable

special cases the stability for the distributed order fractionalChen is discussed Numerical solutions were coincident withresults of Table 1 described in Section 4 All numerical resultsare obtained using MATLAB 78

References

[1] A A Kilbas H M Srivastava and J J Trujillo Theory andApplication of Fractional Differential Equations Elsevier NewYork NY USA 2006

[2] K B Oldham and J SpanierThe Fractional Calculus AcadmicPress New York NY USA 1974

[3] E Reyes-Melo J Martinez-Vega C Guerrero-Salazar andU Ortiz-Mendez ldquoApplication of fractional calculus to themodeling of dielectric relaxation phenomena in polymericmaterialsrdquo Journal of Applied Polymer Science vol 98 no 2 pp923ndash935 2005

[4] R Schumer D A Benson M M Meerschaert and SW Wheatcraft ldquoEulerian derivation of the fractional

12 The Scientific World Journal

advection-dispersion equationrdquo Journal of ContaminantHydrology vol 48 no 1-2 pp 69ndash88 2001

[5] B I Henry and S L Wearne ldquoExistence of turing instabilitiesin a two-species fractional reaction-diffusion systemrdquo SIAMJournal on Applied Mathematics vol 62 no 3 pp 870ndash8872002

[6] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportvol 339 no 1 pp 1ndash77 2000

[7] S Picozzi andB JWest ldquoFractional Langevinmodel ofmemoryin financialmarketsrdquo Physical Review E vol 66 no 4 Article ID046118 pp 0461181ndash04611812 2002

[8] A Ansari A Refahi Sheikhani and S Kordrostami ldquoOn thegenerating function 119890119909119905+119910120593(119905) and its fractional calculusrdquo CentralEuropean Journal of Physics 2013

[9] A Ansari A Refahi Sheikhani and H Saberi Najafi ldquoSolutionto system of partial fractional differential equations using thefractional exponential operatorsrdquoMathematical Methods in theApplied Sciences vol 35 no 1 pp 119ndash123 2012

[10] D Matignon ldquoStability results for fractional differential equa-tions with applications to control processingrdquo in Proceedingsof the Computational Engineering in Systems and ApplicationMulticonference (IMACS IEEE-SMC rsquo96) vol 2 pp 963ndash968Lille France 1996

[11] M S Tavazoei and M Haeri ldquoA note on the stability offractional order systemsrdquoMathematics and Computers in Simu-lation vol 79 no 5 pp 1566ndash1576 2009

[12] V Daftardar-Gejji and A Babakhani ldquoAnalysis of a systemof fractional differential equationsrdquo Journal of MathematicalAnalysis and Applications vol 293 no 2 pp 511ndash522 2004

[13] K Diethelm and N J Ford ldquoAnalysis of fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 265 no 2 pp 229ndash248 2002

[14] J Sabatier M Moze and C Farges ldquoLMI stability conditionsfor fractional order systemsrdquo Computers and Mathematics withApplications vol 59 no 5 pp 1594ndash1609 2010

[15] Y Li Y Chen and I Podlubny ldquoMittag-Leffler stability offractional order nonlinear dynamic systemsrdquo Automatica vol45 no 8 pp 1965ndash1969 2009

[16] Y Li Y Chen and I Podlubny ldquoStability of fractional-ordernonlinear dynamic systems lyapunov directmethod and gener-alizedMittag-Leffler stabilityrdquoComputers andMathematics withApplications vol 59 no 5 pp 1810ndash1821 2010

[17] M Lakshmanan and K Murali Chaos in Nonlinear OscillatorsControlling and Synchronization World Scientific Singapore1996

[18] S KHan C Kurrer andY Kuramoto ldquoDephasing and burstingin coupled neural oscillatorsrdquo Physical Review Letters vol 75no 17 pp 3190ndash3193 1995

[19] B Blasius A Huppert and L Stone ldquoComplex dynamics andphase synchronization in spatially extended ecological systemsrdquoNature vol 399 no 6734 pp 354ndash359 1999

[20] C Sparrow The Lorenz Equations Bifurcations Chaos andStrange Attractors Springer New York NY USA 1982

[21] K T Alligood T D Sauer and J A Yorke Chaos An Intro-duction to Dynamical Systems Springer New York NY USA2008

[22] G Chen and T Ueta ldquoYet another chaotic attractorrdquo Interna-tional Journal of Bifurcation and Chaos vol 9 no 7 pp 1465ndash1466 1999

[23] T Ueta and G Chen ldquoBifurcation analysis of Chenrsquos equationrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 10 no 8 pp 1917ndash1931 2000

[24] I Grigorenko and E Grigorenko ldquoChaotic dynamics of thefractional Lorenz systemrdquo Physical Review Letters vol 91 no3 pp 0341011ndash0341014 2003

[25] C P Li andG J Peng ldquoChaos in Chenrsquos systemwith a fractionalorderrdquo Chaos Solitons and Fractals vol 22 no 2 pp 443ndash4502004

[26] C G Li and G Chen ldquoChaos in the fractional order Chensystem and its controlrdquo Chaos Solitons and Fractals vol 22 no3 pp 549ndash554 2004

[27] T T Hartley C F Lorenzo and H K Qammer ldquoChaos in afractional order Chuarsquos systemrdquo IEEE Transactions on Circuitsand Systems I vol 42 no 8 pp 485ndash490 1995

[28] C G Li and G Chen ldquoChaos and hyperchaos in the fractional-order Rossler equationsrdquo Physica A vol 341 no 1ndash4 pp 55ndash612004

[29] M Shahiri R Ghaderi A Ranjbar N S H Hosseinnia andS Momani ldquoChaotic fractional-order Coullet system synchro-nization and control approachrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 3 pp 665ndash6742010

[30] A E Matouk ldquoChaos feedback control and synchronization ofa fractional-order modified Autonomous Van der Pol-Duffingcircuitrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 2 pp 975ndash986 2011

[31] X YWang andM JWang ldquoDynamic analysis of the fractional-order Liu system and its synchronizationrdquo Chaos vol 17 no 3Article ID 033106 2007

[32] V Daftardar-Gejji S Bhalekar and P Gade ldquoDynamics offractional-ordered Chen system with delayrdquo Journal of Physicsvol 79 pp 61ndash69 2012

[33] M Caputo Elasticita e Dissipazione Zanichelli Bologna Italy1969

[34] M Caputo ldquoMean fractional-order-derivatives differentialequations and filtersrdquo Annali dellrsquoUniversita di Ferrara vol 41no 1 pp 73ndash84 1995

[35] M Caputo ldquoDistributed order differential equations modellingdielectric induction and diffusionrdquo Fractional Calculus andApplied Analysis vol 4 pp 421ndash442 2001

[36] L Bagley and P J Torvik ldquoOn the existence of the order domainand the solution of distributed order equationsrdquo InternationalJournal of Applied Mathematics vol 1 pp 865ndash882 2000

[37] R L Bagley and P J Torvik ldquoOn the existence of the orderdomain and the solution of distributed order equationsrdquo Inter-national Journal of Applied Mathematics vol 2 pp 965ndash9872000

[38] K Diethelm andN J Ford ldquoNumerical analysis for distributed-order differential equationsrdquo Journal of Computational andApplied Mathematics vol 225 no 1 pp 96ndash104 2009

[39] H Saberi Najafi A Refahi Sheikhani and A Ansari ldquoStabilityanalysis of distributed order fractional differential equationsrdquoAbstract and Applied Analysis vol 2011 Article ID 175323 12pages 2011

[40] A Refahi Sheikhani H Saberi Nadjafi A Ansari and FMehrdoust ldquoAnalytic study on linear system of distributed-order fractional differntional equationsrdquo Le Matematiche vol67 p 313 2012

The Scientific World Journal 13

[41] Z Vukic and L Kuljaca Nonlinear Control Systems CRC PressNew York NY USA 2003

[42] L Dorciak Numerical Models for Simulation the Fractional-Order Control Systems UEF-04-94 The Academy of SciencesInstitute of Experimental Physic Kosiice SlovakRepublic 1994

[43] B M Vinagre Y Q Chen and I Petras ldquoTwo direct Tustindiscretization methods for fractional-order differentiatorintegratorrdquo Journal of the Franklin Institute vol 340 no 5 pp349ndash362 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 11

02

015

01

005

0

0

minus02

minus02

minus04

minus04

minus06

minus08

0

02

x(t)

z(t)

y(t)

(a)

minus03

minus04

02

03

01

minus02

minus01

0

0 500 1000 1500 2000

t

x(t)

(b)

02

01

0 500 1000 1500 2000

t

minus025

minus015

minus005

0

005

015

025

minus01

minus02

y(t)

(c)

minus008

minus006

minus004

minus002

0

002

004

006

008

01

minus01

0 500 1000 1500 2000

t

z(t)

(d)

Figure 7 The equilibrium point 1199091of the distributed order fractional Chen system (22) with (119902

11 11990212 11990221 11990222 11990231 11990232) = (025 085 025

095 025 075) and (119886 119887 119888) = (50 3 20) is asymptotically stable

special cases the stability for the distributed order fractionalChen is discussed Numerical solutions were coincident withresults of Table 1 described in Section 4 All numerical resultsare obtained using MATLAB 78

References

[1] A A Kilbas H M Srivastava and J J Trujillo Theory andApplication of Fractional Differential Equations Elsevier NewYork NY USA 2006

[2] K B Oldham and J SpanierThe Fractional Calculus AcadmicPress New York NY USA 1974

[3] E Reyes-Melo J Martinez-Vega C Guerrero-Salazar andU Ortiz-Mendez ldquoApplication of fractional calculus to themodeling of dielectric relaxation phenomena in polymericmaterialsrdquo Journal of Applied Polymer Science vol 98 no 2 pp923ndash935 2005

[4] R Schumer D A Benson M M Meerschaert and SW Wheatcraft ldquoEulerian derivation of the fractional

12 The Scientific World Journal

advection-dispersion equationrdquo Journal of ContaminantHydrology vol 48 no 1-2 pp 69ndash88 2001

[5] B I Henry and S L Wearne ldquoExistence of turing instabilitiesin a two-species fractional reaction-diffusion systemrdquo SIAMJournal on Applied Mathematics vol 62 no 3 pp 870ndash8872002

[6] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportvol 339 no 1 pp 1ndash77 2000

[7] S Picozzi andB JWest ldquoFractional Langevinmodel ofmemoryin financialmarketsrdquo Physical Review E vol 66 no 4 Article ID046118 pp 0461181ndash04611812 2002

[8] A Ansari A Refahi Sheikhani and S Kordrostami ldquoOn thegenerating function 119890119909119905+119910120593(119905) and its fractional calculusrdquo CentralEuropean Journal of Physics 2013

[9] A Ansari A Refahi Sheikhani and H Saberi Najafi ldquoSolutionto system of partial fractional differential equations using thefractional exponential operatorsrdquoMathematical Methods in theApplied Sciences vol 35 no 1 pp 119ndash123 2012

[10] D Matignon ldquoStability results for fractional differential equa-tions with applications to control processingrdquo in Proceedingsof the Computational Engineering in Systems and ApplicationMulticonference (IMACS IEEE-SMC rsquo96) vol 2 pp 963ndash968Lille France 1996

[11] M S Tavazoei and M Haeri ldquoA note on the stability offractional order systemsrdquoMathematics and Computers in Simu-lation vol 79 no 5 pp 1566ndash1576 2009

[12] V Daftardar-Gejji and A Babakhani ldquoAnalysis of a systemof fractional differential equationsrdquo Journal of MathematicalAnalysis and Applications vol 293 no 2 pp 511ndash522 2004

[13] K Diethelm and N J Ford ldquoAnalysis of fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 265 no 2 pp 229ndash248 2002

[14] J Sabatier M Moze and C Farges ldquoLMI stability conditionsfor fractional order systemsrdquo Computers and Mathematics withApplications vol 59 no 5 pp 1594ndash1609 2010

[15] Y Li Y Chen and I Podlubny ldquoMittag-Leffler stability offractional order nonlinear dynamic systemsrdquo Automatica vol45 no 8 pp 1965ndash1969 2009

[16] Y Li Y Chen and I Podlubny ldquoStability of fractional-ordernonlinear dynamic systems lyapunov directmethod and gener-alizedMittag-Leffler stabilityrdquoComputers andMathematics withApplications vol 59 no 5 pp 1810ndash1821 2010

[17] M Lakshmanan and K Murali Chaos in Nonlinear OscillatorsControlling and Synchronization World Scientific Singapore1996

[18] S KHan C Kurrer andY Kuramoto ldquoDephasing and burstingin coupled neural oscillatorsrdquo Physical Review Letters vol 75no 17 pp 3190ndash3193 1995

[19] B Blasius A Huppert and L Stone ldquoComplex dynamics andphase synchronization in spatially extended ecological systemsrdquoNature vol 399 no 6734 pp 354ndash359 1999

[20] C Sparrow The Lorenz Equations Bifurcations Chaos andStrange Attractors Springer New York NY USA 1982

[21] K T Alligood T D Sauer and J A Yorke Chaos An Intro-duction to Dynamical Systems Springer New York NY USA2008

[22] G Chen and T Ueta ldquoYet another chaotic attractorrdquo Interna-tional Journal of Bifurcation and Chaos vol 9 no 7 pp 1465ndash1466 1999

[23] T Ueta and G Chen ldquoBifurcation analysis of Chenrsquos equationrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 10 no 8 pp 1917ndash1931 2000

[24] I Grigorenko and E Grigorenko ldquoChaotic dynamics of thefractional Lorenz systemrdquo Physical Review Letters vol 91 no3 pp 0341011ndash0341014 2003

[25] C P Li andG J Peng ldquoChaos in Chenrsquos systemwith a fractionalorderrdquo Chaos Solitons and Fractals vol 22 no 2 pp 443ndash4502004

[26] C G Li and G Chen ldquoChaos in the fractional order Chensystem and its controlrdquo Chaos Solitons and Fractals vol 22 no3 pp 549ndash554 2004

[27] T T Hartley C F Lorenzo and H K Qammer ldquoChaos in afractional order Chuarsquos systemrdquo IEEE Transactions on Circuitsand Systems I vol 42 no 8 pp 485ndash490 1995

[28] C G Li and G Chen ldquoChaos and hyperchaos in the fractional-order Rossler equationsrdquo Physica A vol 341 no 1ndash4 pp 55ndash612004

[29] M Shahiri R Ghaderi A Ranjbar N S H Hosseinnia andS Momani ldquoChaotic fractional-order Coullet system synchro-nization and control approachrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 3 pp 665ndash6742010

[30] A E Matouk ldquoChaos feedback control and synchronization ofa fractional-order modified Autonomous Van der Pol-Duffingcircuitrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 2 pp 975ndash986 2011

[31] X YWang andM JWang ldquoDynamic analysis of the fractional-order Liu system and its synchronizationrdquo Chaos vol 17 no 3Article ID 033106 2007

[32] V Daftardar-Gejji S Bhalekar and P Gade ldquoDynamics offractional-ordered Chen system with delayrdquo Journal of Physicsvol 79 pp 61ndash69 2012

[33] M Caputo Elasticita e Dissipazione Zanichelli Bologna Italy1969

[34] M Caputo ldquoMean fractional-order-derivatives differentialequations and filtersrdquo Annali dellrsquoUniversita di Ferrara vol 41no 1 pp 73ndash84 1995

[35] M Caputo ldquoDistributed order differential equations modellingdielectric induction and diffusionrdquo Fractional Calculus andApplied Analysis vol 4 pp 421ndash442 2001

[36] L Bagley and P J Torvik ldquoOn the existence of the order domainand the solution of distributed order equationsrdquo InternationalJournal of Applied Mathematics vol 1 pp 865ndash882 2000

[37] R L Bagley and P J Torvik ldquoOn the existence of the orderdomain and the solution of distributed order equationsrdquo Inter-national Journal of Applied Mathematics vol 2 pp 965ndash9872000

[38] K Diethelm andN J Ford ldquoNumerical analysis for distributed-order differential equationsrdquo Journal of Computational andApplied Mathematics vol 225 no 1 pp 96ndash104 2009

[39] H Saberi Najafi A Refahi Sheikhani and A Ansari ldquoStabilityanalysis of distributed order fractional differential equationsrdquoAbstract and Applied Analysis vol 2011 Article ID 175323 12pages 2011

[40] A Refahi Sheikhani H Saberi Nadjafi A Ansari and FMehrdoust ldquoAnalytic study on linear system of distributed-order fractional differntional equationsrdquo Le Matematiche vol67 p 313 2012

The Scientific World Journal 13

[41] Z Vukic and L Kuljaca Nonlinear Control Systems CRC PressNew York NY USA 2003

[42] L Dorciak Numerical Models for Simulation the Fractional-Order Control Systems UEF-04-94 The Academy of SciencesInstitute of Experimental Physic Kosiice SlovakRepublic 1994

[43] B M Vinagre Y Q Chen and I Petras ldquoTwo direct Tustindiscretization methods for fractional-order differentiatorintegratorrdquo Journal of the Franklin Institute vol 340 no 5 pp349ndash362 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 The Scientific World Journal

advection-dispersion equationrdquo Journal of ContaminantHydrology vol 48 no 1-2 pp 69ndash88 2001

[5] B I Henry and S L Wearne ldquoExistence of turing instabilitiesin a two-species fractional reaction-diffusion systemrdquo SIAMJournal on Applied Mathematics vol 62 no 3 pp 870ndash8872002

[6] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportvol 339 no 1 pp 1ndash77 2000

[7] S Picozzi andB JWest ldquoFractional Langevinmodel ofmemoryin financialmarketsrdquo Physical Review E vol 66 no 4 Article ID046118 pp 0461181ndash04611812 2002

[8] A Ansari A Refahi Sheikhani and S Kordrostami ldquoOn thegenerating function 119890119909119905+119910120593(119905) and its fractional calculusrdquo CentralEuropean Journal of Physics 2013

[9] A Ansari A Refahi Sheikhani and H Saberi Najafi ldquoSolutionto system of partial fractional differential equations using thefractional exponential operatorsrdquoMathematical Methods in theApplied Sciences vol 35 no 1 pp 119ndash123 2012

[10] D Matignon ldquoStability results for fractional differential equa-tions with applications to control processingrdquo in Proceedingsof the Computational Engineering in Systems and ApplicationMulticonference (IMACS IEEE-SMC rsquo96) vol 2 pp 963ndash968Lille France 1996

[11] M S Tavazoei and M Haeri ldquoA note on the stability offractional order systemsrdquoMathematics and Computers in Simu-lation vol 79 no 5 pp 1566ndash1576 2009

[12] V Daftardar-Gejji and A Babakhani ldquoAnalysis of a systemof fractional differential equationsrdquo Journal of MathematicalAnalysis and Applications vol 293 no 2 pp 511ndash522 2004

[13] K Diethelm and N J Ford ldquoAnalysis of fractional differentialequationsrdquo Journal of Mathematical Analysis and Applicationsvol 265 no 2 pp 229ndash248 2002

[14] J Sabatier M Moze and C Farges ldquoLMI stability conditionsfor fractional order systemsrdquo Computers and Mathematics withApplications vol 59 no 5 pp 1594ndash1609 2010

[15] Y Li Y Chen and I Podlubny ldquoMittag-Leffler stability offractional order nonlinear dynamic systemsrdquo Automatica vol45 no 8 pp 1965ndash1969 2009

[16] Y Li Y Chen and I Podlubny ldquoStability of fractional-ordernonlinear dynamic systems lyapunov directmethod and gener-alizedMittag-Leffler stabilityrdquoComputers andMathematics withApplications vol 59 no 5 pp 1810ndash1821 2010

[17] M Lakshmanan and K Murali Chaos in Nonlinear OscillatorsControlling and Synchronization World Scientific Singapore1996

[18] S KHan C Kurrer andY Kuramoto ldquoDephasing and burstingin coupled neural oscillatorsrdquo Physical Review Letters vol 75no 17 pp 3190ndash3193 1995

[19] B Blasius A Huppert and L Stone ldquoComplex dynamics andphase synchronization in spatially extended ecological systemsrdquoNature vol 399 no 6734 pp 354ndash359 1999

[20] C Sparrow The Lorenz Equations Bifurcations Chaos andStrange Attractors Springer New York NY USA 1982

[21] K T Alligood T D Sauer and J A Yorke Chaos An Intro-duction to Dynamical Systems Springer New York NY USA2008

[22] G Chen and T Ueta ldquoYet another chaotic attractorrdquo Interna-tional Journal of Bifurcation and Chaos vol 9 no 7 pp 1465ndash1466 1999

[23] T Ueta and G Chen ldquoBifurcation analysis of Chenrsquos equationrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 10 no 8 pp 1917ndash1931 2000

[24] I Grigorenko and E Grigorenko ldquoChaotic dynamics of thefractional Lorenz systemrdquo Physical Review Letters vol 91 no3 pp 0341011ndash0341014 2003

[25] C P Li andG J Peng ldquoChaos in Chenrsquos systemwith a fractionalorderrdquo Chaos Solitons and Fractals vol 22 no 2 pp 443ndash4502004

[26] C G Li and G Chen ldquoChaos in the fractional order Chensystem and its controlrdquo Chaos Solitons and Fractals vol 22 no3 pp 549ndash554 2004

[27] T T Hartley C F Lorenzo and H K Qammer ldquoChaos in afractional order Chuarsquos systemrdquo IEEE Transactions on Circuitsand Systems I vol 42 no 8 pp 485ndash490 1995

[28] C G Li and G Chen ldquoChaos and hyperchaos in the fractional-order Rossler equationsrdquo Physica A vol 341 no 1ndash4 pp 55ndash612004

[29] M Shahiri R Ghaderi A Ranjbar N S H Hosseinnia andS Momani ldquoChaotic fractional-order Coullet system synchro-nization and control approachrdquo Communications in NonlinearScience and Numerical Simulation vol 15 no 3 pp 665ndash6742010

[30] A E Matouk ldquoChaos feedback control and synchronization ofa fractional-order modified Autonomous Van der Pol-Duffingcircuitrdquo Communications in Nonlinear Science and NumericalSimulation vol 16 no 2 pp 975ndash986 2011

[31] X YWang andM JWang ldquoDynamic analysis of the fractional-order Liu system and its synchronizationrdquo Chaos vol 17 no 3Article ID 033106 2007

[32] V Daftardar-Gejji S Bhalekar and P Gade ldquoDynamics offractional-ordered Chen system with delayrdquo Journal of Physicsvol 79 pp 61ndash69 2012

[33] M Caputo Elasticita e Dissipazione Zanichelli Bologna Italy1969

[34] M Caputo ldquoMean fractional-order-derivatives differentialequations and filtersrdquo Annali dellrsquoUniversita di Ferrara vol 41no 1 pp 73ndash84 1995

[35] M Caputo ldquoDistributed order differential equations modellingdielectric induction and diffusionrdquo Fractional Calculus andApplied Analysis vol 4 pp 421ndash442 2001

[36] L Bagley and P J Torvik ldquoOn the existence of the order domainand the solution of distributed order equationsrdquo InternationalJournal of Applied Mathematics vol 1 pp 865ndash882 2000

[37] R L Bagley and P J Torvik ldquoOn the existence of the orderdomain and the solution of distributed order equationsrdquo Inter-national Journal of Applied Mathematics vol 2 pp 965ndash9872000

[38] K Diethelm andN J Ford ldquoNumerical analysis for distributed-order differential equationsrdquo Journal of Computational andApplied Mathematics vol 225 no 1 pp 96ndash104 2009

[39] H Saberi Najafi A Refahi Sheikhani and A Ansari ldquoStabilityanalysis of distributed order fractional differential equationsrdquoAbstract and Applied Analysis vol 2011 Article ID 175323 12pages 2011

[40] A Refahi Sheikhani H Saberi Nadjafi A Ansari and FMehrdoust ldquoAnalytic study on linear system of distributed-order fractional differntional equationsrdquo Le Matematiche vol67 p 313 2012

The Scientific World Journal 13

[41] Z Vukic and L Kuljaca Nonlinear Control Systems CRC PressNew York NY USA 2003

[42] L Dorciak Numerical Models for Simulation the Fractional-Order Control Systems UEF-04-94 The Academy of SciencesInstitute of Experimental Physic Kosiice SlovakRepublic 1994

[43] B M Vinagre Y Q Chen and I Petras ldquoTwo direct Tustindiscretization methods for fractional-order differentiatorintegratorrdquo Journal of the Franklin Institute vol 340 no 5 pp349ndash362 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

The Scientific World Journal 13

[41] Z Vukic and L Kuljaca Nonlinear Control Systems CRC PressNew York NY USA 2003

[42] L Dorciak Numerical Models for Simulation the Fractional-Order Control Systems UEF-04-94 The Academy of SciencesInstitute of Experimental Physic Kosiice SlovakRepublic 1994

[43] B M Vinagre Y Q Chen and I Petras ldquoTwo direct Tustindiscretization methods for fractional-order differentiatorintegratorrdquo Journal of the Franklin Institute vol 340 no 5 pp349ndash362 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Stochastic AnalysisInternational Journal of