a new simple chaotic lorenz-type system and its digital...

Research ArticleA New Simple Chaotic Lorenz-Type System andIts Digital Realization Using a TFT Touch-ScreenDisplay Embedded System

Rodrigo Meacutendez-Ramiacuterez1 Adrian Arellano-Delgado2

Ceacutesar Cruz-Hernaacutendez1 and Rigoberto Martiacutenez-Clark1

1Electronics and Telecommunications Department Scientific Research and Advanced Studies Center of EnsenadaEnsenada BC Mexico2CONACYT-Autonomous Baja California University (UABC) Ensenada BC Mexico

Correspondence should be addressed to Cesar Cruz-Hernandez ccruzcicesemx

Received 14 February 2017 Revised 5 May 2017 Accepted 31 May 2017 Published 26 July 2017

Academic Editor Giacomo Innocenti

Copyright copy 2017 Rodrigo Mendez-Ramırez et al This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

This paper presents a new three-dimensional autonomous chaotic system The proposed system generates a chaotic attractor withthe variation of two parameters Analytical and numerical studies of the dynamic properties to generate chaos for continuousversion (CV) and discretized version (DV) for the new chaotic system (NCS)were conductedTheCVof theNCSwas implementedby using an electronic circuit with operational amplifiers (OAs) In addition the presence of chaos for DV of the NCS was provedby using the analytical and numerical degradation tests the time series was calculated to determine the behavior of Lyapunovexponents (LEs) Finally the DV of NCS was implemented in real-time by using a novel embedded system (ES) Mikromedia Plusfor PIC32MX7 that includes one microcontroller PIC32 and one thin film transistor touch-screen display (TFTTSD) together withexternal digital-to-analog converters (DACs)

1 Introduction

In recent years chaotic systems have attracted the attention ofthe scientific community due to their potential applicationsin several areas of science and engineering as an interestingnonlinear phenomenon with different applications in biol-ogy secure communication complex networks experimen-tal network synchronization fingerprint encryption amongothers [1ndash15] In 1963 Lorenz proposed a three-dimensionalsystem of two scrolls this is recognized as the first reportedchaotic model [16] Since then many chaotic systems likeLorenzmdashand their chaotic behaviormdashhave been reportedin the literature for example [17ndash23] Currently we canmention some new chaotic systems reported in the literature[24ndash33]

The chaotic systems usually are implemented by usingelectronic circuits in continuous (CV) and discretized ver-sions (DV) The CVs usually are represented by using OAs

[31 34] For DVs Matlab or Labview allows simulatingthe dynamical behaviors of discretized chaotic systems todesirably obtain the less degradationwith respect to CVs andtheir implementations are reproduced by using ESs as FPGAs[35] DSPs [36] or microcontrollers [37ndash39]

In this paper we propose a new simple chaotic systemwhich is derived from the Lorenz system [16] The novelty ofthe proposed chaotic system is the combination of differentcharacteristics that it presents two critical parameters onlytwo nonlinearities low complexity time low iterations persecond and a larger step size for the discretized versionwherechaos is preserved low-cost electronic implementation alsoit is flexible and robust with respect to some recent attractorsreported in the literature see for example [22 23] Asa consequence all these features result in a high ease ofimplementation that may be of great interest in engineer-ing applications for example in cryptography biometric

HindawiComplexityVolume 2017 Article ID 6820492 13 pageshttpsdoiorg10115520176820492

2 Complexity

systems telemedicine and secure communications see forexample [10 12 13 37] To the best of our knowledge theelectronical implementation in a portable TFTTSD device ofDVof chaotic systems for the reproduction of their nonlineardynamics in real-time is new

The paper is organized as follows Section 2 reports basicanalytical proof and extensive numerical tests to verify theexistence of chaos in the proposed NCS Section 3 presentstwo electronic implementations to reproduce the NCS thefirst by means of OAs and the second of a novel proposedES by using one TFTTSD and external DACs In additiona degradation study taking into account the preservationof chaos is conducted a comparison among the NCS andsome other chaotic systems reported in the literature ismade In Section 4 we give a complete description of digitalimplementation process of the DV of NSC with the corre-sponding robustness diagram to determine regions in whichthe existence of chaos is guaranteed Finally in Section 5 wedraw some concluding remarks

2 Basic Analysis and Characterization of NCS

This section presents the state equations of NCS and somenumerical and analytical tests to verify the chaos existencein the proposed system The NCS is built starting frominspecting and modification of Lorenz system [16] Theproposed NCS is described by

= minus119886119909 minus 119887119910119911119910 = minus119909 + 119888119910 = 119889 minus 1199102 minus 119911

(1)

The proposed system has seven terms two quadraticnonlinearities and four parameters 119886 119887 119888 119889 isin R+ where 119887and 119889 are characterized as the bifurcation parameters Thenonlinear NCS (1) is chaotic with 119886 = 2 119887 = 2 119888 = 05 and119889 = 4

System (1) is symmetrical about the 119911-axis due to itsinvariance under the coordinate transformation (119909 119910 119911) rarr(minus119909 minus119910 119911) The symmetry is not associated with the a b cand d parameters

The divergence for a 3-dimensional flow of dynamicalsystem is defined by

nabla119881 = 120597120597119909 + 120597 119910120597119910 + 120597120597119911 = minus119886 + 119888 minus 1 = minus25 lt 0 (2)

Therefore the above analysis proves that our system isdissipative The exponential contraction rate is calculated asfollows 119889119881119889119905 = (nabla119881)119881 997888rarr 119881 = 1198810119890minus25119905 (3)

where each volume containing the system trajectory shrinksto zero as 119905 rarr infin at an exponential rate ofminus25119905 Systemorbitsare ultimately confined into a specific limit set of zero volumeand the asymptotic motion settles onto an attractor Therebythe existence on attractor is proved

The boundness of the chaotic trajectories of system (1) isproved by means of the following theoremThe boundness ofa NCS by using similar approach was reported in [40 41]

Theorem 1 Suppose that the parameters 119886 119887 119888 and 119889 ofsystem (1) are positive Then the orbits of system (1) includingchaotic orbits are confined in a bounded region

Proof Consider the candidate Lyapunov function

119881 (119909 119910 119911) = 12 (1199092 + 1199102 + 1199112) (4)

and the time derivative of 119881(119909 119910 119911) along the trajectories ofthe NCS (1) is given by

(119909 119910 119911) = 119909 + 119910 119910 + 119911 (119909 119910 119911) = minus1198861199092 minus 119887119909119910119911 + 1198881199102 minus 119909119910 + 119889119911 minus 1199111199102 minus 1199112

= minus( 1198871199101199112radic119886 + radic119886119909)2 + ( 1199092radic119888 minus radic119888119910)2minus 11990924119888 minus (119911 minus 1198892)2 + 11988924+ ( 1198871199101199112radic119886 minus radic119886119910119887 )2 minus 11988611991021198872

(5)

Let 1198770 be the sufficiently large region so that for alltrajectories (119909 119910 119911) satisfy 119881(119909 119910 119911) = 119877 for 119877 gt 1198770 with thecondition

( 1198871199101199112radic119886 + radic119886119909)2 + (119911 minus 1198892)2 + 11988611991021198872 + 11990924119888gt ( 1199092radic119888 minus radic119888119910)2 + ( 1198871199101199112radic119886 minus radic119886119910119887 )2 + 11988924

(6)

Consequently on the surface (119909 119910 119911)119881(119909 119910 119911) = 119877Since 119877 gt 1198770 we can write (119909 119910 119911) lt 0 or the set(119909 119910 119911)119881(119909 119910 119911) le 119877 is a confined region for all thetrajectories of chaotic system (1)

The number of the equilibrium points and their stabilitiesdetermine the behavior of system (1) it can be found bysetting = 119910 = = 0 and 119886 119887 119888 119889 gt 0 The proposedsystem (1) has five fixed points 1198750(0 0 0) 1198751 1198752 11987531198754(plusmn119888radic119889 + 119886119888119887 plusmnradic119889 + 119886119888119887 minus119886119888119887) The Jacobean matrixof CV-system (1) is given by

119869CV = (minus119886 minus119887119911 minus119887119910minus1 119888 00 minus2119910 minus1 ) (7)

and the characteristic polynomial of (7) is as follows

det (120582119868 minus 119869) = 1205823 + (119886 minus 119888 + 1) 1205822 + (119886 minus 119888 minus 119886119888 minus 119887119911) 120582+ 21198871199102 minus 119886119888 minus 119887119911 = 0 (8)

Complexity 3

Table 1 Stability analysis equilibrium points for NCS

Point Eigenvalues Stability

1198750 1205821 = minus2 1205821 1205822 lt 0 and 1205823 gt 0 unstable saddle point1205822 = minus11205823 = 051198751 1198752 1198753 1198754 1205821 = minus352387 1205821 lt 0 and the real part of 1205822 1205823 gt 0 unstable saddle points1205822 = 051193 + 2201341198941205823 = 051193 + 220134119894

minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

x

1 15 2 25 3 35 4 45 505b

(a)

1 15 2 25 3 35 4 45 505d

minus20

minus15

minus10

minus5

0

5

10

15

20

25

x

(b)

Figure 1 Bifurcation diagrams of parameters 119887 and 119889 versus state 119909 using the initial conditions 1199090 = 1199100 = 1199110 = 1 of system (1) (a) variationsof 119887 = [05 5] and (b) variations of 119889 = [05 5]Evaluating with parameters 119886 = 2 b = 2 c = 05 and d =4 in (8) the stability in the equilibrium points 11987501234 wasstudied Table 1 shows the stability results of equilibria whereall points of NCS are saddle-focus unstable nodes

The bifurcation diagram is built to visualize the transi-tions between periodic and chaotic motions of the proposedsystem with the variation of the critical parameter b or dof NCS (1) for more details of the numerical algorithm toobtain the bifurcation diagram see [42] Figure 1(a) showsthe bifurcation diagram of system (1) where the parameters119886 = 2 119888 = 2 and 119889 = 4were fixed and 119887 is varied In additionFigure 1(b) shows the bifurcation diagramof system (1) wherethe parameters 119886 = 2 119887 = 2 and 119888 = 05 were fixed andd is varied From Figure 1 we conclude that system (1) is alsorobust because it has a large chaotic behavior for parameters 119887and 119889 to guarantee chaotic behaviors fixed point limit cycleand strange attractor

In this numerical study the parameter 119887 is fixed and theparameter 119889 is choice as bifurcation parameter The initialconditions 1199090 = 1199100 = 1199110 = 1 and parameters 119886 = 2 119887 = 2119888 = 05 and 119889 = 4 were chosen for all numerical andexperimental tests

To prove the presence of chaos on the NCS (1) the LEsare calculated using themethod reported in [43 44] Figure 2shows the results of the LEs where 10000 time units were

considered in the analysis Figure 2(a) shows the evolution ofLEs where the obtained results are 1198711 = 024914 1198712 = 0 and1198713 = minus27497 Figure 2(b) shows evolution of LEs consideringthe variation of bifurcation parameter 119889 = [05 5]

The fractal dimension commonly known as Kaplan-Yorke dimension119863KY of this system is

119863KY = 119895 + 110038161003816100381610038161003816119871119895+110038161003816100381610038161003816119895sum119894=1

119871 119894 = 2 + 1198711 + 1198712100381610038161003816100381611987131003816100381610038161003816 = 20908 (9)

The NCS exhibits complex and abundant dynamicsbehaviors see Figure 3 where chaotic attractors are shown

3 Electronic Implementations

This section presents two electronic implementations forsystem (1) (i) the CV was simulated and implemented withthe design of one circuit by using OAs and (ii) the DV wasimplemented with design of an ES where one degradationstudy of NCS is also given

31 Electronic CircuitDesign forCVofNCS For the electronicimplementation of CV-system (1) the attenuation factor of 20for each of the state variables x = 20u y = 20v and 119911 = 20119908

4 Complexity

L1

L2

L3

10005000

minus4

minus2

0

2

(a)

L1

L2

L3

3 42 505

d

minus4

minus2

0

2

1

(b)

Figure 2 LEs of system (1) (a) the bifurcation parameter fixed in 119889 = 4 and (b) variation of bifurcation parameter 119889 = [05 5]

y

minus10

0

10

200minus20

x

(a)

z

minus10

minus5

0

5

200minus20

x

(b)

z

minus10

minus5

0

5

100minus10

y

(c)

minus10 minus20

z

yx

00

10

20

minus10

minus5

0

5

(d)

Figure 3 Chaotic attractors of system (1) (a) phase plane 119909 versus 119910 (b) phase plane 119909 versus 119911 (c) phase plane 119910 versus 119911 and (d) phasespace 119909 versus 119910 versus 119911

Complexity 5

Vd

R

y

R8

R4

R5

R1

R2

R3

R

R9

R10

R11

R12

R13

R7

R6

C3

C1

+Vcc+Vcc

minusVcc

Vd

minus

+

AD633

AD633

minus

+

minus

+

minus

+

minus

+

minus

+

z

zy

C2

x

minusy

minusy

minusy

minusy

minusxminusx

minusx

minusz

minusz

Figure 4 Schematic diagram of the equivalent circuit of system (11)

was calculated Replacing the new variables on system (1) weobtain the following system = minus2119906 minus 40V119908

V = minus119906 + 05V = 02 minus 20V2 minus 119908

(10)

Replacing the state variables 119909 = 119906 119910 = V and 119911 = 119908 insystem (10) the representation of circuit is

= 1119877119862119897 (minus 119877119877119897 119909 minus 119877101198771198970119910119911) 119910 = 11198771198622 (minus119909 + 1198771198775119910) = 11198771198623 ( 1198771198779 119887 minus 1198771011987781199102 minus 119911)

(11)

where the components are OAs TL084 multipliers AD6331198621 = 1198622 = 1198623 = 100 pF 1198771 = 500 kΩ 1198772 = 47 kΩ 119877 =11987710 = 1MΩ 1198773 = 2MΩ 1198776 = 100 kΩ 1198774 = 1198775 = 1198778 =1198779 = 11987712 = 11987713 = 10 kΩ and 1198777 = 5MΩ the bifurcationparameter is fixed in 119889 = 4with 11987711 = 287 kΩ and the circuitof Figure 4 is powered with +119881cc = 18V and minus119881cc = minus18VTo see a change in the dynamical behavior of system (11) it isrecommend to represent the bifurcation parameter 119889 with avariable resistor of 11987711(VAR) = 1MΩ this voltage was referredto as 119881119889 Figure 4 shows the equivalent circuit of system (11)

In order to compare the experimental with numericalresults Figure 5 shows the comparison on phase planesof system (11) between Multisim simulation and electroniccircuit implementation we can see that the correspondingattractors are similar with respect to those shown in Figure 3

32 DV of NCS and Its Digital Implementation It is wellknown that Eulerrsquosmethod in order to discretize a continuoussystem is derived from the expansion of Taylorrsquos series when

the quadratic and upper order term are truncated The Eulermethod to approximate the ordinary differential equations(ODEs)

x = f (x) x (0) = x0

x isin R119873(12)

is given by

x(119899+1) = x(119899) + 120591f (x(119899)) (13)

where 120591 is the step size and 119899 is the iteration number thatrepresent the time in discrete version Eulerrsquos discretization(13) was considered to obtain the DV of the proposed NCS(1) as follows

119909(119899+1) = 119909(119899) + 120591 (minus119886119909(119899) minus 119887119910(119899)119911(119899)) 119910(119899+1) = 119910(119899) + 120591 (minus119909(119899) + 119888119910(119899)) 119911(119899+1) = 119911(119899) + 120591 (119889 minus 1199102(119899) minus 119911(119899)) (14)

The advantage of Eulerrsquos method is that it is easy tounderstand and simple to execute as numerical algorithmin addition it has low time complexity Even though its lowaccuracy thismethod iswidely used for solving (numerically)ODEs for more details please see [45]

The Matlab simulations of the DV-system (14) werecarried out by using 120591 = 0005 and 119899 = 40000 Figure 6 showsthe phase space 119909(119899) versus 119910(119899) versus 119911(119899) of DV-system (14)

Microchip Technology Inc is the manufacturer of micro-controller PIC32 their numerical results were represented infloating points 32 bits according to the IEEE-754 CompliantFloating Point Routines [46] The standard IEEE-754 also isincluded in Matlab for 32-bit version [45] The microcon-troller PIC32 was programmed by using Mikroc Pro for Pic32 compiler that includes the standard IEEE-754 this means

6 Complexity

(a) (b) (c)

(d) (e) (f)

Figure 5 Comparison on phase planes between simulation and circuit implementation of system (11) Multisim simulation (a) 119909 versus 119910(b) 119909 versus 119911 and (c) 119910 versus 119911 and electronic circuit implementation (d) 119909 versus 119910 (e) 119909 versus 119911 and (f) 119910 versus 119911

minus10 minus20

00

1020

minus10

minus5

0

5

z (n)

y(n) x(n)

Figure 6 Chaotic attractor of DV-system (14) projected on 119909(119899)versus 119910(119899) versus 119911(119899)that the numerical results in simulation by using Matlab torepresent the DV-system (14) and implementation by usingMikroc Pro for Pic 32 compiler are equivalents

We use a novel method reported in [37 38] in orderto reproduce the DV of chaotic system (14) by using anPIC32 microcontroller and external DACs connected bythe serial peripheral interface (SPI) protocol The com-pact ES Mikromedia Plus for PIC32MX7 contains one 32-bit PIC32MX795F512L microcontroller as central part TheMikromedia Plus for PIC32MX7 ES allows developmentapplications with multimedia contents and it comes withseveral internal hardware-devices We use the internal mod-ule TFTTSD (with one screen of 43 inches of 480 times 272resolution) to represent in real-time the three phase planesof DV-system (12) TFT touch and LCD controller unitsare included into TFTTSD Table 2 shows the hardware andthe SPI modes description of the ES and the schematiccircuit diagram is shown in Figure 7 The evolution ofdiscretized states 119909(119899) 119910(119899) and 119911(119899) of DV-system (14) were

reproduced by using the external DACs U1 U2 and U3respectively

System (14) describes 119873 = 3 dimension To understandthe simulation and implementation the calculus of time wascarried out in the algorithm of U1 to reproduce DV-system(14) on the ESThe time period119879Td(119873) was considered as total-decoding-time that the ES requires to process one iteration 119899The maximum number of iterations 119899 that the ES generatesin 1 second (ips) was calculated by frequency 119891Td(119873) that isthe reciprocal of 119879Td(119873) these terms are represented by

119879Td(119873) = 1119891Td(119873) = 119905119888 + 119905Tg(119873) (15)

where the time complexity 119905119888 defines the time that thealgorithm of U1 needs to reproduce one iteration 119899 Thetotal-graphics-time 119905Tg(119873) is the time that U1 needs to enablethe internal device TFTTSD and the DACs U2 U3 and U4to reproduce in real-time one iteration 119899 we proposed thecalculus of119879Td(119873) considering DV-systems for119873 dimensionswhere 119905Tg(119873) was calculated externally of 119905119888 The total timerequired for each DAC is referred to as 119905Tdac(119873) and the timerequired for TFTTSD is referred to as 119905tf t The total-graphics-time is represented by

119905Tg(119873) = 119905tf t + (119905dac(119895) + 119905dac(119895+1) + sdot sdot sdot + 119905dac(119873))= 119905tf t + 119873sum

119895=1

119905Tdac(119895) 119895 = 1 2 119873 (16)

In order to develop the equivalence between simulationand implementation on the ES we defined the total quantity

Complexity 7

34

34

34

7

7

7

PIC3

2MX7

95F5

12L

81CS2

SCKSDI

GND

LDAC65SCK

SDO

EDACxU2

SLAVE 1

81CS2

SCKSDI

GND

LDAC65SCK

SDO

EDACyU3

SLAVE 2

81CS2

SCKSDI

GND

LDAC65SCK

SDO

EDACzU4

SLAVE 3

LCD controllerSSD1963

TFT touchAT043B35-15I-10

EDACx

EDACy

EDACz

SCK

SDO

U1ndashMikromedia Plus for PIC32MX7

RB12

RB11

RB7

RD0

RD10

MASTER

4

3

8

20

18

52

51OSC2

MCLR

TFTTSD

IOTFT-LCD bus control

22 pf 22 pf

10KΩ

10 KΩ

10 KΩ

10 KΩ

16 MHz

OSC1

x(t)

y(t)

z(t)

V＞＞ = 33 V

V＞＞

VＭＭ

V＝＝ = 5 V V＝＝ = 5 V

V＝＝ = 5 V

V＝＝ = 5 V

V＝＝

V＝＝

V＝＝

VＩＯＮ

VＩＯＮ

VＩＯＮ

VＬ

VＬ

VＬ

Figure 7 Schematic circuit design of ES for the implementation of DV-system (14)

Table 2 Main hardware description of ES

Peripheral number SPI mode hardware descriptionU1 Master TFTTSDMikromedia Plus for PIC32MX7U2 Slave 1 DACMCP4921 shows 119909(119905)U3 Slave 2 DACMCP4921 shows 119910(119905)U4 Slave 3 DACMCP4921 shows 119911(119905)of iterations 119876119879 as the maximum number of 119899 iterationsgenerated in 1 second

119876119879(119873=3) = 120591 1119879Td(119873) = 120591119891Td(119873) (17)

The time for one specific number of iterations 119899 generatedfrom the DV-system (14) is calculated by using the followingexpression

119905119899 = 119899 [119905119888 + 119905Tg(119873)] = 119899 [119905119888 + 119905tf t + 119905Tdac(119873)] (18)

Figure 8 and Table 3 show the implementation of system(14) to exemplify (15)ndash(18) Finally we obtained 119876119879 = 917considering 120591 = 001 this means that in 1 second we obtained917 time units Figure 8(d) shows 119876119879 = 917 for 119905 = 10 s

33 Degradation Study for DV of NCS To prove the presenceof chaos on the NCS (1) the LEs for discretized system (14)were calculated by using time series [43 44] The result ofJacobean matrix for the discretized system (14) is

119869DV = (1 minus 119886120591 minus119887120591119911(119899) minus119887120591119910(119899)minus120591 1 + 119888120591 00 minus2120591119910(119899) 1 minus 120591 ) (19)

Table 3 Implementation results of DV-system (14) on the proposedES

Parameter Value120591 001119905Tg(3) 27 120583s119905tf t 24 120583s119891Td(3) 917 ips119876119879(3) 917119905119888 1063 120583s119879Td(3) 1090 120583s119899 917119905Tdac(3) 3 120583s119905(119899=917) 099953 s

where the step size 120591 was modified as parameter to prove thechaotic behavior of theDV-system (14) 120591 ismodified by usingan increase of step size 120591 = 0001 regarding 10000 time unitsuntil the sign of the LEs changes and the discretized system(14) diverges The LEs and fractal dimension of discretizedsystem (14) are referred to as 11987110158401 11987110158402 11987110158403 and1198631015840KY respectivelyFigure 9 and Table 4 show the result of chaos degradationcorresponding to DV-system (14) for 2 cases

8 Complexity

(a) (b) (c)

(d) (e)

Figure 8 Implementation of DV-system (14) on proposed ES (a) phase plane 119909(119899) versus 119910(119899) (b) phase plane 119909(119899) versus 119911(119899) (c) phase plane119910(119899) versus 119911(119899) (d) time evolution of states 119909(119905) and 119911(119905) by using 120591 = 001 and 119905 = 10 s and (e) representation of phase planes 119909(119899) versus 119910(119899)119909(119899) versus 119911(119899) and 119910(119899) versus 119911(119899) on TFTTSD

L1

L2

L3

1000050000minus04

minus02

0

02

(a)

L㰀

1

L㰀

2

L㰀

3

2001000

minus05

0

05

1

(b)

Figure 9 LEs of discretized system (14) for (a) 120591 = 0085 and (b) 120591 = 0086

Table 4 Analysis of chaos degradation for DV-system (14) by using LEs

Case 120591 LEs Result

1 (0 0085]

11987110158401 = +0050763Chaotic behavior shown in Figure 9(a)11987110158402 = minus000006458811987110158403 = minus0243471198631015840KY = 20996

2 [0086 +infin)

11987110158401 = no valid

No displayed chaos see Figure 9(b)11987110158402 = no valid11987110158403 = no valid1198631015840KY = no valid

Complexity 9

Table 5 Comparison of the proposed NCS with some chaotic systems reported in the literature

Chaotic system Parameters Critical parameters Nonlinearities Step size 120591 Total time 120583119904 Iterations per second Time units 119876119879Lorenz 120590 b c 120590 2 le0024 1090 917 22Rossler a b c 119888 1 le0005 1073 932 47Chen a b c 119886 2 le0002 1090 917 18Liu and Chen 119886 119887 119888 1198891 1198892 1198893 119888 3 le0002 1096 912 18Proposed NCS a b c d 119887 119889 2 le0085 1090 917 78

For case 1 the discretized system (14) conserves thechaotic behavior This result was compared with the LEscalculated for CV of system (1) where the numerical resultsof 1198711 and 119863KY were similar with respect to 11987110158401 and 1198631015840KY Themaximum step size 120591 = 0085 was found For case 2 the stepsize was increased until obtaining 120591 = 0086 whereby LEscan not be calculated in the DV-system (14) For values of120591 ge 0086 the discretized system (14) diverges and the statetrajectories 119909(119899) 119910(119899) and 119911(119899) can not display chaos

34 Comparison of the Proposed NCS with Some ChaoticSystems In order to compare the performance of the NCS(14) in DV we studied the chaotic degradation of four 3DLorenz Rossler Chen and Liu and Chen classical chaoticsystems where their DVs were obtained by using the sameEuler discretization (13) and the LEs were calculated by usingthe same method as in [43 44] Table 5 shows the results ofthe step sizes 120591 intervals of the five Lorenz Rossler Chenand Liu and Chen CSs using the Euler numerical algorithm(13) where the chaotic behavior is conserved in these chaoticsystems [16 17 22 23] According to Table 5 the proposedNCS in DV (14) presents a higher step size with respect tothe other four 3D Lorenz Rossler Chen and Liu and Chenchaotic systems in DV This means that for implementationthe NCS in DV has more compacts dynamics to digitalimplementations then the NCS in DV is a good alternativeusing ESs where the main part has less processing capacityfor example 8-bit microcontrollers family The novelty of theproposed chaotic system is the combination of the differentcharacteristics that it presents which results in a high easeof implementation for its use in different applications aspreviously mentioned

4 Digital Implementation Process

In this section we present the flow chart and the descriptionof the electronicaldigital implementation process that con-tains the proposed programming algorithm for the imple-mentation of the NSC in DV (14) In addition we presentsome aspects of implementation robustness from the point ofview of software and hardware a study regarding the robust-ness of the critical parameters and comparative advantagesof the implementation for the NSC in DV (14)

41 Flow Chart Digital Implementation In Figure 10 weillustrate the flow chart of the general electronicaldigitalimplementation process The description of each step isdescribed below

Step 1 Set initial calibration of the TFTTSDU1TheTFTTSDis initialized and an internal program that allows calibratingthe internal TFT touch and LCD controllers of the TFTTSDis executed the four edges of the TFT screen are used

Step 2 Set graphic environment variables and parameters ofNCS in DV (14) on PIC32MX795F512L microcontroller Thefloating point and decimal-base constants to be used in theprogramming algorithm of the NSC in DV (14) are defined

Step 3 Initialization of ports and SPI protocol the SPIprotocol of the main PIC32MX795F512L microcontroller isconfigured in master mode and the SPI of the external DACsU1 U2 and U3 are configured in slave mode

Step 4 Set the critical parameters initial conditions1199090 1199100 1199110 and step size 120591 for NCS in DV (14) The criticalparameters a = 2 b = 2 c = 05 and d = 4 initial conditions1199090 = 1199100 = 1199110 = 1 and step size 120591 = 0004 corresponding tothe initial iteration of the NCS in DV (14) are defined

Step 5 Definition of the NCS in DV (14) using Eulerrsquosnumerical algorithm The discretized NCS is defined by theEulerrsquos numerical algorithm Delay time in this stage is 14 120583sStep 6 Storing the current values of state variables 119909(119899) 119910(119899)and 119911(119899) this value corresponds to the next iteration of theNCS in DV (14) Delay time in this stage is 02 120583sStep 7 Rescaling the state variables 119909(119899) 119910(119899) and 119911(119899) inpositive scale Representation of the state variables 119909(119899) 119910(119899)and 119911(119899) is rescaled since the numerical representation in theTFTTSD and the DACs is positive Delay time in this stage is10337 120583sStep 8 Rescaling the values of state variables 119909(119899) 119910(119899) and119911(119899) for the TFTTSD in 480 times 272 resolution to displayimages in the TFTTSD Visual TFT software is used to designa template that displays graphics and text In our case theevolution of the phase planes 119909(119899) versus 119910(119899) 119909(119899) versus 119911(119899)and 119910(119899) versus 119911(119899) is shown in real-time In addition thenames of the authors are shown Delay time in this stage is75 120583sStep 9 Write theTFTTSDusing theTFT library to draw a dotat certain coordinates for each phase plane 119909(119899) versus 119910(119899) inred color 119909(119899) versus 119911(119899) in green color and 119910(119899) versus 119911(119899) inblue color Once the values are rescaled within the TFTTSDresolution the ldquoTFT__Dotrdquo library is used to display a point

10 Complexity

Start

(3)Initialization of ports and SPI

protocol

(5)Definition of the NCS in DV (14) using Eulerrsquos numerical

algorithm

(2) Set graphic environment

variables and parameters ofNCS in DV (14) onPIC32MX795F512L

microcontroller

(1)Set initial calibration of the

TFTTSD U1

Loop

Loop

(4)Set the critical parameters

step size for NCS in DV

(6)Storing the current values of the state variables x(n) y(n)

and z(n)

(7)Rescaling the state variables

scalex(n) y(n) and z(n) in positive

(8)Rescaling the values of state

resolution respectively

variables x(n) y(n) and z(n) forthe TFTTSD in 480 times 272

(9)Writing the TFTTSD using the

TFT library to draw a dot at certain coordinates for each


12 bits for the DACs U2 U3 and U4

variables x(n) y(n) and z(n) in

(11)Writing the external DACs U2

U3 and U4 using the SPI protocol to reproduce the

state variables x(n) y(n) andz(n)

initial conditionsz0 andx0 y0 and

phase plane x(n) versus y(n) inz(n)red x(n) versus in green andz(n) in bluey(n) versus

Figure 10 Flow chart of the general electronicaldigital implementation process

with a different color according to the coordinates indicatedby the phase planes of each state variable Delay time in thisstage is 24120583sStep 10 Rescaling the values of state variables 119909(119899) 119910(119899)and 119911(119899) in 12 bits for the DACs U2 U3 and U4 for theimplementation of the state variables 119909(119899) 119910(119899) and 119911(119899) apositive scale is used with a maximum resolution interval of12 bits from0 to 4095which is theworking range of theDACsand the SPI protocol resulting in 16 Mbps Delay time in thisstage is 76120583s

Step 11 Write the external DACs U2 U3 and U4 using theSPI protocol to reproduce the state variables 119909(119899) 119910(119899) and119911(119899) The microcontroller PIC32MX795F512L configured inmaster mode is used to enable the select chip and writethe DACs U1 U2 and U3 (configured in slave mode)where the state variables 119909(119899) 119910(119899) and 119911(119899) are reproducedsimultaneously Delay time in this stage is 3 120583s

Finally a loop from Steps 11ndash5 is performed where theparameter values of theNSC inDV (14) and 120591were previouslydefined according to Table 4

Complexity 11

b

0

1

2

3

4

5

1 2 3 4 50

d

Figure 11 Robustness diagram to determine chaos existence for bversus d at intervals of 001 and with 120591 = 0085 chaos (red) no chaos(blue)

42 Robustness in the Implementation of the NCS DigitalVersion According to [47] software robustness is the abil-ity of a product to stay in service and function correctlyeven with the occurrence of errors that are attributable tohardware software or even external influences The imple-mented software in the TFTTSD is designed from graphicalinterface tools using the Visual TFT in conjunction withprogramming code designed in C language that is storedin the PIC32MX795F512L microcontroller flash memoryThe accuracy of the programming algorithm calculationsdepends on the IEEE-754 AN575 standardization of thePIC32MX795F512L microcontroller [46] With regard tohardware the TFTTSD has two possible forms of ener-gization the first is through the USB port connected to alaptop or desktop PC the second is via an external lithiumbattery The possibility that the TFTTSD can be energizedthrough an external battery makes it portable which allowsthe autonomy of the equipment

On the other hand in order to show the robustness ofchaos presence in the discretized system (14) a robustnessdiagram based on the variation of critical parameters b andd was carried out In this diagram it is possible to determinethe regions in which the existence of chaos is guaranteedconsidering 120591 = 0085 Figure 11 shows the regions of chaosexistence for b versus d (intervals of 001 are used for bothparameters b and d) where each point in the graph representsthe maximum Lyapunov exponent (1198711015840max) If we have 1198711015840max gt0 that is if the dynamics are chaotic the red color is usedotherwise the blue color is used From the robustness dia-gram in Figure 11 it can be seen that the chaotic dynamics arepreserved for wide intervals of the parameter values b and d

Furthermore it is easy to note that if a value of 120591 lessthan 0085 is considered then the chaos regions increaseTaking into account the fact that the preservation of chaosin the discretized version of the NCS proposed is robustfor the parameters b and d considering the software andhardware characteristics of the proposed ES and the benefitsof digital systems as the elimination of the typical wear ofthe analog systems it is stated that the electronicaldigitalimplementation presented in this work is robust

On the other hand to the best of our knowledge theelectronical implementation in a portable TFTTSD deviceof DV of chaotic systems for the reproduction of theirnonlinear dynamics in real-time is new By having a graph-ical interface and given certain potential applications in theengineering field such as biometric systems telemedicinecryptography and secure communications the proposeddigital implementation makes the interaction between thedevice and the end user very friendly One of the mostrelevant advantages of the NSC in DV (14) is the increase instep size compared with other chaotic systems which allowsimplementation in slower microcontrollers for example in8-bit low-end microcontrollers microchip PIC microcon-trollers Motorola M68HC05 microcontrollers AVR micro-controllers ATmega328 and 8051 from the manufacturerAtmel In the same way there are alternative families of 16-bit mid-range microcontrollers to implement the NSC in DV(14) such as the dsPIC family of manufacturer microchipMSP430 of Texas Instruments

Finally we can find the high-endmicrocontrollers whichare those used in the implementation presented in this workA microchip PIC32 microcontroller was used which showsgreat benefits in the use of TFTTSD along with this micro-controller there are other alternatives such as the STM32microcontrollers of the manufacturer STMicroelectronicsor the FT900 microcontrollers of the manufacturer FutureTechnology Devices International Limited

5 Conclusions

We have proposed a new chaotic system (NCS) whichgenerates chaotic dynamics varying two parameters

Analytical and numerical studies to confirm the chaosgeneration for continuous and discretized version (DV) werepresented Also a degradation analysis on the discretizedversion of the NCS was carried out to find the maximum stepsize The results showed that the NCS is flexible and robustwhich allows obtaining different chaotic behaviors

In addition the NCS was implemented electronically forcontinuous version with operational amplifiers and for DVwe used a novel embedded system that shows dynamicalbehaviors in real-time

As future work the authors will concentrate on carryingout a complete analysis of the proposed chaotic systemproviding rigorous mathematical proofs to estimate theultimate bound and positively invariant set as is reported inthe current literature [11 48ndash50] and in addition to applythese analytical results to synchronize the proposed chaoticsystem via approach reported in [51]

Conflicts of Interest

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

Acknowledgments

This work is supported by the CONACYT Mexico underResearch Grant 166654

12 Complexity

References

[1] J Guckenheimer andPHolmesNonlinearOscillationsDynam-ical Systems and Bifurcation of Vector Fields Springer NewYork NY USA 1982

[2] S Wiggins Introduction to Applied Nonlinear Dynamical Sys-tems and Chaos Springer Berlin Germany 1990

[3] S H Strogatz Nonlinear dynamics and chaos with applicationsto physics biology chemistry and engineering Perseus BooksMassachusetts USA 1994

[4] W Xingyuan and L Chao ldquoResearches on chaos phenomenonof EEG dynamics modelrdquo Applied Mathematics and Computa-tion vol 183 no 1 pp 30ndash41 2006

[5] K-Z Li M-C Zhao and X-C Fu ldquoProjective synchroniza-tion of driving-response systems and its application to securecommunicationrdquo IEEE Transactions on Circuits and Systems IRegular Papers vol 56 no 10 pp 2280ndash2291 2009

[6] H O Wang E H Abed and A M A Hamdan ldquoBifurcationschaos and crises in voltage collapse of a model power systemrdquoIEEE Transactions on Circuits and Systems I FundamentalTheory and Applications vol 41 no 4 pp 294ndash302 1994

[7] F-Y Lin and J-M Liu ldquoChaotic radar using nonlinear laserdynamicsrdquo IEEE Journal of Quantum Electronics vol 40 no 6pp 815ndash820 2004

[8] R V Donner J Heitzig J F Donges Y Zou N Marwan andJ Kurths ldquoThe geometry of chaotic dynamicsmdasha complex net-work perspectiverdquoThe European Physical Journal B CondensedMatter and Complex Systems vol 84 no 4 pp 653ndash672 2011

[9] A Arellano-Delgado R M Lopez-Gutierrez C Cruz-Hernandez C Posadas-Castillo L Cardoza-Avendano and HSerrano-Guerrero ldquoExperimental network synchronization viaplastic optical fiberrdquo Optical Fiber Technology vol 19 no 2 pp93ndash98 2013

[10] M A Murillo-Escobar C Cruz-Hernandez F Abundiz-PerezR M Lopez-Gutierrez and O R Acosta Del Campo ldquoARGB image encryption algorithm based on total plain imagecharacteristics and chaosrdquo Signal Processing vol 109 pp 109ndash131 2015

[11] H Saberi Nik S Effati and J Saberi-Nadjafi ldquoUltimate boundsets of a hyperchaotic system and its application in chaossynchronizationrdquo Complexity vol 20 no 4 pp 30ndash44 2015

[12] M A Murillo-Escobar C Cruz-Hernandez F Abundiz-Perezand R M Lopez-Gutierrez ldquoA robust embedded biometricauthentication system based on fingerprint and chaotic encryp-tionrdquo Expert Systems with Applications vol 42 no 21 pp 8198ndash8211 2015

[13] M A Murillo-Escobar L Cardoza-Avendano R M Lopez-Gutierrez and C Cruz-Hernandez ldquoA Double Chaotic LayerEncryption Algorithm for Clinical Signals in TelemedicinerdquoJournal of Medical Systems vol 41 p 59 2017

[14] Y Yan ldquoSynchronization for a class of uncertain fractional orderchaotic systems with unknown parameters using a robust adap-tive sliding mode controllerrdquo Hindawi Publishing CorporationMathematical Problems in Engineering vol 2016 Article ID7404652 7 pages 2016

[15] J Zhang D Hou and H Ren ldquoImage encryption algorithmbased on dynamic DNA coding and Chenrsquos hyperchaotic sys-temrdquo Mathematical Problems in Engineering vol 2016 ArticleID 6408741 11 pages 2016

[16] E Lorenz ldquoDeterministic nonperiodic flowrdquo Journal of theAtmospheric Sciences vol 20 pp 130ndash141 1963

[17] O E Rossler ldquoAn equation for continuous chaosrdquoPhysics LettersA vol 57 no 5 pp 397-398 1976

[18] J Lu and G Chen ldquoA new chaotic attractor coinedrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 12 no 3 pp 659ndash661 2002

[19] L O Chua ldquoThe Double Scroll Familyrdquo IEEE Transactions onCircuits and Systems vol 33 no 11 pp 1072ndash1118 1986

[20] C Liu T Liu L Liu andK Liu ldquoAnew chaotic attractorrdquoChaosSolitons and Fractals vol 22 no 5 pp 1031ndash1038 2004

[21] J C Sprott ldquoSome simple chaotic flowsrdquo Physical Review EStatistical Nonlinear and SoftMatter Physics vol 50 no 2 partA pp R647ndashR650 1994

[22] G Chen and T Ueta ldquoYet another chaotic attractorrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 9 no 7 pp 1465-1466 1999

[23] W B Liu and G Chen ldquoA new chaotic system and itsgenerationrdquo International Journal of Bifurcation and Chaos vol12 pp 261ndash267 2002

[24] J C Sprott Elegant Chaos Algebraically Simple Chaotic FlowsWorld Scientific Singapore 2010

[25] C Gissinger ldquoA new deterministic model for chaotic reversalsrdquoEuropean Physical Journal B vol 85 no 137 2012

[26] C Li and J C Sprott ldquoMultistability in a butterfly flowrdquo Interna-tional Journal of Bifurcation and Chaos in Applied Sciences andEngineering vol 23 no 12 pp 1350199ndash1350209 2013

[27] W T Verkley and C A Severijns ldquoThemaximum entropy prin-ciple applied to a dynamical system proposed by Lorenzrdquo TheEuropean Physical Journal B Condensed Matter and ComplexSystems vol 87 no 7 2014

[28] J Wu L Wang G Chen and S Duan ldquoA memristive chaoticsystem with heart-shaped attractors and its implementationChaosrdquo Solitons Fractals vol 92 pp 20ndash29 2016

[29] A LrsquoHer P Amil N Rubido A C Marti and C CabezaldquoElectronically-implemented coupled logistic mapsrdquoThe Euro-pean Physical Journal B Condensed Matter and Complex Sys-tems vol 89 no 81 2016

[30] L J Ontanon-Garcıa and E Campos-Canton ldquoPreservation ofa two-wing Lorenz-like attractor with stable equilibriardquo Journalof the Franklin Institute Engineering and Applied Mathematicsvol 350 no 10 pp 2867ndash2880 2013

[31] A T Azar C Volos N Gerodimos et al ldquoA novel chaoticsystem without equilibrium dynamics synchronization andcircuit realizationrdquo Hindawi Publishing Corporation Complex-ity vol 2017 Article ID 7871467 11 pages 2017

[32] X Wang V-T Pham and C Volos ldquoDynamics circuit designand synchronization of a new chaotic system with closed curveequilibriumrdquo Hindawi Publishing Corporation Complexity vol2017 Article ID 7138971 9 pages 2017

[33] M P Mareca and B Bordel ldquoImproving the complexity of theLorenz dynamicsrdquoHindawi Publishing Corporation Complexityvol 2017 Article ID 3204073 16 pages 2017

[34] C Cruz-Hernandez D Lopez-Mancilla V Garcıa-Gradilla HSerrano-Guerrero and R Nunez-Perez ldquoExperimental realiza-tion of binary signals transmission using chaosrdquo in Proceedingsof the 1st International Conference on Communications Circuitsand Systems (ICCCAS rsquo02) pp 146ndash149 July 2002

[35] QWang S Yu C Li et al ldquoTheoretical design and FPGA-basedimplementation of higher-dimensional digital chaotic systemsrdquoIEEE Transactions on Circuits and Systems I Regular Papersvol 63 no 3 pp 401ndash412 2016

Complexity 13

[36] B Cai GWang and F Yuan ldquoPseudo random sequence gener-ation from a new chaotic systemrdquo in Proceedings of the 16th IEEEInternational Conference on Communication Technology (ICCTrsquo15) pp 863ndash867 October 2015

[37] RMendez-Ramırez A Arellano-Delgado C Cruz-HernandezF Abundiz-Perez and R Martınez-Clark ldquoChaotic DigitalCryptosystem by using SPI Protocol and its dsPICs Implemen-tationrdquo Frontiers of Information Technology Electronic Engineer-ing

[38] RMendez-Ramirez AArellano-DelgadoCCruz-Hernandezand R M Lopez-Gutierrez ldquoDegradation analysis of general-ized Chuarsquos circuit generator of multi-scroll chaotic attractorsand its implementation on PIC32rdquo in Proceedings of the FutureTechnologies Conference (FTC) pp 1034ndash1039 San FranciscoCA USA December 2016

[39] L Acho ldquoA discrete-time chaotic oscillator based on the logisticmap a secure communication scheme and a simple experimentusing Arduinordquo Journal of the Franklin Institute Engineeringand Applied Mathematics vol 352 no 8 pp 3113ndash3121 2015

[40] Q Yang andGChen ldquoA chaotic systemwith one saddle and twostable node-focirdquo International Journal of Bifurcation and Chaosin Applied Sciences and Engineering vol 18 no 5 pp 1393ndash14142008

[41] H S Nik andM Golchaman ldquoChaos Control of a Bounded 4DChaotic Systemrdquo Neural Comput Applic vol 25 no 3 pp 683ndash692 2014

[42] M Suneel ldquoElectronic circuit realization of the logistic maprdquoSadhana vol 31 no 1 pp 69ndash78 2006

[43] A Wolf J B Swift H L Swinney and J A Vastano ldquoDeter-mining Lyapunov exponents from a time seriesrdquo Physica DNonlinear Phenomena vol 16 no 3 pp 285ndash317 1985

[44] K Briggs ldquoAn improved method for estimating Liapunovexponents of chaotic time seriesrdquo Physics Letters A vol 151 no1-2 pp 27ndash32 1990

[45] W Y Yang W Cao T-S Chung and J Morris Applied numer-ical methods using Matlab John Wiley and Sons Inc 2005

[46] Microchip Technology Inc ldquoAN575 IEEE-754 CompliantFloating Point Routinesrdquo in DS00575B pp 1ndash155 1997

[47] S Fraser D Campara C Chilley et al ldquoFostering softwarerobustness in an increasingly hostile worldrdquo in Proceedings ofthe Companion to the 20th annual ACM SIGPLAN conferencep 378 San Diego CA USA October 2005

[48] GA LeonovA I Bunin andNKoksch ldquoAttraktorlokalisierungdes Lorenz-Systemsrdquo Zeitschrift fur Angewandte Mathematikund Mechanik vol 67 no 12 pp 649ndash656 1987

[49] A Y Pogromsky G Santoboni and H Nijmeijer ldquoAn ultimatebound on the trajectories of the Lorenz system and its applica-tionsrdquo Nonlinearity vol 16 no 5 pp 1597ndash1605 2003

[50] D Li J Lu XWu and G Chen ldquoEstimating the bounds for theLorenz family of chaotic systems Chaosrdquo Solitons Fractals vol23 pp 529ndash534 2005

[51] H Sira-Ramırez and C Cruz-Hernandez ldquoSynchronization ofchaotic systems a generalized Hamiltonian systems approachrdquoInternational Journal of Bifurcation and Chaos in Applied Sci-ences and Engineering vol 11 no 5 pp 1381ndash1395 2001

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Journal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Complexity

systems telemedicine and secure communications see forexample [10 12 13 37] To the best of our knowledge theelectronical implementation in a portable TFTTSD device ofDVof chaotic systems for the reproduction of their nonlineardynamics in real-time is new

The paper is organized as follows Section 2 reports basicanalytical proof and extensive numerical tests to verify theexistence of chaos in the proposed NCS Section 3 presentstwo electronic implementations to reproduce the NCS thefirst by means of OAs and the second of a novel proposedES by using one TFTTSD and external DACs In additiona degradation study taking into account the preservationof chaos is conducted a comparison among the NCS andsome other chaotic systems reported in the literature ismade In Section 4 we give a complete description of digitalimplementation process of the DV of NSC with the corre-sponding robustness diagram to determine regions in whichthe existence of chaos is guaranteed Finally in Section 5 wedraw some concluding remarks

2 Basic Analysis and Characterization of NCS

This section presents the state equations of NCS and somenumerical and analytical tests to verify the chaos existencein the proposed system The NCS is built starting frominspecting and modification of Lorenz system [16] Theproposed NCS is described by

= minus119886119909 minus 119887119910119911119910 = minus119909 + 119888119910 = 119889 minus 1199102 minus 119911

(1)

The proposed system has seven terms two quadraticnonlinearities and four parameters 119886 119887 119888 119889 isin R+ where 119887and 119889 are characterized as the bifurcation parameters Thenonlinear NCS (1) is chaotic with 119886 = 2 119887 = 2 119888 = 05 and119889 = 4

System (1) is symmetrical about the 119911-axis due to itsinvariance under the coordinate transformation (119909 119910 119911) rarr(minus119909 minus119910 119911) The symmetry is not associated with the a b cand d parameters

The divergence for a 3-dimensional flow of dynamicalsystem is defined by

nabla119881 = 120597120597119909 + 120597 119910120597119910 + 120597120597119911 = minus119886 + 119888 minus 1 = minus25 lt 0 (2)

Therefore the above analysis proves that our system isdissipative The exponential contraction rate is calculated asfollows 119889119881119889119905 = (nabla119881)119881 997888rarr 119881 = 1198810119890minus25119905 (3)

where each volume containing the system trajectory shrinksto zero as 119905 rarr infin at an exponential rate ofminus25119905 Systemorbitsare ultimately confined into a specific limit set of zero volumeand the asymptotic motion settles onto an attractor Therebythe existence on attractor is proved

The boundness of the chaotic trajectories of system (1) isproved by means of the following theoremThe boundness ofa NCS by using similar approach was reported in [40 41]

Theorem 1 Suppose that the parameters 119886 119887 119888 and 119889 ofsystem (1) are positive Then the orbits of system (1) includingchaotic orbits are confined in a bounded region

Proof Consider the candidate Lyapunov function

119881 (119909 119910 119911) = 12 (1199092 + 1199102 + 1199112) (4)

and the time derivative of 119881(119909 119910 119911) along the trajectories ofthe NCS (1) is given by

(119909 119910 119911) = 119909 + 119910 119910 + 119911 (119909 119910 119911) = minus1198861199092 minus 119887119909119910119911 + 1198881199102 minus 119909119910 + 119889119911 minus 1199111199102 minus 1199112

= minus( 1198871199101199112radic119886 + radic119886119909)2 + ( 1199092radic119888 minus radic119888119910)2minus 11990924119888 minus (119911 minus 1198892)2 + 11988924+ ( 1198871199101199112radic119886 minus radic119886119910119887 )2 minus 11988611991021198872

(5)

Let 1198770 be the sufficiently large region so that for alltrajectories (119909 119910 119911) satisfy 119881(119909 119910 119911) = 119877 for 119877 gt 1198770 with thecondition

( 1198871199101199112radic119886 + radic119886119909)2 + (119911 minus 1198892)2 + 11988611991021198872 + 11990924119888gt ( 1199092radic119888 minus radic119888119910)2 + ( 1198871199101199112radic119886 minus radic119886119910119887 )2 + 11988924

(6)

Consequently on the surface (119909 119910 119911)119881(119909 119910 119911) = 119877Since 119877 gt 1198770 we can write (119909 119910 119911) lt 0 or the set(119909 119910 119911)119881(119909 119910 119911) le 119877 is a confined region for all thetrajectories of chaotic system (1)

The number of the equilibrium points and their stabilitiesdetermine the behavior of system (1) it can be found bysetting = 119910 = = 0 and 119886 119887 119888 119889 gt 0 The proposedsystem (1) has five fixed points 1198750(0 0 0) 1198751 1198752 11987531198754(plusmn119888radic119889 + 119886119888119887 plusmnradic119889 + 119886119888119887 minus119886119888119887) The Jacobean matrixof CV-system (1) is given by

119869CV = (minus119886 minus119887119911 minus119887119910minus1 119888 00 minus2119910 minus1 ) (7)

and the characteristic polynomial of (7) is as follows

det (120582119868 minus 119869) = 1205823 + (119886 minus 119888 + 1) 1205822 + (119886 minus 119888 minus 119886119888 minus 119887119911) 120582+ 21198871199102 minus 119886119888 minus 119887119911 = 0 (8)

Complexity 3




minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

x

1 15 2 25 3 35 4 45 505b

(a)

1 15 2 25 3 35 4 45 505d

minus20

minus15

minus10

minus5

0

5

10

15

20

25

x

(b)







119863KY = 119895 + 110038161003816100381610038161003816119871119895+110038161003816100381610038161003816119895sum119894=1

119871 119894 = 2 + 1198711 + 1198712100381610038161003816100381611987131003816100381610038161003816 = 20908 (9)





4 Complexity

L1

L2

L3

10005000

minus4

minus2

0

2

(a)

L1

L2

L3

3 42 505

d

minus4

minus2

0

2

1

(b)


y

minus10

0

10

200minus20

x

(a)

z

minus10

minus5

0

5

200minus20

x

(b)

z

minus10

minus5

0

5

100minus10

y

(c)

minus10 minus20

z

yx

00

10

20

minus10

minus5

0

5

(d)


Complexity 5

Vd

R

y

R8

R4

R5

R1

R2

R3

R

R9

R10

R11

R12

R13

R7

R6

C3

C1

+Vcc+Vcc

minusVcc

Vd

minus

+

AD633

AD633

minus

+

minus

+

minus

+

minus

+

minus

+

z

zy

C2

x

minusy

minusy

minusy

minusy

minusxminusx

minusx

minusz

minusz




(10)



(11)





x = f (x) x (0) = x0

x isin R119873(12)

is given by

x(119899+1) = x(119899) + 120591f (x(119899)) (13)






6 Complexity

(a) (b) (c)

(d) (e) (f)


minus10 minus20

00

1020

minus10

minus5

0

5

z (n)

y(n) x(n)





119879Td(119873) = 1119891Td(119873) = 119905119888 + 119905Tg(119873) (15)



119895=1

119905Tdac(119895) 119895 = 1 2 119873 (16)


Complexity 7

34

34

34

7

7

7

PIC3

2MX7

95F5

12L

81CS2

SCKSDI

GND

LDAC65SCK

SDO

EDACxU2

SLAVE 1

81CS2

SCKSDI

GND

LDAC65SCK

SDO

EDACyU3

SLAVE 2

81CS2

SCKSDI

GND

LDAC65SCK

SDO

EDACzU4

SLAVE 3



EDACx

EDACy

EDACz

SCK

SDO


RB12

RB11

RB7

RD0

RD10

MASTER

4

3

8

20

18

52

51OSC2

MCLR

TFTTSD


22 pf 22 pf

10KΩ

10 KΩ

10 KΩ

10 KΩ

16 MHz

OSC1

x(t)

y(t)

z(t)

V＞＞ = 33 V

V＞＞

VＭＭ

V＝＝ = 5 V V＝＝ = 5 V

V＝＝ = 5 V

V＝＝ = 5 V

V＝＝

V＝＝

V＝＝

VＩＯＮ

VＩＯＮ

VＩＯＮ

VＬ

VＬ

VＬ




119876119879(119873=3) = 120591 1119879Td(119873) = 120591119891Td(119873) (17)


119905119899 = 119899 [119905119888 + 119905Tg(119873)] = 119899 [119905119888 + 119905tf t + 119905Tdac(119873)] (18)







8 Complexity

(a) (b) (c)

(d) (e)


L1

L2

L3

1000050000minus04

minus02

0

02

(a)

L㰀

1

L㰀

2

L㰀

3

2001000

minus05

0

05

1

(b)




1 (0 0085]


2 [0086 +infin)

11987110158401 = no valid


Complexity 9













10 Complexity

Start


protocol


algorithm



microcontroller


TFTTSD U1

Loop

Loop




and z(n)




















Complexity 11

b

0

1

2

3

4

5

1 2 3 4 50

d







5 Conclusions







Acknowledgments


12 Complexity

References




































Complexity 13
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 3




minus25

minus20

minus15

minus10

minus5

0

5

10

15

20

25

x

1 15 2 25 3 35 4 45 505b

(a)

1 15 2 25 3 35 4 45 505d

minus20

minus15

minus10

minus5

0

5

10

15

20

25

x

(b)







119863KY = 119895 + 110038161003816100381610038161003816119871119895+110038161003816100381610038161003816119895sum119894=1

119871 119894 = 2 + 1198711 + 1198712100381610038161003816100381611987131003816100381610038161003816 = 20908 (9)





4 Complexity

L1

L2

L3

10005000

minus4

minus2

0

2

(a)

L1

L2

L3

3 42 505

d

minus4

minus2

0

2

1

(b)


y

minus10

0

10

200minus20

x

(a)

z

minus10

minus5

0

5

200minus20

x

(b)

z

minus10

minus5

0

5

100minus10

y

(c)

minus10 minus20

z

yx

00

10

20

minus10

minus5

0

5

(d)


Complexity 5

Vd

R

y

R8

R4

R5

R1

R2

R3

R

R9

R10

R11

R12

R13

R7

R6

C3

C1

+Vcc+Vcc

minusVcc

Vd

minus

+

AD633

AD633

minus

+

minus

+

minus

+

minus

+

minus

+

z

zy

C2

x

minusy

minusy

minusy

minusy

minusxminusx

minusx

minusz

minusz




(10)



(11)





x = f (x) x (0) = x0

x isin R119873(12)

is given by

x(119899+1) = x(119899) + 120591f (x(119899)) (13)






6 Complexity

(a) (b) (c)

(d) (e) (f)


minus10 minus20

00

1020

minus10

minus5

0

5

z (n)

y(n) x(n)





119879Td(119873) = 1119891Td(119873) = 119905119888 + 119905Tg(119873) (15)



119895=1

119905Tdac(119895) 119895 = 1 2 119873 (16)


Complexity 7

34

34

34

7

7

7

PIC3

2MX7

95F5

12L

81CS2

SCKSDI

GND

LDAC65SCK

SDO

EDACxU2

SLAVE 1

81CS2

SCKSDI

GND

LDAC65SCK

SDO

EDACyU3

SLAVE 2

81CS2

SCKSDI

GND

LDAC65SCK

SDO

EDACzU4

SLAVE 3



EDACx

EDACy

EDACz

SCK

SDO


RB12

RB11

RB7

RD0

RD10

MASTER

4

3

8

20

18

52

51OSC2

MCLR

TFTTSD


22 pf 22 pf

10KΩ

10 KΩ

10 KΩ

10 KΩ

16 MHz

OSC1

x(t)

y(t)

z(t)

V＞＞ = 33 V

V＞＞

VＭＭ

V＝＝ = 5 V V＝＝ = 5 V

V＝＝ = 5 V

V＝＝ = 5 V

V＝＝

V＝＝

V＝＝

VＩＯＮ

VＩＯＮ

VＩＯＮ

VＬ

VＬ

VＬ




119876119879(119873=3) = 120591 1119879Td(119873) = 120591119891Td(119873) (17)


119905119899 = 119899 [119905119888 + 119905Tg(119873)] = 119899 [119905119888 + 119905tf t + 119905Tdac(119873)] (18)







8 Complexity

(a) (b) (c)

(d) (e)


L1

L2

L3

1000050000minus04

minus02

0

02

(a)

L㰀

1

L㰀

2

L㰀

3

2001000

minus05

0

05

1

(b)




1 (0 0085]


2 [0086 +infin)

11987110158401 = no valid


Complexity 9













10 Complexity

Start


protocol


algorithm



microcontroller


TFTTSD U1

Loop

Loop




and z(n)




















Complexity 11

b

0

1

2

3

4

5

1 2 3 4 50

d







5 Conclusions







Acknowledgments


12 Complexity

References




































Complexity 13
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




4 Complexity

L1

L2

L3

10005000

minus4

minus2

0

2

(a)

L1

L2

L3

3 42 505

d

minus4

minus2

0

2

1

(b)


y

minus10

0

10

200minus20

x

(a)

z

minus10

minus5

0

5

200minus20

x

(b)

z

minus10

minus5

0

5

100minus10

y

(c)

minus10 minus20

z

yx

00

10

20

minus10

minus5

0

5

(d)


Complexity 5

Vd

R

y

R8

R4

R5

R1

R2

R3

R

R9

R10

R11

R12

R13

R7

R6

C3

C1

+Vcc+Vcc

minusVcc

Vd

minus

+

AD633

AD633

minus

+

minus

+

minus

+

minus

+

minus

+

z

zy

C2

x

minusy

minusy

minusy

minusy

minusxminusx

minusx

minusz

minusz




(10)



(11)





x = f (x) x (0) = x0

x isin R119873(12)

is given by

x(119899+1) = x(119899) + 120591f (x(119899)) (13)






6 Complexity

(a) (b) (c)

(d) (e) (f)


minus10 minus20

00

1020

minus10

minus5

0

5

z (n)

y(n) x(n)





119879Td(119873) = 1119891Td(119873) = 119905119888 + 119905Tg(119873) (15)



119895=1

119905Tdac(119895) 119895 = 1 2 119873 (16)


Complexity 7

34

34

34

7

7

7

PIC3

2MX7

95F5

12L

81CS2

SCKSDI

GND

LDAC65SCK

SDO

EDACxU2

SLAVE 1

81CS2

SCKSDI

GND

LDAC65SCK

SDO

EDACyU3

SLAVE 2

81CS2

SCKSDI

GND

LDAC65SCK

SDO

EDACzU4

SLAVE 3



EDACx

EDACy

EDACz

SCK

SDO


RB12

RB11

RB7

RD0

RD10

MASTER

4

3

8

20

18

52

51OSC2

MCLR

TFTTSD


22 pf 22 pf

10KΩ

10 KΩ

10 KΩ

10 KΩ

16 MHz

OSC1

x(t)

y(t)

z(t)

V＞＞ = 33 V

V＞＞

VＭＭ

V＝＝ = 5 V V＝＝ = 5 V

V＝＝ = 5 V

V＝＝ = 5 V

V＝＝

V＝＝

V＝＝

VＩＯＮ

VＩＯＮ

VＩＯＮ

VＬ

VＬ

VＬ




119876119879(119873=3) = 120591 1119879Td(119873) = 120591119891Td(119873) (17)


119905119899 = 119899 [119905119888 + 119905Tg(119873)] = 119899 [119905119888 + 119905tf t + 119905Tdac(119873)] (18)







8 Complexity

(a) (b) (c)

(d) (e)


L1

L2

L3

1000050000minus04

minus02

0

02

(a)

L㰀

1

L㰀

2

L㰀

3

2001000

minus05

0

05

1

(b)




1 (0 0085]


2 [0086 +infin)

11987110158401 = no valid


Complexity 9













10 Complexity

Start


protocol


algorithm



microcontroller


TFTTSD U1

Loop

Loop




and z(n)




















Complexity 11

b

0

1

2

3

4

5

1 2 3 4 50

d







5 Conclusions







Acknowledgments


12 Complexity

References




































Complexity 13
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 5

Vd

R

y

R8

R4

R5

R1

R2

R3

R

R9

R10

R11

R12

R13

R7

R6

C3

C1

+Vcc+Vcc

minusVcc

Vd

minus

+

AD633

AD633

minus

+

minus

+

minus

+

minus

+

minus

+

z

zy

C2

x

minusy

minusy

minusy

minusy

minusxminusx

minusx

minusz

minusz




(10)



(11)





x = f (x) x (0) = x0

x isin R119873(12)

is given by

x(119899+1) = x(119899) + 120591f (x(119899)) (13)






6 Complexity

(a) (b) (c)

(d) (e) (f)


minus10 minus20

00

1020

minus10

minus5

0

5

z (n)

y(n) x(n)





119879Td(119873) = 1119891Td(119873) = 119905119888 + 119905Tg(119873) (15)



119895=1

119905Tdac(119895) 119895 = 1 2 119873 (16)


Complexity 7

34

34

34

7

7

7

PIC3

2MX7

95F5

12L

81CS2

SCKSDI

GND

LDAC65SCK

SDO

EDACxU2

SLAVE 1

81CS2

SCKSDI

GND

LDAC65SCK

SDO

EDACyU3

SLAVE 2

81CS2

SCKSDI

GND

LDAC65SCK

SDO

EDACzU4

SLAVE 3



EDACx

EDACy

EDACz

SCK

SDO


RB12

RB11

RB7

RD0

RD10

MASTER

4

3

8

20

18

52

51OSC2

MCLR

TFTTSD


22 pf 22 pf

10KΩ

10 KΩ

10 KΩ

10 KΩ

16 MHz

OSC1

x(t)

y(t)

z(t)

V＞＞ = 33 V

V＞＞

VＭＭ

V＝＝ = 5 V V＝＝ = 5 V

V＝＝ = 5 V

V＝＝ = 5 V

V＝＝

V＝＝

V＝＝

VＩＯＮ

VＩＯＮ

VＩＯＮ

VＬ

VＬ

VＬ




119876119879(119873=3) = 120591 1119879Td(119873) = 120591119891Td(119873) (17)


119905119899 = 119899 [119905119888 + 119905Tg(119873)] = 119899 [119905119888 + 119905tf t + 119905Tdac(119873)] (18)







8 Complexity

(a) (b) (c)

(d) (e)


L1

L2

L3

1000050000minus04

minus02

0

02

(a)

L㰀

1

L㰀

2

L㰀

3

2001000

minus05

0

05

1

(b)




1 (0 0085]


2 [0086 +infin)

11987110158401 = no valid


Complexity 9













10 Complexity

Start


protocol


algorithm



microcontroller


TFTTSD U1

Loop

Loop




and z(n)




















Complexity 11

b

0

1

2

3

4

5

1 2 3 4 50

d







5 Conclusions







Acknowledgments


12 Complexity

References




































Complexity 13
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




6 Complexity

(a) (b) (c)

(d) (e) (f)


minus10 minus20

00

1020

minus10

minus5

0

5

z (n)

y(n) x(n)





119879Td(119873) = 1119891Td(119873) = 119905119888 + 119905Tg(119873) (15)



119895=1

119905Tdac(119895) 119895 = 1 2 119873 (16)


Complexity 7

34

34

34

7

7

7

PIC3

2MX7

95F5

12L

81CS2

SCKSDI

GND

LDAC65SCK

SDO

EDACxU2

SLAVE 1

81CS2

SCKSDI

GND

LDAC65SCK

SDO

EDACyU3

SLAVE 2

81CS2

SCKSDI

GND

LDAC65SCK

SDO

EDACzU4

SLAVE 3



EDACx

EDACy

EDACz

SCK

SDO


RB12

RB11

RB7

RD0

RD10

MASTER

4

3

8

20

18

52

51OSC2

MCLR

TFTTSD


22 pf 22 pf

10KΩ

10 KΩ

10 KΩ

10 KΩ

16 MHz

OSC1

x(t)

y(t)

z(t)

V＞＞ = 33 V

V＞＞

VＭＭ

V＝＝ = 5 V V＝＝ = 5 V

V＝＝ = 5 V

V＝＝ = 5 V

V＝＝

V＝＝

V＝＝

VＩＯＮ

VＩＯＮ

VＩＯＮ

VＬ

VＬ

VＬ




119876119879(119873=3) = 120591 1119879Td(119873) = 120591119891Td(119873) (17)


119905119899 = 119899 [119905119888 + 119905Tg(119873)] = 119899 [119905119888 + 119905tf t + 119905Tdac(119873)] (18)







8 Complexity

(a) (b) (c)

(d) (e)


L1

L2

L3

1000050000minus04

minus02

0

02

(a)

L㰀

1

L㰀

2

L㰀

3

2001000

minus05

0

05

1

(b)




1 (0 0085]


2 [0086 +infin)

11987110158401 = no valid


Complexity 9













10 Complexity

Start


protocol


algorithm



microcontroller


TFTTSD U1

Loop

Loop




and z(n)




















Complexity 11

b

0

1

2

3

4

5

1 2 3 4 50

d







5 Conclusions







Acknowledgments


12 Complexity

References




































Complexity 13
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 7

34

34

34

7

7

7

PIC3

2MX7

95F5

12L

81CS2

SCKSDI

GND

LDAC65SCK

SDO

EDACxU2

SLAVE 1

81CS2

SCKSDI

GND

LDAC65SCK

SDO

EDACyU3

SLAVE 2

81CS2

SCKSDI

GND

LDAC65SCK

SDO

EDACzU4

SLAVE 3



EDACx

EDACy

EDACz

SCK

SDO


RB12

RB11

RB7

RD0

RD10

MASTER

4

3

8

20

18

52

51OSC2

MCLR

TFTTSD


22 pf 22 pf

10KΩ

10 KΩ

10 KΩ

10 KΩ

16 MHz

OSC1

x(t)

y(t)

z(t)

V＞＞ = 33 V

V＞＞

VＭＭ

V＝＝ = 5 V V＝＝ = 5 V

V＝＝ = 5 V

V＝＝ = 5 V

V＝＝

V＝＝

V＝＝

VＩＯＮ

VＩＯＮ

VＩＯＮ

VＬ

VＬ

VＬ




119876119879(119873=3) = 120591 1119879Td(119873) = 120591119891Td(119873) (17)


119905119899 = 119899 [119905119888 + 119905Tg(119873)] = 119899 [119905119888 + 119905tf t + 119905Tdac(119873)] (18)







8 Complexity

(a) (b) (c)

(d) (e)


L1

L2

L3

1000050000minus04

minus02

0

02

(a)

L㰀

1

L㰀

2

L㰀

3

2001000

minus05

0

05

1

(b)




1 (0 0085]


2 [0086 +infin)

11987110158401 = no valid


Complexity 9













10 Complexity

Start


protocol


algorithm



microcontroller


TFTTSD U1

Loop

Loop




and z(n)




















Complexity 11

b

0

1

2

3

4

5

1 2 3 4 50

d







5 Conclusions







Acknowledgments


12 Complexity

References




































Complexity 13
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




8 Complexity

(a) (b) (c)

(d) (e)


L1

L2

L3

1000050000minus04

minus02

0

02

(a)

L㰀

1

L㰀

2

L㰀

3

2001000

minus05

0

05

1

(b)




1 (0 0085]


2 [0086 +infin)

11987110158401 = no valid


Complexity 9













10 Complexity

Start


protocol


algorithm



microcontroller


TFTTSD U1

Loop

Loop




and z(n)




















Complexity 11

b

0

1

2

3

4

5

1 2 3 4 50

d







5 Conclusions







Acknowledgments


12 Complexity

References




































Complexity 13
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 9













10 Complexity

Start


protocol


algorithm



microcontroller


TFTTSD U1

Loop

Loop




and z(n)




















Complexity 11

b

0

1

2

3

4

5

1 2 3 4 50

d







5 Conclusions







Acknowledgments


12 Complexity

References




































Complexity 13
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




10 Complexity

Start


protocol


algorithm



microcontroller


TFTTSD U1

Loop

Loop




and z(n)




















Complexity 11

b

0

1

2

3

4

5

1 2 3 4 50

d







5 Conclusions







Acknowledgments


12 Complexity

References




































Complexity 13
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 11

b

0

1

2

3

4

5

1 2 3 4 50

d







5 Conclusions







Acknowledgments


12 Complexity

References




































Complexity 13
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




12 Complexity

References




































Complexity 13
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




Complexity 13
























Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of











Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




a new simple chaotic lorenz-type system and its digital...

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