Dynamic Analysis and Circuit Implementation of a …downloads.hindawi.com/journals/mpe/2019/6581586.pdflayer secure communication systems and ultra-fast physi-cal random bits generation

Research ArticleDynamic Analysis and Circuit Implementation of a New 4DLorenz-Type Hyperchaotic System

A Al-khedhairi 1 A Elsonbaty23 A H Abdel Kader3 and A A Elsadany 24

1Department of Statistics and Operations Researches College of Science King Saud UniversityPO Box 2455 Riyadh 11451 Saudi Arabia

2Mathematics Department College of Science and Humanities Studies Prince Sattam Bin Abdulaziz UniversityAl-Kharj Saudi Arabia

3Department of Engineering Mathematics and Physics Faculty of Engineering Mansoura University Mansoura 35516 Egypt4Basic Science Department Faculty of Computers and Informatics Ismailia 41522 Suez Canal University Egypt

Correspondence should be addressed to A Al-khedhairi akhediriksuedusa

Received 1 December 2018 Accepted 19 January 2019 Published 6 February 2019

Academic Editor Mingshu Peng

Copyright copy 2019 A Al-khedhairi et alThis 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

This paper attempts to further extend the results of dynamical analysis carried out on a recent 4D Lorenz-type hyperchaotic systemwhile exploring new analytical results concerns its local and global dynamics In particular the equilibrium points of the systemalong with solutionrsquos continuous dependence on initial conditions are examinedThen a detailed 1198852 symmetrical Bogdanov-Takensbifurcation analysis of the hyperchaotic system is performed Also the possible first integrals and global invariant surfaces whichexist in systemrsquos phase space are analytically found Theoretical results reveal the rich dynamics and the complexity of systembehavior Finally numerical simulations and a proposed circuit implementation of the hyperchaotic system are provided to validatethe present analytical study of the system

1 Introduction

Nonlinear dynamics analysis of various phenomena andsystems of physics engineering biology chemistry economyand industry has attracted a great interest among scientistsand considered a very active area of research from 1960s inthe last century till now [1ndash4] There are two key reasonswhich interpret this great interest The first one is that thedynamical systems tools help scientists better comprehendand analyze the varieties of nonlinear characteristics and newphenomena exhibited by systems from different disciplinesIn particular the tools of dynamical systems such as theapplied bifurcation theories are successfully employed toinvestigate the qualitative behaviors of nonlinear systems[5ndash7] This includes investigation of equilibrium points andtheir stability creation destruction and stability of periodicorbits quasiperiodic behavior homoclinic orbits creationor destruction of chaotic attractors and chaos control andsynchronization The second reason is that engineers andscientists can utilize some of the fascinating features of

nonlinear dynamical systems in wide range of interestingapplications

The spellbinding chaotic dynamics as an example is rec-ognized by high sensitivity to initial conditions and positiveLyapunov exponents The generation of chaos for practicalapplications can be achieved by exploiting nonlinear elec-tronic circuits nonlinearities of laser systems with feedbacksor via digital platforms such as FPGAs and DSPs The noise-like behavior of chaotic systems their wide spectrum andthe possibility of attaining chaos synchronization betweentwo chaotic systems render them essential for cutting edgeapplications related to cryptography and robust physical-layer secure communication systems and ultra-fast physi-cal random bits generation [4ndash11] The other applicationsof dynamical systems methods and chaos theory includefinancial systems mathematical biology nonlinear circuitsnonlinear mechanical systems plasma physics chaos controlefficient image encryption neuroscience research and geo-physics [12ndash33]

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 6581586 17 pageshttpsdoiorg10115520196581586

2 Mathematical Problems in Engineering

Themore complex hyperchaotic system possesses at leasttwo positive Lyapunov exponents and it has a phase spaceof dimension at least four Clearly the hyperchaotic systemshavemore randomness and higher unpredictability than sim-ple chaotic systems Therefore hyperchaos is more preferredthan simple chaos and its applications have recently becomea central topic in research including chaos-based securecommunications image encryption and cryptography

In the last two decades some interesting high-dimensional hyperchaotic systems in science and engineer-ing have been explored and their dynamics have beenextensively investigated [18 24 27 28 34 35] In fact it isof great importance from theoretical and practical aspectsto explain complicated phenomena and internal structuralcharacteristics of hyperchaotic systems This research linefocuses on applying codimension two or three bifurcationanalysis to system under investigation For exampleanalysis of Bogdanov-Takens bifurcation degenerate Hopfbifurcation and Heteroclinic and Homoclinic bifurcationscan be undertaken Also it is crucial to determine whetherthere exists any set of parameters values for which theconsidered system is integrable and find the correspondinginvariant surfaces if exist As it is practically unattainablegoal to apply exhaustive numerical investigation to acquirevalues of parameters where the hyperchaotic system isintegrable one needs a powerful analytical method thatenables achieving this goal easily

Recently theoretical analysis based on Fishing Principleis applied to study some global features of a new four-dimensional Lorenz-type hyperchaotic system [36] Morespecifically conditions for existence of homoclinic orbits areobtained This work aims at extending the aforementionedwork and exploring other aspects of complicated dynami-cal behaviors of the 4D Lorenz-type hyperchaotic systemAnalytical bifurcation structure of the model which includes1198852 symmetrical Bogdanov-Takens Pitchfork Andronov-Hopf bifurcation and homoclinic bifurcation are obtainedExistence uniqueness and continuous dependence on initialconditions for the solution of 4D hyperchaotic system are allinvestigated It is important to ask whether there is any set ofparameter values in which system dynamics are regular andthe studied system is integrable The different integrable casesof the system are studied and the closed-form expressionsfor invariant surfaces corresponding to first integrals of thesystem are also obtained Moreover the hyperchaotic systemhas been simulated using a proposed electronic circuit real-ization and numerical simulations are performed confirmingthe new results of theoretical analysis Note that attractorsfound in dynamical system are classified as being self-excitedor hidden attractors [37 38] The key difference betweenthem is that the self-excited attractor has a basin of attractionexcited from unstable equilibria The hidden attractor on theother hand has a basin of attraction with no intersection withneighborhoods of dynamical systems equilibrium points Inthis work we focus on self-excited attractors exist in thehyperchaotic system

The rest of the paper is structured as follows In Section 2we introduce the 4D Lorenz-type system and discuss theequilibrium pointsrsquo existence A sufficient condition for

continuous dependence on initial condition is determinedPhase portraits bifurcation diagrams and Lyapunov charac-teristics spectrum are obtained In Section 3 the analysis ofsome possible codimension two bifurcations is performed Itis shown that the 4D Lorenz system undergoes Bogdanov-Takens bifurcation Andronov-Hopf bifurcation Pitchforkbifurcation and homoclinic bifurcation The integrabilityanalysis of system is investigated in Section 4 A practicalapplication to engineering will be realized by an electroniccircuit in Section 5 Finally Section 6 concludes the paper

2 The 4D Lorenz-Type Hyperchaotic System

The following 4D hyperchaotic system of Lorenz type waspresented in [36] = 119886 (119910 minus 119909) 119910 = 119888119909 minus 119889119910 minus 119909119911 = minus119887119911 + 119909119910 + 119908 = minus119903119908 + 119896119911 (1)

where state variables of the system are denoted by 119909 119910 119911and 119908 the parameters of the system are represented by119886 119887 119888 119889 119903 and 119896 and the dot above state variables refers totime derivative of state variables

The following subsections examine themain properties ofsystem (1) and provide elementary dynamical analysis of themodel

21 Equilibrium Points of the System The fixed points ofsystem (1) can be obtained when 119909 = = = = 0 iethrough solving 119886 (119910 minus 119909) = 0119888119909 minus 119889119910 minus 119909119911 = 0minus119887119911 + 119909119910 + 119908 = 0minus119903119908 + 119896119911 = 0 (2)

Therefore system (1) has the following equilibrium solutions(I) when (119889 minus 119888)(119896 minus 119887119903)119903 le 0 the fixed point is given by119909 = 119910 = 119911 = 119908 = 0 (3)

(II) when (119889 minus 119888)(119896 minus 119887119903)119903 gt 0 the fixed points are119909 = 119910 = 119911 = 119908 = 0119909 = 119910 = plusmnradic (119889 minus 119888) (119896 minus 119887119903)119903 119911 = 119888 minus 119889119908 = 119896 (119888 minus 119889)119903

(4)

Mathematical Problems in Engineering 3

22 Existence and Uniqueness of the Solution The hyper-chaotic system (1) can be put in the form

X (t) = Φ (X (t)) 119905 isin (0 119879] (5)

where

X = [[[[[[1199091119909211990931199094

]]]]]] X0 = [[[[[[

11990901119909021199090311990904]]]]]]

Φ (X) = [[[[[[119886 (1199092 minus 1199091) 1198881199091 minus 1198891199092 minus 11990911199093minus1198871199093 + 11990911199092 + 1199094minus1199031199094 + 1198961199093

]]]]]] (6)

and the initial conditions are given by

X (0) = X0 (7)

For the class of continuous functions 119865(119905) isin 119862[0 119879] weuse the following norm in subsequent analysis119865 = sup

119905isin[0119879]

|119865 (119905)| (8)

while the matrix 119870 = [119896119894119895[119905]] of continuous functionsemploys the norm 119870 = sum

119894119895

sup119905isin[0119879]

10038161003816100381610038161003816119896119894119895 [119905]10038161003816100381610038161003816 (9)

It is obvious that 4D system (1) is dissipative if nablaF(X) =minus(119886 + 119887 + 119889 + 119903) lt 0 Now solution of the system is examinedin specific region Γ times 119869 where 119869 = [0 119879] andΓ = (1199091 1199092 1199093 1199094) max |119909| 100381610038161003816100381611990921003816100381610038161003816 100381610038161003816100381611990931003816100381610038161003816 and 100381610038161003816100381611990941003816100381610038161003816le 119872 119872 gt 0 (10)

Parameter 119872 is utilized to lay a boundary for the phase spaceregion where existence and uniqueness of the solution areinvestigated

The solution of (5) and (7) can be represented by

X (t) = X0 + int1199050F (X (s)) ds (11)

The equivalence of the integral equation (11) and system (5)-(7) is obvious Now denoting the right hand side of (11) by

H(X) then for X1 = [ 11990911119909121199091311990914

] and X2 = [ 11990921119909221199092311990924

] we get

H (X1) minus H (X2) = int1199050

(Φ (X1 (s)) minusΦ (X2 (s))) ds (12)

and therefore1003816100381610038161003816H (X1) minus H (X2)1003816100381610038161003816le int1199050

1003816100381610038161003816(Φ (X1 (s)) minusΦ (X2 (s)))1003816100381610038161003816 ds (13)

Then we obtain1003817100381710038171003817H (X1) minus H (X2)1003817100381710038171003817 le 119879max |119886| + |119888| + 2119872 |119886| + |119889|+ 119872 |119887| + |119896| + 119872 1 + |119903| 1003817100381710038171003817X1 minus X21003817100381710038171003817 le 119870 1003817100381710038171003817X1minus X2

1003817100381710038171003817 (14)

where119870 = 119879max |119886| + |119888| + 2119872 |119886| + |119889| + 119872 |119887| + |119896|+ 119872 1 + |119903| gt 0 (15)

Thus X = H(X) for 0 lt 119870 lt 1 as sufficient condition is acontraction mapping

Theorem 1 Assume that 0 lt 119879max|119886| + |119888| + 2119872 |119886| +|119889| + 119872 |119887| + |119896| + 119872 1 + |119903| lt 1 then a unique solution ofhyperchaotic system (1) starting from initial condition X(0) =X0 in the region Ω times 119869 exists23 Continuous Dependence on Initial Conditions The con-tinuous dependence on initial conditions means that solutiontrajectories of the system which start close to each other stillclose to each other with evolution of time This property iscontrary to sensitive dependence on initial conditions whichspecifies chaotic dynamics The goal of the next analysisis to find the particular parameters and time range wherecontinuous dependence on state variables initial conditions ispersevered ie system (1) does not exhibit chaotic dynamics

Assume that there are two points of initial conditions ofsystem (5)-(7) namely X01 and X02 satisfy1003817100381710038171003817X01 minus X02

1003817100381710038171003817 le 120575 (16)First suppose that the condition of Theorem 1 holds Thus

X1 (119905) = X01 + int1199050H (X1 (s)) ds

X2 (119905) = X02 + int1199050H (X2 (s)) ds (17)

and also we obtain1003817100381710038171003817X1 minus X21003817100381710038171003817 le 1003817100381710038171003817X01 minus X02

1003817100381710038171003817 + 119870 1003817100381710038171003817X1 minus X21003817100381710038171003817 (18)

and (1 minus 119870) 1003817100381710038171003817X1 minus X21003817100381710038171003817 le 1003817100381710038171003817X01 minus X02

1003817100381710038171003817 (19)where 0 lt 119870 lt 1 (20)is defined by (15) Finally we get1003817100381710038171003817X1 minus X2

1003817100381710038171003817 le 120598 (21)where 120598 = 120575(1 minus 119870) Thus we can formulate the followingtheorem

Theorem 2 e solution of hyperchaotic system (1) exhibitscontinuous dependence on initial conditions if system (1)


satisfies the condition of eorem 1 More specifically forall120598 gt0exist120575(120598) = (1 minus 119870)120598 gt 0 such that whenever X01 minus X02 le 120575 itimplies that X1 minus X2 le 1205983 1198852 Symmetry Bogdanov-Takens Bifurcation

Now we study the case where 119888 = 119888lowast = minus119886 and119889 = 119889lowast = minus119886 that implies Jacobian matrix has two real

zero eigenvalues 12058212 = 0 and two negative eigenvalues ifeither 1198872minus2119887119903+4119896+1199032 lt 119887+119903 or 1198872minus2119887119903+4119896+1199032 lt 0 evaluatedat the origin The following coordinatesrsquo transformation isapplied to (1) in order to put the system in standard form

(1199091119909211990931199094) = ((

1 minus 1119886 0 01 0 0 00 0 minus119887 + 119903 minus 1206032119896 minus119887 + 119903 + radic1198872 minus 2119903119887 + 1199032 + 411989621198960 0 1 1))

(1199061119906211990631199064) (22)

which yields

(1234) = (1234

) (23)

where

1 = 119910411991021206032119886119896 minus 119910311991021206032119886119896 minus 119887119910311991022119886119896 minus 119887119910411991022119886119896 + 119903119910311991022119886119896+ 119903119910411991022119886119896 + 119910111991031206032119896 minus 119910111991041206032119896 + 119887119910111991032119896+ 119887119910111991042119896 minus 119903119910111991032119896 minus 119903119910111991042119896 + 11991022 = 119886119910111991031206032119896 minus 119886119910111991041206032119896 + 119886119887119910111991032119896 + 119886119887119910111991042119896minus 119886119903119910111991032119896 minus 119886119903119910111991042119896 minus 119910211991031206032119896 + 119910211991041206032119896minus 119887119910211991032119896 minus 119887119910211991042119896 + 119903119910211991032119896 + 119903119910211991042119896 3 = 11989611991011199102119886120603 minus 119887211991034120603 minus 119887211991044120603 + 11988711990311991032120603 + 11988711990311991042120603 minus 141199103120603+ 141199104120603 minus 11989611991021120603 minus 1198961199103120603 minus 1198961199104120603 minus 119903211991034120603 minus 119903211991044120603minus 11988711991032 minus 11990311991032 4 = minus11989611991011199102119886120603 + 119887211991034120603 + 119887211991044120603 minus 11988711990311991032120603 minus 11988711990311991042120603 + 11989611991021120603minus 141199103120603 + 1198961199103120603 + 119903211991034120603 + 141199104120603 + 1198961199104120603+ 119903211991044120603 minus 11988711991042 minus 11990311991042

(24)

such that 120603 = radic1198872 minus 2119887119903 + 4119896 + 1199032

The center manifold is assumed in the form of second-order polynomial for sufficiently small 1199061 and 1199062 as1199063 = 1205722011990621 + 1205721111990611199062 + 1205720211990622 (25a)1199064 = 1205732011990621 + 1205731111990611199062 + 1205730211990622 (25b)

then substituting from ((25a)-(25b)) and the first two equa-tions of (23) into the last two equation of (23) By comparingthe coefficient of 11990611989421199061198953 119894 + 119895 = 2 in both sides of last twoequations of (23) after substitution in ((25a)-(25b)) the valuesof center manifold coefficients are obtained as follows12057220 = minus 21198961198872 + 119887 (120603 minus 2119903) + 119903 (120603 + 119903) + 4119896

12057211 = 2119896 (4119886120603 + 1198872 + 119887 (120603 minus 2119903) + 119903 (120603 + 119903) + 4119896)119886 (1198872 + 119887 (120603 minus 2119903) + 119903 (120603 + 119903) + 4119896)2 12057202= minus 4119896 ((119887 minus 119903)2 + 4119896) (4119886 + 120603 + 119887 + 119903)119886 (1198872 + 119887 (radic(119887 minus 119903)2 + 4119896 minus 2119903) + 119903 (120603 + 119903) + 4119896)3 12057320 = minus119896 (120603 + 119887 + 119903)2120603 (119896 minus 119887119903) 12057311 = 119896 (minus4119886 + 120603 + 119887 + 119903)2119886120603 (119896 minus 119887119903) 12057302 = 119896 (119886120603 + 119886 (minus (119887 + 119903)) minus 119887119903 + 119896)119886120603 (119896 minus 119887119903)2

(26)

The dynamics of the system on the center manifold is thendescribed by

(12) = (11990620 ) + 1Ψ ( 119866 (1199062 1199063)119886119866 (1199062 1199063)) (27)

where


Ψ = 2 ( 11988724120603 minus 1198871199032120603 + 14120603 + 119896120603 + 11990324120603 + 1198872 + 1199032) 119866 (1199062 1199063) = 21206031199062111990621199035119886 (119887119903 minus 119896) (1199032 + 119896)2 minus 120603119906311199035(119887119903 minus 119896) (1199032 + 119896)2 minus 12060311990611199062211990351198862 (119887119903 minus 119896) (1199032 + 119896)2 + 61206031199061119906221199034119886 (119887119903 minus 119896) (1199032 + 119896)2 + 61199061119906221199034119886 (1199032 + 119896)2 120603minus 41206031199062111990621199034(119887119903 minus 119896) (1199032 + 119896)2 minus 21206031199063211990341198862 (119887119903 minus 119896) (1199032 + 119896)2 minus 41199062111990621199034(1199032 + 119896)2 120603 minus 21199063211990341198862 (1199032 + 119896)2 120603 + 21198871199063211990331198862 (1199032 + 119896)2 120603+ 41206031199061119906221199033(119887119903 minus 119896) (1199032 + 119896)2 + 41199061119906221199033(1199032 + 119896)2 120603 + 41198961206031199062111990621199033119886 (119887119903 minus 119896) (1199032 + 119896)2 + 41198871199062111990621199033(1199032 + 119896)2 120603 minus 2119896120603119906311199033(119887119903 minus 119896) (1199032 + 119896)2minus 4120603119906321199033119886 (119887119903 minus 119896) (1199032 + 119896)2 minus 211989612060311990611199062211990331198862 (119887119903 minus 119896) (1199032 + 119896)2 minus 4119906321199033119886 (1199032 + 119896)2 120603 minus 61198871199061119906221199033119886 (1199032 + 119896)2 120603 + 4119887119906321199032119886 (1199032 + 119896)2 120603+ 21198961199063211990321198862 (1199032 + 119896)2 120603 + 61198961206031199061119906221199032119886 (119887119903 minus 119896) (1199032 + 119896)2 + 41198961199062111990621199032(1199032 + 119896)2 120603 minus 41198961206031199062111990621199032(119887119903 minus 119896) (1199032 + 119896)2 minus 21198961206031199063211990321198862 (119887119903 minus 119896) (1199032 + 119896)2minus 41198871199061119906221199032(1199032 + 119896)2 120603 minus 61198961199061119906221199032119886 (1199032 + 119896)2 120603 + 411989612060311990632119903119886 (119887119903 minus 119896) (1199032 + 119896)2 + 2011989611990632119903119886 (1199032 + 119896)2 120603 + 2119887119896119906321199031198862 (1199032 + 119896)2 120603 + 2120603119906111990622119903(119887119903 minus 119896)2+ 21198962120603119906211199062119903119886 (119887119903 minus 119896) (1199032 + 119896)2 + 4119887119896119906211199062119903(1199032 + 119896)2 120603 minus 212060311990632119903119886 (119887119903 minus 119896)2 minus 119896212060311990631119903(119887119903 minus 119896) (1199032 + 119896)2 minus 4119896120603119906111990622119903(119887119903 minus 119896) (1199032 + 119896)2minus 11989621206031199061119906221199031198862 (119887119903 minus 119896) (1199032 + 119896)2 minus 20119896119906111990622119903(1199032 + 119896)2radic1198872 minus 2119903119887 + 1199032 + 4119896 minus 6119887119896119906111990622119903119886 (1199032 + 119896)2 120603 + 41198962119906321198862 (1199032 + 119896)2 120603 + 4119887119896119906111990622(1199032 + 119896)2 120603+ 81198962119906211199062(1199032 + 119896)2 120603 + minus1198862119903211990631 minus 119886211989611990631 + 21198861199032119906211990621 + 2119886119896119906211990621 + 21198862119906221199061 minus 1199032119906221199061 minus 119896119906221199061 minus 2119886119906321198862 (119887119903 minus 119896)

minus 1199061 (1198861199061 minus 1199062)21198862 minus 2 (minus119903211990632 minus 11989611990632 + 1198861199032119906111990622 + 119886119896119906111990622)119886 (119887119903 minus 119896)2 minus 411988711989611990632119886 (1199032 + 119896)2 120603 minus 121198962119906111990622119886 (1199032 + 119896)2 120603

(28)

Conditional normal form at bifurcation values of param-eters can be obtained by following Kuznetsov approachwhich implies that the simplified systemwhich represents theconditional normal form at bifurcation value is the followingsystem

V1 = V2V2 = 1198861199031 minus 119887119903V31+ 119903 (119886 (minus31198872 + 6119887119903 minus 31199032 minus 12) minus 31198872 + 6119887119903 minus 31199032 minus 12)11198873119903 + 1198872 (minus21199032 minus 1) + 119887119903 (11199032 + 6) minus 11199032 minus 4sdot V21V2

(29)

The next step in this analysis is to obtain the universalunfolding of the BT bifurcation In order to put the systemin an appropriate form the bifurcation parameters should be

perturbed around bifurcation value such that 119888 = minus119886 + Δ 1and 119889 = minus119886 + Δ 2 The following system is attained1 = 119886 (minus1199091 + 1199092) 2 = (minus119886 + Δ 1) 1199092 minus (minus119886 + Δ 2) 1199092 minus 119909111990933 = 1199094 + 11990911199092 minus 11988711990934 = minus1199031199094 + 1198961199093Δ 1 = 0Δ 2 = 0

(30)

Now applying transformation (22) to get

(1234) = (1234

) (31)

6 Mathematical Problems in EngineeringΔ 1 = 0Δ 2 = 0 (32)

where

1 = 119906411990621206032119886119896 minus 119906311990621206032119886119896 minus 119887119906311990622119886119896 minus 119887119906411990622119886119896 + 119903119906311990622119886119896+ 119903119906411990622119886119896 minus Δ 11199062119886 + 119906111990631206032119896 minus 119906111990641206032119896+ 119887119906111990632119896 + 119887119906111990642119896 minus 119903119906111990632119896 minus 119903119906111990642119896 + Δ 11199061minus Δ 21199061 + 11990622 = 119886119906111990631206032119896 minus 119886119906111990641206032119896 + 119886119887119906111990632119896 + 119886119887119906111990642119896minus 119886119903119906111990632119896 minus 119886119903119906111990642119896 + 119886Δ 11199061 minus 119886Δ 21199061minus 119906211990631206032119896 + 119906211990641206032119896 minus 119887119906211990632119896 minus 119887119906211990642119896+ 119903119906211990632119896 + 119903119906211990642119896 minus Δ 111990623 = 11989611990611199062119886120603 minus 119887211990634120603 minus 119887211990644120603 + 11988711990311990632120603 + 11988711990311990642120603 minus 141199063120603+ 141199064120603 minus 11989611990621120603 minus 1198961199063120603 minus 1198961199064120603 minus 119903211990634120603 minus 119903211990644120603minus 11988711990632 minus 11990311990632 4 = minus11989611990611199062119886120603 + 119887211990634120603 + 119887211990644120603 minus 11988711990311990632120603 minus 11988711990311990642120603 + 11989611990621120603minus 141199063120603 + 1198961199063120603 + 119903211990634120603 + 141199064120603 + 1198961199064120603+ 119903211990644120603 minus 11988711990642 minus 11990311990642

(33)

The center manifold is assumed as follows1199063 = 119894+119895+119896+119897=2sum119894119895119896119897=0

11989411989511989611989711990611989411199061198952Δ1198961Δ11989721199064 = 119894+119895+119896+119897=2sum

119894119895119896119897=0

12057311989411989511989611989711990611989411199061198952Δ1198961Δ1198972 (34)

where parameters 119894119895119896119897 and 120573119894119895119896119897 can be attained through thesame way employed in the first part of this subsection Thefollowing reduced system on centermanifold is obtained aftersome simplifications(12) = 119860 (11990611199062) + (1198661 (1199061 1199062 Δ 1 Δ 2)1198662 (1199061 1199062 Δ 1 Δ 2))

119860 = (0 10 0) (35)

where1198661 (1199061 1199062 Δ 1 Δ 2)= 1199062111990621198873119886 (1198872 + (120603lowast minus 2119903) 119887 + 119903 (119903 + 120603lowast) + 16)2+ 120603lowast1199062111990621198872119886 (1198872 + (120603lowast minus 2119903) 119887 + 119903 (119903 + 120603lowast) + 16)2minus 31199031199062111990621198872119886 (1198872 + (120603lowast minus 2119903) 119887 + 119903 (119903 + 120603lowast) + 16)2+ 119906311198872 (119887119903 minus 4) + radic1198872 minus 2119903119887 + 1199032 + 161199062111990621198872119886120603lowast (119887119903 minus 4)+ 4120603lowast119906211199062119887(1198872 + (120603lowast minus 2119903) 119887 + 119903 (119903 + 120603lowast) + 16)2+ 31199032119906211199062119887119886 (1198872 + (120603lowast minus 2119903) 119887 + 119903 (119903 + 120603lowast) + 16)2+ 16119906211199062119887119886 (1198872 + (120603lowast minus 2119903) 119887 + 119903 (119903 + 120603lowast) + 16)2minus 1199062111990621198872119886 (119887119903 minus 4) minus radic1198872 minus 2119903119887 + 1199032 + 16119906311198872120603lowast (119887119903 minus 4)+ 119903radic1198872 minus 2119903119887 + 1199032 + 161199062111990622119886120603lowast (119887119903 minus 4) + 1199031199062111990622119886 (119887119903 minus 4)minus 119903119906312 (119887119903 minus 4) minus 119903radic1198872 minus 2119903119887 + 1199032 + 16119906312120603lowast (119887119903 minus 4)minus 4120603lowast119903119906211199062(1198872 + (120603lowast minus 2119903) 119887 + 119903 (119903 + 120603lowast) + 16)2minus 16119903119906211199062119886 (1198872 + (120603lowast minus 2119903) 119887 + 119903 (119903 + 120603lowast) + 16)2 1198662 (1199061 1199062 Δ 1 Δ 2)= minus11988611988711990631radic1198872 minus 2119887119903 + 1199032 + 162120603lowast (119887119903 minus 4)minus 11988611990311990631radic1198872 minus 2119887119903 + 1199032 + 162120603lowast (119887119903 minus 4)+ 4119886119887119906211199062120603lowast(1198872 + 119887 (120603lowast minus 2119903) + 119903 (120603lowast + 119903) + 16)2+ 119886119887119906312 (119887119903 minus 4) minus 119886119903119906312 (119887119903 minus 4)+ 119887119906211199062radic1198872 minus 2119887119903 + 1199032 + 162120603lowast (119887119903 minus 4)


+ 119903119906211199062radic1198872 minus 2119887119903 + 1199032 + 162120603lowast (119887119903 minus 4)+ 1198872119906211199062120603lowast(1198872 + 119887 (120603lowast minus 2119903) + 119903 (120603lowast + 119903) + 16)2+ 16119887119906211199062(1198872 + 119887 (120603lowast minus 2119903) + 119903 (120603lowast + 119903) + 16)2minus 16119903119906211199062(1198872 + 119887 (120603lowast minus 2119903) + 119903 (120603lowast + 119903) + 16)2+ 1198873119906211199062(1198872 + 119887 (120603lowast minus 2119903) + 119903 (120603lowast + 119903) + 16)2minus 1198871199062111990622 (119887119903 minus 4) + 1199031199062111990622 (119887119903 minus 4) (36)

such that 120603lowast = radic(119887 minus 119903)2 + 16Now it necessary to define the next vector valued func-

tions119862 (119904 V 119908)= 3sum119894119895119896=1

1205973119866 (1199061 1199062 Δ 1 Δ 2)120597119906119894120597119906119895120597119906119896 1003816100381610038161003816100381610038161003816100381610038161003816 1199061=1199062=0Δ 1=Δ 2=0

119904119894V1198951199081198961198601 (119904 Λ) = 2sum

119894=1

2sum119895=1

1205972119866 (1199061 1199062 Δ 1 Δ 2)120597119906119894120597Δ 119895 1003816100381610038161003816100381610038161003816100381610038161003816 1199062=1199063=0Δ 1=Δ 2=0

119904119894Λ 1198951198602 (119904 Λ Ω)= 2sum119894=1

2sum119895119896=1

1205972119866 (1199062 1199063 Δ 1 Δ 2)120597119906119894120597Δ 119895120597Δ 119896 1003816100381610038161003816100381610038161003816100381610038161003816 1199062=1199063=0Δ 1=Δ 2=0

119904119894Λ 119895Ω119896(37)

where 119866 (1199062 1199063 Δ 1 Δ 2) = (1198661 (1199061 1199062 Δ 1 Δ 2)1198662 (1199061 1199062 Δ 1 Δ 2)) (38)

The exact expression for these vector valued functions isomitted for brevity Now following the procedure describedin [39] we assume that the following relation holds betweenunfolding parameters 120578 = (1205781 1205782)119879 and perturbation valuesΔ = (Δ 1 Δ 2)119879Δ 119894 = 1205881198941205781 + 1205921198941205782 + 12057411989412057821 + 12057511989412057811205782 + 12057711989412057822 + 119874 (100381710038171003817100381712057810038171003817100381710038173) 119894 = 1 2 (39)

or equivalently120578119894 = 120588119894Δ 1 + 120592119894Δ 2 + 120574119894Δ21 + 120575119894Δ 1Δ 2 + 120577119894Δ22+ 119874 (Δ3) 119894 = 1 2 (40)

The two real linearly independent generalized eigenvec-tors 1199021 and 1199022 that satisfy 1198601199021 = 0 and 1198601199022 = 1199021 can be foundas 1199021 = [10]

1199022 = [01] (41)

Also for the transposed matrix 119860119879 there exist eigenvectors1199011 and 1199012 that have the properties1198601198791199012 = 01198601198791199011 = 11990121199011198791 1199021 = 1199011198792 1199022 = 11199011198791 1199022 = 1199011198792 1199021 = 0(42)

and they are computed as1199011 = [10] 1199012 = [01] (43)

After some calculations the linear part coefficients of (39)can be evaluated as 1205881 = 1119886 1205882 = 01205921 = minus11205922 = minus1

(44)

Then the transversality condition is verified by evaluating

det(1119886 minus10 minus1) = minus 1119886 = 0 (45)

which implies that transversality condition is fulfilled Theother coefficients of (39) can be evaluated by solving a set ofsystems of linear equations described in [39] which results in1205741 = 1205742 = 01205751 = 1205752 = 01205771 = 1205772 = 0 (46)

By substituting from (39) into (40) and comparing thecoefficients of 12057811989411205781198952 119894 + 119895 = 1 2 on both sides of (40)the unfolding parameters 1205781(Δ 1 Δ 2) and 1205782(Δ 1 Δ 2) can beobtained in the following expression1205781 (Δ 1 Δ 2) = 119886Δ 1 minus 119886Δ 21205782 (Δ 1 Δ 2) = minus119886Δ 2 (47)


The system representing the universal unfolding of BTbifurcation is thus written as1 = 11990622 = (119886Δ 1 minus 119886Δ 2) 1199061 minus 119886Δ 21199062 + 1198861199031 minus 11988711990311990631

+ 119903 (119886 (minus31198872 + 6119887119903 minus 31199032 minus 12) minus 31198872 + 6119887119903 minus 31199032 minus 12)11198873119903 + 1198872 (minus21199032 minus 1) + 119887119903 (11199032 + 6) minus 11199032 minus 4sdot 119906211199062(48)

Let 12059930 = 119886119903(1 minus 119887119903) and 12059921 = 119903(119886(minus31198872 + 6119887119903 minus 31199032 minus 12) minus31198872+6119887119903minus31199032minus12)(11198873119903+1198872(minus21199032minus1)+119887119903(11199032+6)minus11199032minus4)then apply the following rescaling of coordinates and time tofurther simplify system (48)

1199061 = 1199041radic10038161003816100381610038161205993010038161003816100381610038161003816100381610038161003816120599211003816100381610038161003816 12058511199062 = minus11990411199042 10038161003816100381610038161205993010038161003816100381610038163210038161003816100381610038161205992110038161003816100381610038162 1205852

119905 = minus 120599211003816100381610038161003816120599301003816100381610038161003816 120591(49)

After simplification we can obtain1205851 = 12058521205852 = 12058311205851 + 12058321205852 + 119904112058531 minus 120585211205852 (50)

where derivative is considered with respect to 120591The bifurcation structure of (50) is summarized in the

next theorem [39ndash41]

Theorem 3 e bifurcation diagram of (50) in 1205831 minus 1205832 planeinvolves the following bifurcation curves

(i) 119875119861 = (1205831 1205832) 1205831 = 0 1205832 = 0(ii) 1198671198611 = (1205831 1205832) 1205832 = 0 1205831 lt 0(iii) 1198671198612 = (1205831 1205832) 1205832 = 1205831 + 119874(12058321) 1205831 gt 0(iv) 119867119871119861 = (1205831 1205832) 1205832 = (45)1205831 + 119874(120583321 ) 1205831 gt 0(v) 119878119873119861 = (1205831 1205832) 1205782 = 07521205831 + 119874(120583321 ) 1205831 gt0 where 1205831 = 9(119886Δ 1 + 1198862(Δ21 + Δ 1Δ 2)) and 1205832 =3(119886(Δ 1 + Δ 2) + 1198862(2Δ21 + 3Δ 1Δ 2 + Δ22)) where 119875119861 represents

a pitchfork bifurcation of system (50) 1198671198611 denotes Hopfbifurcation of the origin equilibrium point 1198671198612 refers to theHopf bifurcation of the nontrivial equilibrium points 119867119871119861corresponds to Homoclinic connection to the origin and finally119878119873119861 denotes the saddle-node bifurcation of symmetric periodicorbits

The phase portraits of system (1) are obtained vianumerical simulations and illustrate examples of differentdynamics that can be exhibited by the hyperchaotic systemIn particular Figure 1 verifies the occurrences of Hopf andpitchfork bifurcations of system (2) whereas Figure 2 shows2D and 3D portraits of hyperchaotic attractor of the system

Bifurcation diagrams are used to provide useful illus-trations for variations of a particular system dynamics withchanges in its parameters Figure 3 shows examples ofbifurcation diagrams of system (1) with respect to differentparameters in the system Figure 4 depicts the associatedLyapunov exponent plots to the cases presented in Figure 3

4 First Integrals of the System

It is well known that if the analyzed system has chaoticdynamics the behaviors of the system are irregular behaviorsSo it is impossible to predict the behavior of the system ina long time scale Therefore it is important to ask whetherthere is any set of parameter values in which system dynamicsare regular and that the studied system is integrable In thissection we provide the question of global analytic integrabil-ity investigated for hyperchaotic system (1) In particular weprovide conditions of analytic integrability in hyperchaoticsystem (1) and obtain the possible first integrals of it

Theorem 4 System (1) has the following first integrals

(1) 1198681(119909 119910 119911 119908 119905) = 1199082 minus 119896(1199102 + 1199112) when 119888 = 119887 = 119889 =119903 = 0(2) 1198682(119909 119910 119911 119908 119905) = 1198902119887119905(1198871199082 minus 119896119887(1199102 + 1199112) + 1198881198961199092) when119886 = 119889 = 119903 = 119887(3) 1198683(119909 119910 119911 119908 119905) = 1198902119887119905(21198883(1199082 + 1198881198871199092 minus 1198872(1199102 + 1199112)) +1198882(minus2119887119911 + 2119908 + 1199092)) when 119886 = 119888 = 119889 = 119903 = 119887 and119896 = 1198872(4) 1198684(119909 119910 119911 119908 119905) = 1198902119886119905(1199092(119903 minus 2119886) + 2119886119911(2119886 minus 119903) minus 2119886119908)

when 119896 = (2119886 minus 119887)(2119886 minus 119903)(5) 1198685(119909 119910 119911 119908 119905) = 1198902119887119905(2119888119887(119887119911 minus 119908) minus 1198872(1199102 + 1199112) + 1199082)

when 119888 = 119889 = 119903 = 119887 and 119896 = 1198872where 1198882 and 1198883 are arbitrary constantsProof The function 119868(119909(119905) 119910(119905) 119911(119905) 119908(119905) 119905) is said to be thefirst integral of system (1) if it satisfies the condition119889119868119889119909 = 119868119905 + 119868119909 + 119868119910 + 119868119911 + 119868119908 = 0 (51)

Substituting (1) into (51) we obtain119868119905 + 119886 (119910 minus 119909) 119868119909 + (119888119909 minus 119889119910 minus 119909119911) 119868119910+ (minus119887119911 + 119909119910 + 119908) 119868119911 + (minus119903119908 + 119896119911) 119868119908 = 0 (52)

Assume that the solution of (52) takes the form119868 (119909 119910 119911 119908 119905) = 119877 (119909 119910 119911 119908) 119890119904119905 (53)

where 119904 is a constant Substituting (53) into (52) we obtain119904119877 + 119886 (119910 minus 119909) 119877119909 + (119888119909 minus 119889119910 minus 119909119911) 119877119910+ (minus119887119911 + 119909119910 + 119908) 119877119911 + (minus119903119908 + 119896119911) 119877119908 = 0 (54)

Let the solution of (54) take the form119877 (119909 119910 119911 119908) = 119860 (119909) + 119861 (119910) + 119879 (119911) + 119876 (119908) (55)


20 22 24 26 28 30

20

22

24

26

28

30

y(t)

x(t)

(a)

minus30 minus28 minus26 minus24 minus22 minus20

minus30

minus28

minus26

minus24

minus22

minus20

y(t)

x(t)

(b)

minus2 minus1 0 1 2

minus2

minus1

0

1

2

y(t)

x(t)

(c)

10 15 20 25 3010

15

20

25

30y(

t)

x(t)

(d)

minus30 minus25 minus20 minus15

minus30

minus25

minus20

minus15

y(t)

x(t)

(e)

Figure 1 Periodic orbit of system (1) is created around nontrivial equilibrium points via Hopf bifurcation for a=10 b=73 c=-98 d=-99k=-95 r=01 in (ab) stable equilibrium point at the origin for a = 10 b = 73 c = -999 d = -99 k = -95 r = 01 is shown in (c) the originloses its stability and two stable nontrivial equilibrium points are created via pitchfork bifurcation and shown in (de) for a = 7 b = 73 c =-7 d = -69 k = -95 r = 01


50

0

minus50

40

20

0

minus20

minus20

0

20

40

Z(t)

y(t)

x(t)

(a)

50

0

minus50

minus20

0

20

40

0

minus100

minus200

minus300

minus400

w(t)

y(t)

x(t)

(b)

40

20

0

minus20

minus40

40200minus20

y(t)

x(t)

(c)

40

20

0

minus20

40200minus20

z(t)

x(t)

(d)

40

20

0

minus20

40200minus20minus40

z(t)

y(t)

(e)

0

minus100

minus200

minus300

minus400

40200minus20minus40

w(t)

y(t)

(f)

Figure 2 Hyperchaotic attractor of system (1) is illustrated via phase portraits on (a) 3D 119909 minus 119910 minus 119911 space (b) 119909 minus 119910 minus 119908 space (c) 2D 119909 minus 119910plane (d) 2D 119909 minus 119911 plane (e) 2D 119910 minus 119911 plane and (f) 2D 119910 minus 119908 plane at the values of parameters 119886 = 217 119887 = 73 119888 = 66 119889 = minus2 119903 = 01and 119896 = 95


x

minus

minus

a

(a)

x

minus

minus

b

(b)x

minus

minus

c

(c)

x

minus

minus

minus

kminus minus minus minus minus

(d)

Figure 3 Bifurcation diagrams of hyperchaotic system (1) obtained for the following values of parameters (a) 0 le 119886 le 30 119887 = 73 119888 = 66 119889 =minus2 119903 = 01 and 119896 = 95 (b) 119886 = 217 0 le 119887 le 12 119888 = 66 119889 = minus2 119903 = 01 and 119896 = 95 (c) 119886 = 217 119887 = 73 0 le 119888 = 10 119889 = minus2 119903 = 01 andminus12 le 119896 le 0Substituting (55) into (54) we obtain119904 (119860 (119909) + 119861 (119910) + 119879 (119911) + 119876 (119908)) + 119886 (119910 minus 119909) 1198601015840 (119909)+ (119888119909 minus 119889119910 minus 119909119911) 1198611015840 (119910)+ (minus119887119911 + 119909119910 + 119908) 1198791015840 (119911) + (minus119903119908 + 119896119911) 1198761015840 (119908)= 0

(56)

By derivative (56) twice with respect to 119908 we obtain119896119911119876101584010158401015840 (119908) minus 119903119908119876101584010158401015840 (119908) + (119904 minus 2119903) 11987610158401015840 (119908) = 0 (57)

Equating the coefficient of 119911 with zero and solving theobtained equation we obtain119876 (119908) = 1198881 + 1198882119908 + 11988831199082 (58)

where 1198881 are arbitrary constant Substituting (58) into (56) weobtain119904119879 (119911) + 1199091199101198791015840 (119911) + 119886 (119910 minus 119909) 1198601015840 (119909) + 119904119860 (119909)minus 1198871199111198791015840 (119911) + 1198611015840 (119910) (119888119909 minus 119889119910 minus 119909119911) + 119904119861 (119910)+ 1198881119904 + 119908 (minus1198882119903 + 1198882119904 + 2c3119896119911 + 1198791015840 (119911)) + 1198882119896119911+ 1199082 (1198883119904 minus 21198883119903) = 0

(59)


1050 15 20 25 30a

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

2Ly

apun

ov ex

pone

nts

(a)

1086420 12b

minus20

minus15

minus10

minus5

0

5

10

Lyap

unov

expo

nent

s

(b)

108 96 74 521 30c

minus20

minus15

minus10

minus5

0

5

Lyap

unov

expo

nent

s

(c)

minus12 minus10 minus8 minus6 minus4 minus2 0k

minus25

minus20

minus15

minus10

minus5

0

5

Lyap

unov

expo

nent

s

(d)

Figure 4 Lyapunov exponent spectrum corresponding to Figure 3

Equating the coefficient of 119908 with zero and solving theobtained equation we obtain119879 (119911) = 1198882 (119903 minus 119904) 119911 minus 11988831198961199112 + 1198884 (60)

where 1198884 are arbitrary constant Substituting (60) into (59) weobtain1198884119904 + 119886 (119910 minus 119909) 1198601015840 (119909) + 119904 (119860 (119909) + 119861 (119910))+ 119911 (1198882 ((119887 minus 119904) (119904 minus 119903) + 119896)) + 11988831198961199112 (2119887 minus 119904)+ 1198611015840 (119910) (119888119909 minus 119889119910) + 119909119911 (minus1198611015840 (119910) minus 21198883119896119910) + 1198881119904+ 1198882119909119910 (119903 minus 119904) + 11988831199082 (119904 minus 2119903) = 0 (61)

Equating the coefficient of 119909119911 with zero and solving theobtained equation we obtain119861 (119910) = 1198885 minus 11988831198961199102 (62)

where 1198885 are arbitrary constant Substituting (62) into (61) weobtain119910 (1198861198601015840 (119909) + 119909 (1198882 (119903 minus 119904) minus 21198881198883119896)) minus 1198861199091198601015840 (119909)+ 119904119860 (119909) + 1198882119911 ((119887 minus 119904) (119904 minus 119903) + 119896)

+ 11988831198961199112 (2119887 minus 119904) + 1198881119904 + 11988831198961199102 (2119889 minus 119904) minus 211988831199031199082+ 11988831199041199082 + 1198884119904 + 1198885119904 = 0(63)

Equating the coefficient of 119910 with zero and solving theobtained equation we obtain119860 (119909) = 121198861199092 (21198881198883119896 + 1198882 (119904 minus 119903)) + 1198886 (64)

where 1198886 are arbitrary constant Substituting (64) into (63) weobtain 121198861199092 (119904 minus 2119886) (21198881198883119896 + 1198882 (119904 minus 119903))+ 1198882119911 ((119887 minus 119904) (119904 minus 119903) + 119896) + 11988831198961199112 (2119887 minus 119904)+ 119904 (1198881 + 1198884 + 1198885 + 1198886) + 11988831198961199102 (2119889 minus 119904)+ 11988831199082 (119904 minus 2119903)

(65)

Equation (65) satisfies when


(1) 119904 = 0 119887 = 119888 = 119889 = 119903 = 0 and 1198882 = 0(2) 119904 = 2119887 119886 = 119889 = 119903 = 119887 1198886 = minus1198881 minus 1198884 minus 1198885 and 1198882 = 0(3) 119904 = 2119887 119886 = 119889 = 119903 = 119887 1198886 = minus1198881 minus 1198884 minus 1198885 and 119896 = 1198872(4) 119904 = 2119886 119896 = (2119886 minus 119887)(2119886 minus 119903) 1198886 = minus1198881 minus 1198884 minus 1198885 and1198883 = 0(5) 119904 = 2119887 119889 = 119903 = 119887 1198886 = minus1198881 minus 1198884 minus 1198885 119896 = 1198872 and1198883 = minus11988822119887119888

So when we substitute (58) (60) (62) and (64) into (53) weobtain the following first integrals of system (1) respectively

(1) 1198681(119909 119910 119911 119908 119905) = 1199082 minus 119896(1199102 + 1199112)(2) 1198682(119909 119910 119911 119908 119905) = 1198902119887119905(1198871199082 minus 119896119887(1199102 + 1199112) + 1198881198961199092)(3) 1198683(119909 119910 119911 119908 119905) = 1198902119887119905(21198883(1199082 + 1198881198871199092 minus 1198872(1199102 + 1199112)) +1198882(minus2119887119911 + 2119908 + 1199092))(4) 1198684(119909 119910 119911 119908 119905) = 1198902119886119905(1199092(119903 minus 2119886) + 2119886119911(2119886 minus 119903) minus 2119886119908)(5) 1198685(119909 119910 119911 119908 119905) = 1198902119887119905(2119888119887(119887119911 minus 119908) minus 1198872(1199102 + 1199112) + 1199082)

Hence the proof is completed

41 First Integrals Containing the Origin Equilibrium PointIn this subsection the invariant surfaces in phase space ofhyperchaotic system (1) which contain the origin equilibriumpoint are attained If we substitute the equilibrium points intothe first integrals of the system (1) we obtain1198681 = 1199082 minus 119896 (1199102 + 1199112) = 0 (66)1198682 = 1199102 + 1199112 minus 1198881198871199092 minus 11198961199082 = 0 (67)1198683 = 21198883 (1199082 + 1198881198871199092 minus 1198872 (1199102 + 1199112))+ 1198882 (minus2119887119911 + 2119908 + 1199092) = 0 (68)

1198684 = 1199092 (119903 minus 2119886) + 2119886119911 (2119886 minus 119903) minus 2119886119908 = 0 (69)1198685 = 2119888119887 (119887119911 minus 119908) minus 1198872 (1199102 + 1199112) + 1199082 = 0 (70)

Hence we will classify the solutions (66)-(70) as follows

(1) For (66) when 119896 gt 0 (66) becomes circular cone (seeFigure 5)

(2) For (67) there are some cases

(i) when 119888 = 0 and 119896 gt 0 (67) becomes circularcone (see Figure 6(a))

(ii) when 119888119887 gt 0 119896 gt 0 and 119908 = 120573 (67) becomeshyperboloid of one sheet (see Figure 6(b))

(iii) when 119888119887 lt 0 119896 gt 0 and 119908 = 120573 (67) becomesellipsoid (see Figure 6(c))

(iv) when 119888 = 119887 119896 gt 0 and 119908 = 120573 (67) becomessphere (see Figure 6(d))

(v) when 119888119887 gt 0 119896 gt 0 and 119908 = 120573 (67) becomeshyperboloid of two sheets (see Figure 6(e))

10

10

10

0

0

0

minus10

minus10

minus10minus20

minus20

20

20

y

w

z

Figure 5 Plot the equations 1198681 = 0 for 119896 = 05(3) Let 1198882 = 21198883 (68) becomes1199102 + (1119887 + 119911)2 minus (119887119888 + 1)1198872 1199092 minus 11198872 (119908 + 1)2 = 0 (71)

which is classified as follows

(i) when 119887119888 = minus1 (71) becomes circular cone (seeFigure 7(a))

(ii) when 119887119888 gt minus1 and 119908 = 120573 (71) becomeshyperboloid of one sheet (see Figure 7(b))

(iii) when 119887119888 lt minus1 and 119908 = 120573 (71) becomes ellipsoid(see Figure 7(c))

(iv) when 119888 = minus119887 minus 1119887 and 119908 = 120573 (71) becomessphere (see Figure 7(d))

(4) Equation (69) represents parabola (see Figure 8)(5) Equation (70)can be rewritten in the form1199102 + (119911 minus 119888)2 minus (119908 minus 119887119888)21198872 = 0 (72)

which represents circular cone (see Figure 9)120573 is a constant

5 Circuit Implementation

The circuit implementation of hyperchaotic system (1) canbe designed using operational amplifier analog multipliersresistors and capacitors [17 18] This hyperchaotic non-linear circuit is equivalent to system (1) and enables theuseful experimental study of complex dynamics exhibitedby system (1) and verify the results of theoretical analysiscarried out upon it The Multisim package is used to carryout circuit simulations and depict the circuit outputs onthe oscilloscope The nonlinear circuit implementation ofhyperchaotic system can be used as a central in a varietyof advanced applications eg chaos based physical-layersecure communications and physical random number gener-ators which outperform pseudo random number generators


10

10

10

0

0

0

minus10

minus10

minus10minus20

minus20

20

20

y

w

z

(a)

20

minus2

5

0

minus5

x

y

20

minus2

z

(b)1

1

0

0minus1

minus1

minus10

minus05

10

05

00

x

z

y

(c)

minus10minus10

minus05

minus05

00

00z

x 05

05

10

10minus10

minus0500

y 0510

(d)

10

10

10

0

0

0

minus10

minus10

minus10

x

minus20minus20

minus2020

20

20y

z

(e)

Figure 6 Plot of the equations 1198682 = 0 at different values of the parameters (a) 119888 = 0 and 119896 = 05 (b) 119888119887 = 02 and 120573 = radic119896 119896 gt 0 (c)119888119887 = minus05 and 120573 = radic119896 119896 gt 0 (d) 119888119887 = minus1 and 120573 = radic119896 119896 gt 0 (e) 119888119887 = 1 and 120573 = radicminus119896 119896 lt 010

10

0

0

0

minus10minus10

minus10minus5

minus5

minus5

5

5

5y

w

z

(a)

10

10

0

0

0

minus10

minus10minus5

minus5

minus5

5

5

5y

x

z

(b)

2

2

4

4

minus2 minus20

0minus4

minus4

x

y

2

0

minus2

z

(c)2

2

21

1

1

minus2minus2

minus2

minus1

minus1

minus1

0

0

0

x

y

z

(d)

Figure 7 Plot the equations 1198683 = 0 at different values of the parameters (a) 119888 = minus1 and 119887 = 1 (b) 119888 = 1 119887 = 2 and 120573 = 2 (c) 119888 = 2 119887 = minus1and 120573 = 2 (d) 119888 = minus25 119887 = 2 and 120573 = 2

10

5

5

5

0

0

0

x

w

z

minus5

minus5 minus5

minus10minus10

minus1010

10

Figure 8 Plot of the equation 1198684 = 0 for 119903 = 119886 = 1

minus10

10

105

5

5

0

0

0

minus5

minus5

minus5

minus10

y

w

z



(a)

(b) (c)

(d) (e)

Figure 10 (a) Circuit implementation of system (1) (bndashe) circuit simulations outputs representing119909minus119910 119910minus119911 119909minus119911 and 119909minus119908 phase portraitsof the model at 119886 = 217 119887 = 73 119888 = 66 119889 = minus2 119903 = 01 119896 = 95Figure 10 shows schematics of proposed the circuit realizationof hyperchaotic system (1) for the values of parameters givenby 119886 = 217 119887 = 73 119888 = 66 119889 = minus2 119903 = 01 and 119896 = 95

Regarding digital realization of chaotic circuits it isimportant to note that dynamics degradation of digital

chaos occurs due to impossibility of the chaotic map toreach an ideal chaotic state under the limitations of digitaldomain with finite-precision Various methods are proposedto counteract dynamics degradation see for example [42]and references therein


6 Conclusion

In this work dynamical and bifurcation analysis are appliedto explore local and global characteristics of 4D Lorenz-typehyperchaotic system Through investigation of codimensiontwo local bifurcation of the hyperchaotic system it is foundthat the 4D hyperchaotic system undergoes various typesof bifurcations including BT bifurcation Hopf bifurcationpitchfork bifurcation and homoclinic bifurcation Moreoverthe paper successfully acquires for first time the closed formsof first integrals and regular invariant surfaces in solutionspace which pass through the origin equilibrium points andrelates these forms to the parameters of the system Thus itpredicts some analytical solutions of the system and estimatesthe associated parameters values Finally a circuit realizationfor 4D hyperchaotic system is proposed as an engineeringapplication and numerical examples are presented to verifytheoretical results Future work can include analytical andnumerical investigations of hidden attractors if exist inhyperchaotic 4D Lorenz system Also other possible firstintegrals and invariant structures of the state space of thesystem may be explored in future work

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

Theauthors would like to extend their sincere appreciation tothe Deanship of Scientific Research at King Saud Universityfor funding this research (Group no RG-1438-046)

References

[1] E M Izhikevich Dynamical Systems in Neuroscience theGeometry of Excitability And Bursting MIT Press CambridgeMass USA 2007

[2] P Stavroulakis Chaos Applications in Telecommunications CRCPress 2006

[3] S H Strogatz Nonlinear dynamics and chaos with applicationsto physics Biology chemistry and engineering Biology chem-istry and engineering Westview Press 2001

[4] P N V Tu Dynamical Systems An Introduction with Applica-tions in Economics And Biology Springer Science amp BusinessMedia 2012

[5] M Han and P Yu Normal Forms Melnikov Functions andBifurcations of Limit Cycles vol 181 Springer Scienceamp BusinessMedia 2012

[6] J D Meiss Differential Dynamical Systems vol 14 SIAM 2007[7] S Wiggins Introduction to Applied Nonlinear Dynamical Sys-

tems and Chaos vol 2 Springer New York NY USA 2003[8] A Elsonbaty S F Hegazy and S S A Obayya ldquoSimultaneous

concealment of time delay signature in chaotic nanolaser with

hybrid feedbackrdquo Optics and Lasers in Engineering vol 107 pp342ndash351 2018

[9] A Elsonbaty S F Hegazy and S S A Obayya ldquoNumericalanalysis of ultrafast physical random number generator usingdual-channel optical chaosrdquo Optical Engineering vol 55 no 92016

[10] A Elsonbaty S F Hegazy and S S A Obayya ldquoSimultaneoussuppression of time-delay signature in intensity and phase ofdual-channel chaos communicationrdquo IEEE Journal of QuantumElectronics vol 51 pp 1ndash9 2015

[11] L Kocarev and S Lian Chaos-Based Cryptography Springer2011

[12] A A Elsadany A M Yousef and A Elsonbaty ldquoFurtheranalytical bifurcation analysis and applications of coupledlogistic mapsrdquo Applied Mathematics and Computation vol 338pp 314ndash336 2018

[13] A A Elsadany A Elsonbaty and H N Agiza ldquoQualitativedynamical analysis of chaotic plasma perturbations modelrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 59 pp 409ndash423 2018

[14] A Al-khedhairi A A Elsadany A Elsonbaty and A GAbdelwahab ldquoDynamical study of a chaotic predator-preymodel with an omnivorerdquo e European Physical Journal Plusvol 133 no 29 2018

[15] A Elsonbaty andAA Elsadany ldquoBifurcation analysis of chaoticgeomagnetic field modelrdquo Chaos Solitons amp Fractals vol 103pp 325ndash335 2017

[16] A Elsonbaty and A M El-Sayed ldquoAnalytical study of globalbifurcations stabilization and chaos synchronization of jerksystem with multiple attractorsrdquo Nonlinear Dynamics vol 90no 4 pp 2637ndash2655 2017

[17] A R Elsonbaty andAM El-Sayed ldquoFurthernonlinear dynam-ical analysis of simple jerk system with multiple attractorsrdquoNonlinear Dynamics vol 87 no 2 pp 1169ndash1186 2017

[18] A M El-Sayed H M Nour A Elsaid A E Matouk andA Elsonbaty ldquoDynamical behaviors circuit realization chaoscontrol and synchronization of a new fractional order hyper-chaotic systemrdquo Applied Mathematical Modelling Simulationand Computation for Engineering and Environmental Systemsvol 40 no 5-6 pp 3516ndash3534 2016

[19] QWang S Yu and C Li ldquoTheoretical design and FPGA-basedimplementation of higher-dimensional digital chaotic systemsrdquoIEEE Transactions on Circuits and Systems I-Regular Papers vol63 no 3 pp 401ndash412 2016

[20] Q Hong Q Xie Y Shen and X Wang ldquoGenerating multi-double-scroll attractors via nonautonomous approachrdquo ChaosAn Interdisciplinary Journal of Nonlinear Science vol 26 no 8Article ID 083110 2016

[21] C Li ldquoCracking a hierarchical chaotic image encryption algo-rithm based on permutationrdquo Signal Processing vol 118 pp203ndash210 2016

[22] EHarjanto and JM Tuwankotta ldquoBifurcation of periodic solu-tion in a PredatorndashPrey type of systems with non-monotonicresponse function and periodic perturbationrdquo InternationalJournal of Non-Linear Mechanics vol 85 pp 188ndash196 2016

[23] L Barreira C Valls and J Llibre ldquoIntegrability and limit cyclesof the Moon-Rand systemrdquo International Journal of Non-LinearMechanics vol 69 pp 129ndash136 2015

[24] J Kengne ldquoCoexistence of chaos with hyperchaos period-3doubling bifurcation and transient chaos in the hyperchaoticoscillator with gyratorsrdquo International Journal of Bifurcation andChaos vol 25 no 4 Article ID 1550052 2015


[25] A P Kuznetsov S P Kuznetsov E Mosekilde and N VStankevich ldquoCo-existing hidden attractors in a radio-physicaloscillator systemrdquo Journal of Physics A Mathematical andeoretical vol 48 no 12 Article ID 125101 2015

[26] J Kengne J C Chedjou T Fonzin Fozin K Kyamakya andG Kenne ldquoOn the analysis of semiconductor diode-basedchaotic and hyperchaotic generatorsmdasha case studyrdquo NonlinearDynamics vol 77 no 1-2 pp 373ndash386 2014

[27] A M El-Sayed H M Nour A Elsaid A E Matouk andA Elsonbaty ldquoCircuit realization bifurcations chaos andhyperchaos in a new 4D systemrdquo Applied Mathematics andComputation vol 239 pp 333ndash345 2014

[28] H M Nour A Elsaid and A Elsonbaty ldquoCircuit realizationchaos synchronization and estimation of parameters of ahyperchaotic system with unknown parametersrdquo Journal of theEgyptian Mathematical Society vol 22 no 3 pp 550ndash557 2014

[29] A M El-Sayed A Elsaid H M Nour and A ElsonbatyldquoSynchronization of different dimensional chaotic systems withtime varying parameters disturbances and input nonlineari-tiesrdquo Journal of Applied Analysis and Computation vol 4 no4 pp 323ndash338 2014

[30] A M El-Sayed A Elsaid H M Nour and A ElsonbatyldquoDynamical behavior chaos control and synchronization of amemristor-based ADVP circuitrdquoCommunications in NonlinearScience and Numerical Simulation vol 18 no 1 pp 148ndash1702013

[31] C Li Y Liu T Xie and M Z Q Chen ldquoBreaking a novelimage encryption scheme based on improved hyperchaoticsequencesrdquo Nonlinear Dynamics vol 73 no 3 pp 2083ndash20892013

[32] B Muthuswamy and L O Chua ldquoSimplest chaotic circuitrdquoInternational Journal of Bifurcation and Chaos vol 20 no 5 pp1567ndash1580 2010

[33] A H Nayfeh and B Balachandran Applied Nonlinear Dynam-ics Analytical Computational and Experimental Methods JohnWiley amp Sons 2008

[34] NAA Fataf S K Palit SMukherjeeM RM SaidDH Sonand S Banerjee ldquoCommunication scheme using a hyperchaoticsemiconductor laser model Chaos shift key revisitedrdquo eEuropean Physical Journal Plus vol 132 no 492 2017

[35] S Vaidyanathan A Akgul S Kacar and U Cavusoglu ldquoAnew 4-D chaotic hyperjerk system its synchronization circuitdesign and applications in RNG image encryption and chaos-based steganographyrdquo e European Physical Journal Plus vol133 no 46 2018

[36] Y Chen ldquoThe existence of homoclinic orbits in a 4D Lorenz-type hyperchaotic systemrdquo Nonlinear Dynamics vol 87 no 3pp 1445ndash1452 2017

[37] G A Leonov N V Kuznetsov and V I Vagaitsev ldquoHiddenattractor in smooth Chua systemsrdquo Physica D Nonlinear Phe-nomena vol 241 no 18 pp 1482ndash1486 2012

[38] C Li and J C Sprott ldquoCoexisting hidden attractors in a 4-Dsimplified lorenz systemrdquo International Journal of Bifurcationand Chaos vol 24 no 3 Article ID 1450034 pp 1ndash12 2014

[39] G Peng and Y Jiang ldquoComputation of universal unfoldingof the double-zero bifurcation in symmetric systems by ahomological methodrdquo Journal of Difference Equations andApplications vol 19 no 9 pp 1501ndash1512 2013

[40] Y A Kuznetsov Elements of Applied Bifurcation eorySpringer Berlin Germany 2nd edition 1997

[41] JGuckenheimer andPHolmesNonlinearOscillationsDynam-ical Systems and Bifurcation of Vector Fields Springer NewYork NY USA 1983

[42] C Li B Feng S Li J Kurths and G Chen ldquoDynamic analysisof digital chaotic maps via state-mapping networksrdquo IEEETransactions on Circuits and Systems I Regular Papers pp 1ndash142018

Hindawiwwwhindawicom Volume 2018

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

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Submit your manuscripts atwwwhindawicom


Themore complex hyperchaotic system possesses at leasttwo positive Lyapunov exponents and it has a phase spaceof dimension at least four Clearly the hyperchaotic systemshavemore randomness and higher unpredictability than sim-ple chaotic systems Therefore hyperchaos is more preferredthan simple chaos and its applications have recently becomea central topic in research including chaos-based securecommunications image encryption and cryptography

In the last two decades some interesting high-dimensional hyperchaotic systems in science and engineer-ing have been explored and their dynamics have beenextensively investigated [18 24 27 28 34 35] In fact it isof great importance from theoretical and practical aspectsto explain complicated phenomena and internal structuralcharacteristics of hyperchaotic systems This research linefocuses on applying codimension two or three bifurcationanalysis to system under investigation For exampleanalysis of Bogdanov-Takens bifurcation degenerate Hopfbifurcation and Heteroclinic and Homoclinic bifurcationscan be undertaken Also it is crucial to determine whetherthere exists any set of parameters values for which theconsidered system is integrable and find the correspondinginvariant surfaces if exist As it is practically unattainablegoal to apply exhaustive numerical investigation to acquirevalues of parameters where the hyperchaotic system isintegrable one needs a powerful analytical method thatenables achieving this goal easily

Recently theoretical analysis based on Fishing Principleis applied to study some global features of a new four-dimensional Lorenz-type hyperchaotic system [36] Morespecifically conditions for existence of homoclinic orbits areobtained This work aims at extending the aforementionedwork and exploring other aspects of complicated dynami-cal behaviors of the 4D Lorenz-type hyperchaotic systemAnalytical bifurcation structure of the model which includes1198852 symmetrical Bogdanov-Takens Pitchfork Andronov-Hopf bifurcation and homoclinic bifurcation are obtainedExistence uniqueness and continuous dependence on initialconditions for the solution of 4D hyperchaotic system are allinvestigated It is important to ask whether there is any set ofparameter values in which system dynamics are regular andthe studied system is integrable The different integrable casesof the system are studied and the closed-form expressionsfor invariant surfaces corresponding to first integrals of thesystem are also obtained Moreover the hyperchaotic systemhas been simulated using a proposed electronic circuit real-ization and numerical simulations are performed confirmingthe new results of theoretical analysis Note that attractorsfound in dynamical system are classified as being self-excitedor hidden attractors [37 38] The key difference betweenthem is that the self-excited attractor has a basin of attractionexcited from unstable equilibria The hidden attractor on theother hand has a basin of attraction with no intersection withneighborhoods of dynamical systems equilibrium points Inthis work we focus on self-excited attractors exist in thehyperchaotic system

The rest of the paper is structured as follows In Section 2we introduce the 4D Lorenz-type system and discuss theequilibrium pointsrsquo existence A sufficient condition for

continuous dependence on initial condition is determinedPhase portraits bifurcation diagrams and Lyapunov charac-teristics spectrum are obtained In Section 3 the analysis ofsome possible codimension two bifurcations is performed Itis shown that the 4D Lorenz system undergoes Bogdanov-Takens bifurcation Andronov-Hopf bifurcation Pitchforkbifurcation and homoclinic bifurcation The integrabilityanalysis of system is investigated in Section 4 A practicalapplication to engineering will be realized by an electroniccircuit in Section 5 Finally Section 6 concludes the paper

2 The 4D Lorenz-Type Hyperchaotic System

The following 4D hyperchaotic system of Lorenz type waspresented in [36] = 119886 (119910 minus 119909) 119910 = 119888119909 minus 119889119910 minus 119909119911 = minus119887119911 + 119909119910 + 119908 = minus119903119908 + 119896119911 (1)

where state variables of the system are denoted by 119909 119910 119911and 119908 the parameters of the system are represented by119886 119887 119888 119889 119903 and 119896 and the dot above state variables refers totime derivative of state variables

The following subsections examine themain properties ofsystem (1) and provide elementary dynamical analysis of themodel

21 Equilibrium Points of the System The fixed points ofsystem (1) can be obtained when 119909 = = = = 0 iethrough solving 119886 (119910 minus 119909) = 0119888119909 minus 119889119910 minus 119909119911 = 0minus119887119911 + 119909119910 + 119908 = 0minus119903119908 + 119896119911 = 0 (2)

Therefore system (1) has the following equilibrium solutions(I) when (119889 minus 119888)(119896 minus 119887119903)119903 le 0 the fixed point is given by119909 = 119910 = 119911 = 119908 = 0 (3)

(II) when (119889 minus 119888)(119896 minus 119887119903)119903 gt 0 the fixed points are119909 = 119910 = 119911 = 119908 = 0119909 = 119910 = plusmnradic (119889 minus 119888) (119896 minus 119887119903)119903 119911 = 119888 minus 119889119908 = 119896 (119888 minus 119889)119903

(4)



X (t) = Φ (X (t)) 119905 isin (0 119879] (5)

where

X = [[[[[[1199091119909211990931199094

]]]]]] X0 = [[[[[[

11990901119909021199090311990904]]]]]]


]]]]]] (6)


X (0) = X0 (7)


119905isin[0119879]

|119865 (119905)| (8)


119894119895

sup119905isin[0119879]

10038161003816100381610038161003816119896119894119895 [119905]10038161003816100381610038161003816 (9)




X (t) = X0 + int1199050F (X (s)) ds (11)


H(X) then for X1 = [ 11990911119909121199091311990914

] and X2 = [ 11990921119909221199092311990924

] we get

H (X1) minus H (X2) = int1199050

(Φ (X1 (s)) minusΦ (X2 (s))) ds (12)


1003816100381610038161003816(Φ (X1 (s)) minusΦ (X2 (s)))1003816100381610038161003816 ds (13)


1003817100381710038171003817 (14)

where119870 = 119879max |119886| + |119888| + 2119872 |119886| + |119889| + 119872 |119887| + |119896|+ 119872 1 + |119903| gt 0 (15)





X1 (119905) = X01 + int1199050H (X1 (s)) ds

X2 (119905) = X02 + int1199050H (X2 (s)) ds (17)


1003817100381710038171003817 + 119870 1003817100381710038171003817X1 minus X21003817100381710038171003817 (18)









(1199091119909211990931199094) = ((


(1199061119906211990631199064) (22)

which yields

(1234) = (1234

) (23)

where


(24)





(26)


(12) = (11990620 ) + 1Ψ ( 119866 (1199062 1199063)119886119866 (1199062 1199063)) (27)

where




(28)



(29)



(30)


(1234) = (1234

) (31)


where


(33)


11989411989511989611989711990611989411199061198952Δ1198961Δ11989721199064 = 119894+119895+119896+119897=2sum

119894119895119896119897=0

12057311989411989511989611989711990611989411199061198952Δ1198961Δ1198972 (34)


119860 = (0 10 0) (35)





tions119862 (119904 V 119908)= 3sum119894119895119896=1

1205973119866 (1199061 1199062 Δ 1 Δ 2)120597119906119894120597119906119895120597119906119896 1003816100381610038161003816100381610038161003816100381610038161003816 1199061=1199062=0Δ 1=Δ 2=0

119904119894V1198951199081198961198601 (119904 Λ) = 2sum

119894=1

2sum119895=1

1205972119866 (1199061 1199062 Δ 1 Δ 2)120597119906119894120597Δ 119895 1003816100381610038161003816100381610038161003816100381610038161003816 1199062=1199063=0Δ 1=Δ 2=0

119904119894Λ 1198951198602 (119904 Λ Ω)= 2sum119894=1

2sum119895119896=1

1205972119866 (1199062 1199063 Δ 1 Δ 2)120597119906119894120597Δ 119895120597Δ 119896 1003816100381610038161003816100381610038161003816100381610038161003816 1199062=1199063=0Δ 1=Δ 2=0

119904119894Λ 119895Ω119896(37)

where 119866 (1199062 1199063 Δ 1 Δ 2) = (1198661 (1199061 1199062 Δ 1 Δ 2)1198662 (1199061 1199062 Δ 1 Δ 2)) (38)




1199022 = [01] (41)




(44)









1199061 = 1199041radic10038161003816100381610038161205993010038161003816100381610038161003816100381610038161003816120599211003816100381610038161003816 12058511199062 = minus11990411199042 10038161003816100381610038161205993010038161003816100381610038163210038161003816100381610038161205992110038161003816100381610038162 1205852

119905 = minus 120599211003816100381610038161003816120599301003816100381610038161003816 120591(49)




















20 22 24 26 28 30

20

22

24

26

28

30

y(t)

x(t)

(a)


minus30

minus28

minus26

minus24

minus22

minus20

y(t)

x(t)

(b)

minus2 minus1 0 1 2

minus2

minus1

0

1

2

y(t)

x(t)

(c)

10 15 20 25 3010

15

20

25

30y(

t)

x(t)

(d)


minus30

minus25

minus20

minus15

y(t)

x(t)

(e)



50

0

minus50

40

20

0

minus20

minus20

0

20

40

Z(t)

y(t)

x(t)

(a)

50

0

minus50

minus20

0

20

40

0

minus100

minus200

minus300

minus400

w(t)

y(t)

x(t)

(b)

40

20

0

minus20

minus40

40200minus20

y(t)

x(t)

(c)

40

20

0

minus20

40200minus20

z(t)

x(t)

(d)

40

20

0

minus20

40200minus20minus40

z(t)

y(t)

(e)

0

minus100

minus200

minus300

minus400

40200minus20minus40

w(t)

y(t)

(f)



x

minus

minus

a

(a)

x

minus

minus

b

(b)x

minus

minus

c

(c)

x

minus

minus

minus


(d)


(56)




(59)


1050 15 20 25 30a

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

2Ly

apun

ov ex

pone

nts

(a)

1086420 12b

minus20

minus15

minus10

minus5

0

5

10

Lyap

unov

expo

nent

s

(b)

108 96 74 521 30c

minus20

minus15

minus10

minus5

0

5

Lyap

unov

expo

nent

s

(c)


minus25

minus20

minus15

minus10

minus5

0

5

Lyap

unov

expo

nent

s

(d)









(65)

















10

10

10

0

0

0

minus10

minus10

minus10minus20

minus20

20

20

y

w

z












10

10

10

0

0

0

minus10

minus10

minus10minus20

minus20

20

20

y

w

z

(a)

20

minus2

5

0

minus5

x

y

20

minus2

z

(b)1

1

0

0minus1

minus1

minus10

minus05

10

05

00

x

z

y

(c)

minus10minus10

minus05

minus05

00

00z

x 05

05

10

10minus10

minus0500

y 0510

(d)

10

10

10

0

0

0

minus10

minus10

minus10

x

minus20minus20

minus2020

20

20y

z

(e)


10

0

0

0

minus10minus10

minus10minus5

minus5

minus5

5

5

5y

w

z

(a)

10

10

0

0

0

minus10

minus10minus5

minus5

minus5

5

5

5y

x

z

(b)

2

2

4

4

minus2 minus20

0minus4

minus4

x

y

2

0

minus2

z

(c)2

2

21

1

1

minus2minus2

minus2

minus1

minus1

minus1

0

0

0

x

y

z

(d)


10

5

5

5

0

0

0

x

w

z

minus5

minus5 minus5

minus10minus10

minus1010

10


minus10

10

105

5

5

0

0

0

minus5

minus5

minus5

minus10

y

w

z



(a)

(b) (c)

(d) (e)





6 Conclusion


Data Availability




Acknowledgments


References




















































Journal of












Journal of







Volume 2018






Volume 2018










X (t) = Φ (X (t)) 119905 isin (0 119879] (5)

where

X = [[[[[[1199091119909211990931199094

]]]]]] X0 = [[[[[[

11990901119909021199090311990904]]]]]]


]]]]]] (6)


X (0) = X0 (7)


119905isin[0119879]

|119865 (119905)| (8)


119894119895

sup119905isin[0119879]

10038161003816100381610038161003816119896119894119895 [119905]10038161003816100381610038161003816 (9)




X (t) = X0 + int1199050F (X (s)) ds (11)


H(X) then for X1 = [ 11990911119909121199091311990914

] and X2 = [ 11990921119909221199092311990924

] we get

H (X1) minus H (X2) = int1199050

(Φ (X1 (s)) minusΦ (X2 (s))) ds (12)


1003816100381610038161003816(Φ (X1 (s)) minusΦ (X2 (s)))1003816100381610038161003816 ds (13)


1003817100381710038171003817 (14)

where119870 = 119879max |119886| + |119888| + 2119872 |119886| + |119889| + 119872 |119887| + |119896|+ 119872 1 + |119903| gt 0 (15)





X1 (119905) = X01 + int1199050H (X1 (s)) ds

X2 (119905) = X02 + int1199050H (X2 (s)) ds (17)


1003817100381710038171003817 + 119870 1003817100381710038171003817X1 minus X21003817100381710038171003817 (18)









(1199091119909211990931199094) = ((


(1199061119906211990631199064) (22)

which yields

(1234) = (1234

) (23)

where


(24)





(26)


(12) = (11990620 ) + 1Ψ ( 119866 (1199062 1199063)119886119866 (1199062 1199063)) (27)

where




(28)



(29)



(30)


(1234) = (1234

) (31)


where


(33)


11989411989511989611989711990611989411199061198952Δ1198961Δ11989721199064 = 119894+119895+119896+119897=2sum

119894119895119896119897=0

12057311989411989511989611989711990611989411199061198952Δ1198961Δ1198972 (34)


119860 = (0 10 0) (35)





tions119862 (119904 V 119908)= 3sum119894119895119896=1

1205973119866 (1199061 1199062 Δ 1 Δ 2)120597119906119894120597119906119895120597119906119896 1003816100381610038161003816100381610038161003816100381610038161003816 1199061=1199062=0Δ 1=Δ 2=0

119904119894V1198951199081198961198601 (119904 Λ) = 2sum

119894=1

2sum119895=1

1205972119866 (1199061 1199062 Δ 1 Δ 2)120597119906119894120597Δ 119895 1003816100381610038161003816100381610038161003816100381610038161003816 1199062=1199063=0Δ 1=Δ 2=0

119904119894Λ 1198951198602 (119904 Λ Ω)= 2sum119894=1

2sum119895119896=1

1205972119866 (1199062 1199063 Δ 1 Δ 2)120597119906119894120597Δ 119895120597Δ 119896 1003816100381610038161003816100381610038161003816100381610038161003816 1199062=1199063=0Δ 1=Δ 2=0

119904119894Λ 119895Ω119896(37)

where 119866 (1199062 1199063 Δ 1 Δ 2) = (1198661 (1199061 1199062 Δ 1 Δ 2)1198662 (1199061 1199062 Δ 1 Δ 2)) (38)




1199022 = [01] (41)




(44)









1199061 = 1199041radic10038161003816100381610038161205993010038161003816100381610038161003816100381610038161003816120599211003816100381610038161003816 12058511199062 = minus11990411199042 10038161003816100381610038161205993010038161003816100381610038163210038161003816100381610038161205992110038161003816100381610038162 1205852

119905 = minus 120599211003816100381610038161003816120599301003816100381610038161003816 120591(49)




















20 22 24 26 28 30

20

22

24

26

28

30

y(t)

x(t)

(a)


minus30

minus28

minus26

minus24

minus22

minus20

y(t)

x(t)

(b)

minus2 minus1 0 1 2

minus2

minus1

0

1

2

y(t)

x(t)

(c)

10 15 20 25 3010

15

20

25

30y(

t)

x(t)

(d)


minus30

minus25

minus20

minus15

y(t)

x(t)

(e)



50

0

minus50

40

20

0

minus20

minus20

0

20

40

Z(t)

y(t)

x(t)

(a)

50

0

minus50

minus20

0

20

40

0

minus100

minus200

minus300

minus400

w(t)

y(t)

x(t)

(b)

40

20

0

minus20

minus40

40200minus20

y(t)

x(t)

(c)

40

20

0

minus20

40200minus20

z(t)

x(t)

(d)

40

20

0

minus20

40200minus20minus40

z(t)

y(t)

(e)

0

minus100

minus200

minus300

minus400

40200minus20minus40

w(t)

y(t)

(f)



x

minus

minus

a

(a)

x

minus

minus

b

(b)x

minus

minus

c

(c)

x

minus

minus

minus


(d)


(56)




(59)


1050 15 20 25 30a

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

2Ly

apun

ov ex

pone

nts

(a)

1086420 12b

minus20

minus15

minus10

minus5

0

5

10

Lyap

unov

expo

nent

s

(b)

108 96 74 521 30c

minus20

minus15

minus10

minus5

0

5

Lyap

unov

expo

nent

s

(c)


minus25

minus20

minus15

minus10

minus5

0

5

Lyap

unov

expo

nent

s

(d)









(65)

















10

10

10

0

0

0

minus10

minus10

minus10minus20

minus20

20

20

y

w

z












10

10

10

0

0

0

minus10

minus10

minus10minus20

minus20

20

20

y

w

z

(a)

20

minus2

5

0

minus5

x

y

20

minus2

z

(b)1

1

0

0minus1

minus1

minus10

minus05

10

05

00

x

z

y

(c)

minus10minus10

minus05

minus05

00

00z

x 05

05

10

10minus10

minus0500

y 0510

(d)

10

10

10

0

0

0

minus10

minus10

minus10

x

minus20minus20

minus2020

20

20y

z

(e)


10

0

0

0

minus10minus10

minus10minus5

minus5

minus5

5

5

5y

w

z

(a)

10

10

0

0

0

minus10

minus10minus5

minus5

minus5

5

5

5y

x

z

(b)

2

2

4

4

minus2 minus20

0minus4

minus4

x

y

2

0

minus2

z

(c)2

2

21

1

1

minus2minus2

minus2

minus1

minus1

minus1

0

0

0

x

y

z

(d)


10

5

5

5

0

0

0

x

w

z

minus5

minus5 minus5

minus10minus10

minus1010

10


minus10

10

105

5

5

0

0

0

minus5

minus5

minus5

minus10

y

w

z



(a)

(b) (c)

(d) (e)





6 Conclusion


Data Availability




Acknowledgments


References




















































Journal of












Journal of







Volume 2018






Volume 2018












(1199091119909211990931199094) = ((


(1199061119906211990631199064) (22)

which yields

(1234) = (1234

) (23)

where


(24)





(26)


(12) = (11990620 ) + 1Ψ ( 119866 (1199062 1199063)119886119866 (1199062 1199063)) (27)

where




(28)



(29)



(30)


(1234) = (1234

) (31)


where


(33)


11989411989511989611989711990611989411199061198952Δ1198961Δ11989721199064 = 119894+119895+119896+119897=2sum

119894119895119896119897=0

12057311989411989511989611989711990611989411199061198952Δ1198961Δ1198972 (34)


119860 = (0 10 0) (35)





tions119862 (119904 V 119908)= 3sum119894119895119896=1

1205973119866 (1199061 1199062 Δ 1 Δ 2)120597119906119894120597119906119895120597119906119896 1003816100381610038161003816100381610038161003816100381610038161003816 1199061=1199062=0Δ 1=Δ 2=0

119904119894V1198951199081198961198601 (119904 Λ) = 2sum

119894=1

2sum119895=1

1205972119866 (1199061 1199062 Δ 1 Δ 2)120597119906119894120597Δ 119895 1003816100381610038161003816100381610038161003816100381610038161003816 1199062=1199063=0Δ 1=Δ 2=0

119904119894Λ 1198951198602 (119904 Λ Ω)= 2sum119894=1

2sum119895119896=1

1205972119866 (1199062 1199063 Δ 1 Δ 2)120597119906119894120597Δ 119895120597Δ 119896 1003816100381610038161003816100381610038161003816100381610038161003816 1199062=1199063=0Δ 1=Δ 2=0

119904119894Λ 119895Ω119896(37)

where 119866 (1199062 1199063 Δ 1 Δ 2) = (1198661 (1199061 1199062 Δ 1 Δ 2)1198662 (1199061 1199062 Δ 1 Δ 2)) (38)




1199022 = [01] (41)




(44)









1199061 = 1199041radic10038161003816100381610038161205993010038161003816100381610038161003816100381610038161003816120599211003816100381610038161003816 12058511199062 = minus11990411199042 10038161003816100381610038161205993010038161003816100381610038163210038161003816100381610038161205992110038161003816100381610038162 1205852

119905 = minus 120599211003816100381610038161003816120599301003816100381610038161003816 120591(49)




















20 22 24 26 28 30

20

22

24

26

28

30

y(t)

x(t)

(a)


minus30

minus28

minus26

minus24

minus22

minus20

y(t)

x(t)

(b)

minus2 minus1 0 1 2

minus2

minus1

0

1

2

y(t)

x(t)

(c)

10 15 20 25 3010

15

20

25

30y(

t)

x(t)

(d)


minus30

minus25

minus20

minus15

y(t)

x(t)

(e)



50

0

minus50

40

20

0

minus20

minus20

0

20

40

Z(t)

y(t)

x(t)

(a)

50

0

minus50

minus20

0

20

40

0

minus100

minus200

minus300

minus400

w(t)

y(t)

x(t)

(b)

40

20

0

minus20

minus40

40200minus20

y(t)

x(t)

(c)

40

20

0

minus20

40200minus20

z(t)

x(t)

(d)

40

20

0

minus20

40200minus20minus40

z(t)

y(t)

(e)

0

minus100

minus200

minus300

minus400

40200minus20minus40

w(t)

y(t)

(f)



x

minus

minus

a

(a)

x

minus

minus

b

(b)x

minus

minus

c

(c)

x

minus

minus

minus


(d)


(56)




(59)


1050 15 20 25 30a

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

2Ly

apun

ov ex

pone

nts

(a)

1086420 12b

minus20

minus15

minus10

minus5

0

5

10

Lyap

unov

expo

nent

s

(b)

108 96 74 521 30c

minus20

minus15

minus10

minus5

0

5

Lyap

unov

expo

nent

s

(c)


minus25

minus20

minus15

minus10

minus5

0

5

Lyap

unov

expo

nent

s

(d)









(65)

















10

10

10

0

0

0

minus10

minus10

minus10minus20

minus20

20

20

y

w

z












10

10

10

0

0

0

minus10

minus10

minus10minus20

minus20

20

20

y

w

z

(a)

20

minus2

5

0

minus5

x

y

20

minus2

z

(b)1

1

0

0minus1

minus1

minus10

minus05

10

05

00

x

z

y

(c)

minus10minus10

minus05

minus05

00

00z

x 05

05

10

10minus10

minus0500

y 0510

(d)

10

10

10

0

0

0

minus10

minus10

minus10

x

minus20minus20

minus2020

20

20y

z

(e)


10

0

0

0

minus10minus10

minus10minus5

minus5

minus5

5

5

5y

w

z

(a)

10

10

0

0

0

minus10

minus10minus5

minus5

minus5

5

5

5y

x

z

(b)

2

2

4

4

minus2 minus20

0minus4

minus4

x

y

2

0

minus2

z

(c)2

2

21

1

1

minus2minus2

minus2

minus1

minus1

minus1

0

0

0

x

y

z

(d)


10

5

5

5

0

0

0

x

w

z

minus5

minus5 minus5

minus10minus10

minus1010

10


minus10

10

105

5

5

0

0

0

minus5

minus5

minus5

minus10

y

w

z



(a)

(b) (c)

(d) (e)





6 Conclusion


Data Availability




Acknowledgments


References




















































Journal of












Journal of







Volume 2018






Volume 2018











(28)



(29)



(30)


(1234) = (1234

) (31)


where


(33)


11989411989511989611989711990611989411199061198952Δ1198961Δ11989721199064 = 119894+119895+119896+119897=2sum

119894119895119896119897=0

12057311989411989511989611989711990611989411199061198952Δ1198961Δ1198972 (34)


119860 = (0 10 0) (35)





tions119862 (119904 V 119908)= 3sum119894119895119896=1

1205973119866 (1199061 1199062 Δ 1 Δ 2)120597119906119894120597119906119895120597119906119896 1003816100381610038161003816100381610038161003816100381610038161003816 1199061=1199062=0Δ 1=Δ 2=0

119904119894V1198951199081198961198601 (119904 Λ) = 2sum

119894=1

2sum119895=1

1205972119866 (1199061 1199062 Δ 1 Δ 2)120597119906119894120597Δ 119895 1003816100381610038161003816100381610038161003816100381610038161003816 1199062=1199063=0Δ 1=Δ 2=0

119904119894Λ 1198951198602 (119904 Λ Ω)= 2sum119894=1

2sum119895119896=1

1205972119866 (1199062 1199063 Δ 1 Δ 2)120597119906119894120597Δ 119895120597Δ 119896 1003816100381610038161003816100381610038161003816100381610038161003816 1199062=1199063=0Δ 1=Δ 2=0

119904119894Λ 119895Ω119896(37)

where 119866 (1199062 1199063 Δ 1 Δ 2) = (1198661 (1199061 1199062 Δ 1 Δ 2)1198662 (1199061 1199062 Δ 1 Δ 2)) (38)




1199022 = [01] (41)




(44)









1199061 = 1199041radic10038161003816100381610038161205993010038161003816100381610038161003816100381610038161003816120599211003816100381610038161003816 12058511199062 = minus11990411199042 10038161003816100381610038161205993010038161003816100381610038163210038161003816100381610038161205992110038161003816100381610038162 1205852

119905 = minus 120599211003816100381610038161003816120599301003816100381610038161003816 120591(49)




















20 22 24 26 28 30

20

22

24

26

28

30

y(t)

x(t)

(a)


minus30

minus28

minus26

minus24

minus22

minus20

y(t)

x(t)

(b)

minus2 minus1 0 1 2

minus2

minus1

0

1

2

y(t)

x(t)

(c)

10 15 20 25 3010

15

20

25

30y(

t)

x(t)

(d)


minus30

minus25

minus20

minus15

y(t)

x(t)

(e)



50

0

minus50

40

20

0

minus20

minus20

0

20

40

Z(t)

y(t)

x(t)

(a)

50

0

minus50

minus20

0

20

40

0

minus100

minus200

minus300

minus400

w(t)

y(t)

x(t)

(b)

40

20

0

minus20

minus40

40200minus20

y(t)

x(t)

(c)

40

20

0

minus20

40200minus20

z(t)

x(t)

(d)

40

20

0

minus20

40200minus20minus40

z(t)

y(t)

(e)

0

minus100

minus200

minus300

minus400

40200minus20minus40

w(t)

y(t)

(f)



x

minus

minus

a

(a)

x

minus

minus

b

(b)x

minus

minus

c

(c)

x

minus

minus

minus


(d)


(56)




(59)


1050 15 20 25 30a

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

2Ly

apun

ov ex

pone

nts

(a)

1086420 12b

minus20

minus15

minus10

minus5

0

5

10

Lyap

unov

expo

nent

s

(b)

108 96 74 521 30c

minus20

minus15

minus10

minus5

0

5

Lyap

unov

expo

nent

s

(c)


minus25

minus20

minus15

minus10

minus5

0

5

Lyap

unov

expo

nent

s

(d)









(65)

















10

10

10

0

0

0

minus10

minus10

minus10minus20

minus20

20

20

y

w

z












10

10

10

0

0

0

minus10

minus10

minus10minus20

minus20

20

20

y

w

z

(a)

20

minus2

5

0

minus5

x

y

20

minus2

z

(b)1

1

0

0minus1

minus1

minus10

minus05

10

05

00

x

z

y

(c)

minus10minus10

minus05

minus05

00

00z

x 05

05

10

10minus10

minus0500

y 0510

(d)

10

10

10

0

0

0

minus10

minus10

minus10

x

minus20minus20

minus2020

20

20y

z

(e)


10

0

0

0

minus10minus10

minus10minus5

minus5

minus5

5

5

5y

w

z

(a)

10

10

0

0

0

minus10

minus10minus5

minus5

minus5

5

5

5y

x

z

(b)

2

2

4

4

minus2 minus20

0minus4

minus4

x

y

2

0

minus2

z

(c)2

2

21

1

1

minus2minus2

minus2

minus1

minus1

minus1

0

0

0

x

y

z

(d)


10

5

5

5

0

0

0

x

w

z

minus5

minus5 minus5

minus10minus10

minus1010

10


minus10

10

105

5

5

0

0

0

minus5

minus5

minus5

minus10

y

w

z



(a)

(b) (c)

(d) (e)





6 Conclusion


Data Availability




Acknowledgments


References




















































Journal of












Journal of







Volume 2018






Volume 2018









where


(33)


11989411989511989611989711990611989411199061198952Δ1198961Δ11989721199064 = 119894+119895+119896+119897=2sum

119894119895119896119897=0

12057311989411989511989611989711990611989411199061198952Δ1198961Δ1198972 (34)


119860 = (0 10 0) (35)





tions119862 (119904 V 119908)= 3sum119894119895119896=1

1205973119866 (1199061 1199062 Δ 1 Δ 2)120597119906119894120597119906119895120597119906119896 1003816100381610038161003816100381610038161003816100381610038161003816 1199061=1199062=0Δ 1=Δ 2=0

119904119894V1198951199081198961198601 (119904 Λ) = 2sum

119894=1

2sum119895=1

1205972119866 (1199061 1199062 Δ 1 Δ 2)120597119906119894120597Δ 119895 1003816100381610038161003816100381610038161003816100381610038161003816 1199062=1199063=0Δ 1=Δ 2=0

119904119894Λ 1198951198602 (119904 Λ Ω)= 2sum119894=1

2sum119895119896=1

1205972119866 (1199062 1199063 Δ 1 Δ 2)120597119906119894120597Δ 119895120597Δ 119896 1003816100381610038161003816100381610038161003816100381610038161003816 1199062=1199063=0Δ 1=Δ 2=0

119904119894Λ 119895Ω119896(37)

where 119866 (1199062 1199063 Δ 1 Δ 2) = (1198661 (1199061 1199062 Δ 1 Δ 2)1198662 (1199061 1199062 Δ 1 Δ 2)) (38)




1199022 = [01] (41)




(44)









1199061 = 1199041radic10038161003816100381610038161205993010038161003816100381610038161003816100381610038161003816120599211003816100381610038161003816 12058511199062 = minus11990411199042 10038161003816100381610038161205993010038161003816100381610038163210038161003816100381610038161205992110038161003816100381610038162 1205852

119905 = minus 120599211003816100381610038161003816120599301003816100381610038161003816 120591(49)




















20 22 24 26 28 30

20

22

24

26

28

30

y(t)

x(t)

(a)


minus30

minus28

minus26

minus24

minus22

minus20

y(t)

x(t)

(b)

minus2 minus1 0 1 2

minus2

minus1

0

1

2

y(t)

x(t)

(c)

10 15 20 25 3010

15

20

25

30y(

t)

x(t)

(d)


minus30

minus25

minus20

minus15

y(t)

x(t)

(e)



50

0

minus50

40

20

0

minus20

minus20

0

20

40

Z(t)

y(t)

x(t)

(a)

50

0

minus50

minus20

0

20

40

0

minus100

minus200

minus300

minus400

w(t)

y(t)

x(t)

(b)

40

20

0

minus20

minus40

40200minus20

y(t)

x(t)

(c)

40

20

0

minus20

40200minus20

z(t)

x(t)

(d)

40

20

0

minus20

40200minus20minus40

z(t)

y(t)

(e)

0

minus100

minus200

minus300

minus400

40200minus20minus40

w(t)

y(t)

(f)



x

minus

minus

a

(a)

x

minus

minus

b

(b)x

minus

minus

c

(c)

x

minus

minus

minus


(d)


(56)




(59)


1050 15 20 25 30a

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

2Ly

apun

ov ex

pone

nts

(a)

1086420 12b

minus20

minus15

minus10

minus5

0

5

10

Lyap

unov

expo

nent

s

(b)

108 96 74 521 30c

minus20

minus15

minus10

minus5

0

5

Lyap

unov

expo

nent

s

(c)


minus25

minus20

minus15

minus10

minus5

0

5

Lyap

unov

expo

nent

s

(d)









(65)

















10

10

10

0

0

0

minus10

minus10

minus10minus20

minus20

20

20

y

w

z












10

10

10

0

0

0

minus10

minus10

minus10minus20

minus20

20

20

y

w

z

(a)

20

minus2

5

0

minus5

x

y

20

minus2

z

(b)1

1

0

0minus1

minus1

minus10

minus05

10

05

00

x

z

y

(c)

minus10minus10

minus05

minus05

00

00z

x 05

05

10

10minus10

minus0500

y 0510

(d)

10

10

10

0

0

0

minus10

minus10

minus10

x

minus20minus20

minus2020

20

20y

z

(e)


10

0

0

0

minus10minus10

minus10minus5

minus5

minus5

5

5

5y

w

z

(a)

10

10

0

0

0

minus10

minus10minus5

minus5

minus5

5

5

5y

x

z

(b)

2

2

4

4

minus2 minus20

0minus4

minus4

x

y

2

0

minus2

z

(c)2

2

21

1

1

minus2minus2

minus2

minus1

minus1

minus1

0

0

0

x

y

z

(d)


10

5

5

5

0

0

0

x

w

z

minus5

minus5 minus5

minus10minus10

minus1010

10


minus10

10

105

5

5

0

0

0

minus5

minus5

minus5

minus10

y

w

z



(a)

(b) (c)

(d) (e)





6 Conclusion


Data Availability




Acknowledgments


References




















































Journal of












Journal of







Volume 2018






Volume 2018











tions119862 (119904 V 119908)= 3sum119894119895119896=1

1205973119866 (1199061 1199062 Δ 1 Δ 2)120597119906119894120597119906119895120597119906119896 1003816100381610038161003816100381610038161003816100381610038161003816 1199061=1199062=0Δ 1=Δ 2=0

119904119894V1198951199081198961198601 (119904 Λ) = 2sum

119894=1

2sum119895=1

1205972119866 (1199061 1199062 Δ 1 Δ 2)120597119906119894120597Δ 119895 1003816100381610038161003816100381610038161003816100381610038161003816 1199062=1199063=0Δ 1=Δ 2=0

119904119894Λ 1198951198602 (119904 Λ Ω)= 2sum119894=1

2sum119895119896=1

1205972119866 (1199062 1199063 Δ 1 Δ 2)120597119906119894120597Δ 119895120597Δ 119896 1003816100381610038161003816100381610038161003816100381610038161003816 1199062=1199063=0Δ 1=Δ 2=0

119904119894Λ 119895Ω119896(37)

where 119866 (1199062 1199063 Δ 1 Δ 2) = (1198661 (1199061 1199062 Δ 1 Δ 2)1198662 (1199061 1199062 Δ 1 Δ 2)) (38)




1199022 = [01] (41)




(44)









1199061 = 1199041radic10038161003816100381610038161205993010038161003816100381610038161003816100381610038161003816120599211003816100381610038161003816 12058511199062 = minus11990411199042 10038161003816100381610038161205993010038161003816100381610038163210038161003816100381610038161205992110038161003816100381610038162 1205852

119905 = minus 120599211003816100381610038161003816120599301003816100381610038161003816 120591(49)




















20 22 24 26 28 30

20

22

24

26

28

30

y(t)

x(t)

(a)


minus30

minus28

minus26

minus24

minus22

minus20

y(t)

x(t)

(b)

minus2 minus1 0 1 2

minus2

minus1

0

1

2

y(t)

x(t)

(c)

10 15 20 25 3010

15

20

25

30y(

t)

x(t)

(d)


minus30

minus25

minus20

minus15

y(t)

x(t)

(e)



50

0

minus50

40

20

0

minus20

minus20

0

20

40

Z(t)

y(t)

x(t)

(a)

50

0

minus50

minus20

0

20

40

0

minus100

minus200

minus300

minus400

w(t)

y(t)

x(t)

(b)

40

20

0

minus20

minus40

40200minus20

y(t)

x(t)

(c)

40

20

0

minus20

40200minus20

z(t)

x(t)

(d)

40

20

0

minus20

40200minus20minus40

z(t)

y(t)

(e)

0

minus100

minus200

minus300

minus400

40200minus20minus40

w(t)

y(t)

(f)



x

minus

minus

a

(a)

x

minus

minus

b

(b)x

minus

minus

c

(c)

x

minus

minus

minus


(d)


(56)




(59)


1050 15 20 25 30a

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

2Ly

apun

ov ex

pone

nts

(a)

1086420 12b

minus20

minus15

minus10

minus5

0

5

10

Lyap

unov

expo

nent

s

(b)

108 96 74 521 30c

minus20

minus15

minus10

minus5

0

5

Lyap

unov

expo

nent

s

(c)


minus25

minus20

minus15

minus10

minus5

0

5

Lyap

unov

expo

nent

s

(d)









(65)

















10

10

10

0

0

0

minus10

minus10

minus10minus20

minus20

20

20

y

w

z












10

10

10

0

0

0

minus10

minus10

minus10minus20

minus20

20

20

y

w

z

(a)

20

minus2

5

0

minus5

x

y

20

minus2

z

(b)1

1

0

0minus1

minus1

minus10

minus05

10

05

00

x

z

y

(c)

minus10minus10

minus05

minus05

00

00z

x 05

05

10

10minus10

minus0500

y 0510

(d)

10

10

10

0

0

0

minus10

minus10

minus10

x

minus20minus20

minus2020

20

20y

z

(e)


10

0

0

0

minus10minus10

minus10minus5

minus5

minus5

5

5

5y

w

z

(a)

10

10

0

0

0

minus10

minus10minus5

minus5

minus5

5

5

5y

x

z

(b)

2

2

4

4

minus2 minus20

0minus4

minus4

x

y

2

0

minus2

z

(c)2

2

21

1

1

minus2minus2

minus2

minus1

minus1

minus1

0

0

0

x

y

z

(d)


10

5

5

5

0

0

0

x

w

z

minus5

minus5 minus5

minus10minus10

minus1010

10


minus10

10

105

5

5

0

0

0

minus5

minus5

minus5

minus10

y

w

z



(a)

(b) (c)

(d) (e)





6 Conclusion


Data Availability




Acknowledgments


References




















































Journal of












Journal of







Volume 2018






Volume 2018












1199061 = 1199041radic10038161003816100381610038161205993010038161003816100381610038161003816100381610038161003816120599211003816100381610038161003816 12058511199062 = minus11990411199042 10038161003816100381610038161205993010038161003816100381610038163210038161003816100381610038161205992110038161003816100381610038162 1205852

119905 = minus 120599211003816100381610038161003816120599301003816100381610038161003816 120591(49)




















20 22 24 26 28 30

20

22

24

26

28

30

y(t)

x(t)

(a)


minus30

minus28

minus26

minus24

minus22

minus20

y(t)

x(t)

(b)

minus2 minus1 0 1 2

minus2

minus1

0

1

2

y(t)

x(t)

(c)

10 15 20 25 3010

15

20

25

30y(

t)

x(t)

(d)


minus30

minus25

minus20

minus15

y(t)

x(t)

(e)



50

0

minus50

40

20

0

minus20

minus20

0

20

40

Z(t)

y(t)

x(t)

(a)

50

0

minus50

minus20

0

20

40

0

minus100

minus200

minus300

minus400

w(t)

y(t)

x(t)

(b)

40

20

0

minus20

minus40

40200minus20

y(t)

x(t)

(c)

40

20

0

minus20

40200minus20

z(t)

x(t)

(d)

40

20

0

minus20

40200minus20minus40

z(t)

y(t)

(e)

0

minus100

minus200

minus300

minus400

40200minus20minus40

w(t)

y(t)

(f)



x

minus

minus

a

(a)

x

minus

minus

b

(b)x

minus

minus

c

(c)

x

minus

minus

minus


(d)


(56)




(59)


1050 15 20 25 30a

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

2Ly

apun

ov ex

pone

nts

(a)

1086420 12b

minus20

minus15

minus10

minus5

0

5

10

Lyap

unov

expo

nent

s

(b)

108 96 74 521 30c

minus20

minus15

minus10

minus5

0

5

Lyap

unov

expo

nent

s

(c)


minus25

minus20

minus15

minus10

minus5

0

5

Lyap

unov

expo

nent

s

(d)









(65)

















10

10

10

0

0

0

minus10

minus10

minus10minus20

minus20

20

20

y

w

z












10

10

10

0

0

0

minus10

minus10

minus10minus20

minus20

20

20

y

w

z

(a)

20

minus2

5

0

minus5

x

y

20

minus2

z

(b)1

1

0

0minus1

minus1

minus10

minus05

10

05

00

x

z

y

(c)

minus10minus10

minus05

minus05

00

00z

x 05

05

10

10minus10

minus0500

y 0510

(d)

10

10

10

0

0

0

minus10

minus10

minus10

x

minus20minus20

minus2020

20

20y

z

(e)


10

0

0

0

minus10minus10

minus10minus5

minus5

minus5

5

5

5y

w

z

(a)

10

10

0

0

0

minus10

minus10minus5

minus5

minus5

5

5

5y

x

z

(b)

2

2

4

4

minus2 minus20

0minus4

minus4

x

y

2

0

minus2

z

(c)2

2

21

1

1

minus2minus2

minus2

minus1

minus1

minus1

0

0

0

x

y

z

(d)


10

5

5

5

0

0

0

x

w

z

minus5

minus5 minus5

minus10minus10

minus1010

10


minus10

10

105

5

5

0

0

0

minus5

minus5

minus5

minus10

y

w

z



(a)

(b) (c)

(d) (e)





6 Conclusion


Data Availability




Acknowledgments


References




















































Journal of












Journal of







Volume 2018






Volume 2018









20 22 24 26 28 30

20

22

24

26

28

30

y(t)

x(t)

(a)


minus30

minus28

minus26

minus24

minus22

minus20

y(t)

x(t)

(b)

minus2 minus1 0 1 2

minus2

minus1

0

1

2

y(t)

x(t)

(c)

10 15 20 25 3010

15

20

25

30y(

t)

x(t)

(d)


minus30

minus25

minus20

minus15

y(t)

x(t)

(e)



50

0

minus50

40

20

0

minus20

minus20

0

20

40

Z(t)

y(t)

x(t)

(a)

50

0

minus50

minus20

0

20

40

0

minus100

minus200

minus300

minus400

w(t)

y(t)

x(t)

(b)

40

20

0

minus20

minus40

40200minus20

y(t)

x(t)

(c)

40

20

0

minus20

40200minus20

z(t)

x(t)

(d)

40

20

0

minus20

40200minus20minus40

z(t)

y(t)

(e)

0

minus100

minus200

minus300

minus400

40200minus20minus40

w(t)

y(t)

(f)



x

minus

minus

a

(a)

x

minus

minus

b

(b)x

minus

minus

c

(c)

x

minus

minus

minus


(d)


(56)




(59)


1050 15 20 25 30a

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

2Ly

apun

ov ex

pone

nts

(a)

1086420 12b

minus20

minus15

minus10

minus5

0

5

10

Lyap

unov

expo

nent

s

(b)

108 96 74 521 30c

minus20

minus15

minus10

minus5

0

5

Lyap

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expo

nent

s

(c)


minus25

minus20

minus15

minus10

minus5

0

5

Lyap

unov

expo

nent

s

(d)









(65)

















10

10

10

0

0

0

minus10

minus10

minus10minus20

minus20

20

20

y

w

z












10

10

10

0

0

0

minus10

minus10

minus10minus20

minus20

20

20

y

w

z

(a)

20

minus2

5

0

minus5

x

y

20

minus2

z

(b)1

1

0

0minus1

minus1

minus10

minus05

10

05

00

x

z

y

(c)

minus10minus10

minus05

minus05

00

00z

x 05

05

10

10minus10

minus0500

y 0510

(d)

10

10

10

0

0

0

minus10

minus10

minus10

x

minus20minus20

minus2020

20

20y

z

(e)


10

0

0

0

minus10minus10

minus10minus5

minus5

minus5

5

5

5y

w

z

(a)

10

10

0

0

0

minus10

minus10minus5

minus5

minus5

5

5

5y

x

z

(b)

2

2

4

4

minus2 minus20

0minus4

minus4

x

y

2

0

minus2

z

(c)2

2

21

1

1

minus2minus2

minus2

minus1

minus1

minus1

0

0

0

x

y

z

(d)


10

5

5

5

0

0

0

x

w

z

minus5

minus5 minus5

minus10minus10

minus1010

10


minus10

10

105

5

5

0

0

0

minus5

minus5

minus5

minus10

y

w

z



(a)

(b) (c)

(d) (e)





6 Conclusion


Data Availability




Acknowledgments


References




















































Journal of












Journal of







Volume 2018






Volume 2018









50

0

minus50

40

20

0

minus20

minus20

0

20

40

Z(t)

y(t)

x(t)

(a)

50

0

minus50

minus20

0

20

40

0

minus100

minus200

minus300

minus400

w(t)

y(t)

x(t)

(b)

40

20

0

minus20

minus40

40200minus20

y(t)

x(t)

(c)

40

20

0

minus20

40200minus20

z(t)

x(t)

(d)

40

20

0

minus20

40200minus20minus40

z(t)

y(t)

(e)

0

minus100

minus200

minus300

minus400

40200minus20minus40

w(t)

y(t)

(f)



x

minus

minus

a

(a)

x

minus

minus

b

(b)x

minus

minus

c

(c)

x

minus

minus

minus


(d)


(56)




(59)


1050 15 20 25 30a

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

2Ly

apun

ov ex

pone

nts

(a)

1086420 12b

minus20

minus15

minus10

minus5

0

5

10

Lyap

unov

expo

nent

s

(b)

108 96 74 521 30c

minus20

minus15

minus10

minus5

0

5

Lyap

unov

expo

nent

s

(c)


minus25

minus20

minus15

minus10

minus5

0

5

Lyap

unov

expo

nent

s

(d)









(65)

















10

10

10

0

0

0

minus10

minus10

minus10minus20

minus20

20

20

y

w

z












10

10

10

0

0

0

minus10

minus10

minus10minus20

minus20

20

20

y

w

z

(a)

20

minus2

5

0

minus5

x

y

20

minus2

z

(b)1

1

0

0minus1

minus1

minus10

minus05

10

05

00

x

z

y

(c)

minus10minus10

minus05

minus05

00

00z

x 05

05

10

10minus10

minus0500

y 0510

(d)

10

10

10

0

0

0

minus10

minus10

minus10

x

minus20minus20

minus2020

20

20y

z

(e)


10

0

0

0

minus10minus10

minus10minus5

minus5

minus5

5

5

5y

w

z

(a)

10

10

0

0

0

minus10

minus10minus5

minus5

minus5

5

5

5y

x

z

(b)

2

2

4

4

minus2 minus20

0minus4

minus4

x

y

2

0

minus2

z

(c)2

2

21

1

1

minus2minus2

minus2

minus1

minus1

minus1

0

0

0

x

y

z

(d)


10

5

5

5

0

0

0

x

w

z

minus5

minus5 minus5

minus10minus10

minus1010

10


minus10

10

105

5

5

0

0

0

minus5

minus5

minus5

minus10

y

w

z



(a)

(b) (c)

(d) (e)





6 Conclusion


Data Availability




Acknowledgments


References




















































Journal of












Journal of







Volume 2018






Volume 2018









x

minus

minus

a

(a)

x

minus

minus

b

(b)x

minus

minus

c

(c)

x

minus

minus

minus


(d)


(56)




(59)


1050 15 20 25 30a

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

2Ly

apun

ov ex

pone

nts

(a)

1086420 12b

minus20

minus15

minus10

minus5

0

5

10

Lyap

unov

expo

nent

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(b)

108 96 74 521 30c

minus20

minus15

minus10

minus5

0

5

Lyap

unov

expo

nent

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(c)


minus25

minus20

minus15

minus10

minus5

0

5

Lyap

unov

expo

nent

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(d)









(65)

















10

10

10

0

0

0

minus10

minus10

minus10minus20

minus20

20

20

y

w

z












10

10

10

0

0

0

minus10

minus10

minus10minus20

minus20

20

20

y

w

z

(a)

20

minus2

5

0

minus5

x

y

20

minus2

z

(b)1

1

0

0minus1

minus1

minus10

minus05

10

05

00

x

z

y

(c)

minus10minus10

minus05

minus05

00

00z

x 05

05

10

10minus10

minus0500

y 0510

(d)

10

10

10

0

0

0

minus10

minus10

minus10

x

minus20minus20

minus2020

20

20y

z

(e)


10

0

0

0

minus10minus10

minus10minus5

minus5

minus5

5

5

5y

w

z

(a)

10

10

0

0

0

minus10

minus10minus5

minus5

minus5

5

5

5y

x

z

(b)

2

2

4

4

minus2 minus20

0minus4

minus4

x

y

2

0

minus2

z

(c)2

2

21

1

1

minus2minus2

minus2

minus1

minus1

minus1

0

0

0

x

y

z

(d)


10

5

5

5

0

0

0

x

w

z

minus5

minus5 minus5

minus10minus10

minus1010

10


minus10

10

105

5

5

0

0

0

minus5

minus5

minus5

minus10

y

w

z



(a)

(b) (c)

(d) (e)





6 Conclusion


Data Availability




Acknowledgments


References




















































Journal of












Journal of







Volume 2018






Volume 2018









1050 15 20 25 30a

minus16

minus14

minus12

minus10

minus8

minus6

minus4

minus2

0

2Ly

apun

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pone

nts

(a)

1086420 12b

minus20

minus15

minus10

minus5

0

5

10

Lyap

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expo

nent

s

(b)

108 96 74 521 30c

minus20

minus15

minus10

minus5

0

5

Lyap

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expo

nent

s

(c)


minus25

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0

5

Lyap

unov

expo

nent

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(d)









(65)

















10

10

10

0

0

0

minus10

minus10

minus10minus20

minus20

20

20

y

w

z












10

10

10

0

0

0

minus10

minus10

minus10minus20

minus20

20

20

y

w

z

(a)

20

minus2

5

0

minus5

x

y

20

minus2

z

(b)1

1

0

0minus1

minus1

minus10

minus05

10

05

00

x

z

y

(c)

minus10minus10

minus05

minus05

00

00z

x 05

05

10

10minus10

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y 0510

(d)

10

10

10

0

0

0

minus10

minus10

minus10

x

minus20minus20

minus2020

20

20y

z

(e)


10

0

0

0

minus10minus10

minus10minus5

minus5

minus5

5

5

5y

w

z

(a)

10

10

0

0

0

minus10

minus10minus5

minus5

minus5

5

5

5y

x

z

(b)

2

2

4

4

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0minus4

minus4

x

y

2

0

minus2

z

(c)2

2

21

1

1

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minus1

0

0

0

x

y

z

(d)


10

5

5

5

0

0

0

x

w

z

minus5

minus5 minus5

minus10minus10

minus1010

10


minus10

10

105

5

5

0

0

0

minus5

minus5

minus5

minus10

y

w

z



(a)

(b) (c)

(d) (e)





6 Conclusion


Data Availability




Acknowledgments


References




















































Journal of












Journal of







Volume 2018






Volume 2018























10

10

10

0

0

0

minus10

minus10

minus10minus20

minus20

20

20

y

w

z












10

10

10

0

0

0

minus10

minus10

minus10minus20

minus20

20

20

y

w

z

(a)

20

minus2

5

0

minus5

x

y

20

minus2

z

(b)1

1

0

0minus1

minus1

minus10

minus05

10

05

00

x

z

y

(c)

minus10minus10

minus05

minus05

00

00z

x 05

05

10

10minus10

minus0500

y 0510

(d)

10

10

10

0

0

0

minus10

minus10

minus10

x

minus20minus20

minus2020

20

20y

z

(e)


10

0

0

0

minus10minus10

minus10minus5

minus5

minus5

5

5

5y

w

z

(a)

10

10

0

0

0

minus10

minus10minus5

minus5

minus5

5

5

5y

x

z

(b)

2

2

4

4

minus2 minus20

0minus4

minus4

x

y

2

0

minus2

z

(c)2

2

21

1

1

minus2minus2

minus2

minus1

minus1

minus1

0

0

0

x

y

z

(d)


10

5

5

5

0

0

0

x

w

z

minus5

minus5 minus5

minus10minus10

minus1010

10


minus10

10

105

5

5

0

0

0

minus5

minus5

minus5

minus10

y

w

z



(a)

(b) (c)

(d) (e)





6 Conclusion


Data Availability




Acknowledgments


References




















































Journal of












Journal of







Volume 2018






Volume 2018









10

10

10

0

0

0

minus10

minus10

minus10minus20

minus20

20

20

y

w

z

(a)

20

minus2

5

0

minus5

x

y

20

minus2

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(b)1

1

0

0minus1

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10

05

00

x

z

y

(c)

minus10minus10

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x 05

05

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10minus10

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(d)

10

10

10

0

0

0

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minus10

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x

minus20minus20

minus2020

20

20y

z

(e)


10

0

0

0

minus10minus10

minus10minus5

minus5

minus5

5

5

5y

w

z

(a)

10

10

0

0

0

minus10

minus10minus5

minus5

minus5

5

5

5y

x

z

(b)

2

2

4

4

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0minus4

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x

y

2

0

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(c)2

2

21

1

1

minus2minus2

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minus1

minus1

0

0

0

x

y

z

(d)


10

5

5

5

0

0

0

x

w

z

minus5

minus5 minus5

minus10minus10

minus1010

10


minus10

10

105

5

5

0

0

0

minus5

minus5

minus5

minus10

y

w

z



(a)

(b) (c)

(d) (e)





6 Conclusion


Data Availability




Acknowledgments


References




















































Journal of












Journal of







Volume 2018






Volume 2018









(a)

(b) (c)

(d) (e)





6 Conclusion


Data Availability




Acknowledgments


References




















































Journal of












Journal of







Volume 2018






Volume 2018









6 Conclusion


Data Availability




Acknowledgments


References




















































Journal of












Journal of







Volume 2018






Volume 2018


































Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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Dynamic Analysis and Circuit Implementation of a …downloads.hindawi.com/journals/mpe/2019/6581586.pdflayer secure communication systems and ultra-fast physi-cal random bits generation