Hyperchaos in a Conservative System with Nonhyperbolic Fixed Points

Hindawi Complexity Volume 2018, Article ID 9430637, 8 pages https://doi.org/10.1155/2018/9430637

Research Article Hyperchaos in a Conservative System with Nonhyperbolic Fixed Points

Aiguo Wu,1 Shijian Cang ,1,2 Ruiye Zhang,1 Zenghui Wang ,3 and Zengqiang Chen 4

1 School of Electrical Engineering and Automation, Tianjin University, Tianjin 300072, China 2Department of Product Design, Tianjin University of Science and Technology, Tianjin 300457, China 3Department of Electrical and Mining Engineering, University of South Africa, Florida 1710, South Africa 4College of Computer and Control Engineering, Nankai University, Tianjin 300071, China

Correspondence should be addressed to Shijian Cang; [email protected]

Received 8 November 2017; Revised 21 February 2018; Accepted 12 March 2018; Published 22 April 2018

Academic Editor: Jose´ Angel´ Acosta

Copyright © 2018 Aiguo Wu et al. Tis is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Chaotic dynamics exists in many natural systems, such as weather and climate, and there are many applications in diferent disciplines. However, there are few research results about chaotic conservative systems especially the smooth hyperchaotic conservative system in both theory and application. Tis paper proposes a fve-dimensional (5D) smooth autonomous hyperchaotic system with nonhyperbolic fxed points. Although the proposed system includes four linear terms and four quadratic terms, the new system shows complicated dynamics which has been proven by the theoretical analysis. Several notable properties related to conservative systems and the existence of perpetual points are investigated for the proposed system. Moreover, its conservative hyperchaotic behavior is illustrated by numerical techniques including phase portraits and Lyapunov exponents.

1. Introduction white-noise-like, the conservative chaos has no chaotic attractor, which means that the conservative chaos when As a research hotspot in the feld of nonlinear science applied in information security has distinct advantages over over the last fve decades, chaos theory has achieved great dissipative chaos. development since the Ueda attractor [1], the Lorenz attractor It is well known that the nonlinear dynamical systems [2], and Li-Yoke chaos [3] were discovered. So far, chaos theory has been successfully applied in many felds, such can produce various dynamical behaviors, such as periodic as electronic engineering [4], computer science [5], com- motion, quasiperiodic motion, chaos, and hyperchaos. Gen- munication systems [6, 7], complex networks [8], chemical erally, hyperchaos is defned as its dynamical behavior with at engineering [9], and economic models [10]. In the recent least two positive Lyapunov exponents (LEs), and the mini- decade, hundreds of physical chaotic models [11–14] and mal dimension for an autonomous continuous hyperchaotic artifcial chaotic systems [15–19] have been investigated in system is four. In comparison with chaos, hyperchaos, espe- theory and by numerical simulations due to the poten- cially conservative hyperchaos, is preferable for those appli- tial applications of chaotic system in various chaos-based cations that require the chaotic systems showing complicated technologies [20, 21]. By now, numerous dissipative-chaos- dynamics, such as network security and data encryption. Te based encryption algorithms have been developed to ensure frst hyperchaotic system was proposed in 1979 by Rossler¨ the safety of information, but these algorithms are not [22]. Since then, various hyperchaotic systems have been strong enough because the dissipative chaotic attractors can found [23–26], but almost all of these hyperchaotic systems be reconstructed by delay embedding method based on are dissipative and it is difcult to fnd odd-dimensional the sampled data. Besides the general properties of chaos systems that can generate conservative hyperchaotic fows. such as ergodicity, aperiodic, uncorrelated, broadband, and In 1994, Sprott found a 3D nonequilibrium non-Hamiltonian 2 Complexity system with conservative chaos, lately known as the Sprott A �=�̇ V system [15], which is described as V̇=��−�� =�̇ �=−�̇ V −��, �=−�+��̇ (1) (2) �=−�̇ 2 +1, where �, �, �, V,and� are the system variables and � and �∈ R+ � � � are the system’s constant parameters. Obviously, system where , ,and are the system states. It has been proven that (2) does not meet the requirements for classic Hamiltonian system (1) is a special case of the Nose-Hoover´ system [27]. systems. Although system (2) is not Hamiltonian, it does have Moreover, there exist coexisting quasiperiodic and conserva- a conserved quantity, which will be discussed in the following tive chaotic fows under diferent initial conditions. In view of sections. thefactthatthesesystemswithconservativehyperchaosare rare, thus, it is very interesting to fnd hyperchaotic systems 2.1. Invariance and Time-Reversibility. Many of chaotic tra- with conservative fows. jectories are symmetric; if �→+∞,itiseasytoget Trough sorting out the published papers about chaotic the invariance of the system (2) under the coordinate systems of recent years, it is found that hidden attractors and transformation (�,�,�,V,�) → (�,�,−�,−V,−�),which perpetual points become two important new research topics. persists for all values of � and �.However,theinvari- Te properties of hidden attractor are diferent from that of ance of the solution of system (2) in the time-reversible self-excited chaotic attractor [28, 29]. It has been shown that direction can also be achieved under the transformation (�,�,�,�,V, �) → (−�, −�, �, �, V,−�) the dynamical systems having hidden attractors include these . Generally, a dynami- systems without equilibrium, with no unstable equilibrium, cal system is reversible if there is an involution in phase space with one stable equilibrium, and with lines, curves, planes, which reverses the direction of time. It is a very common and surfaces of equilibria [30–35]. Recently, perpetual points thing to fnd the time-reversibility for conservative systems, were introduced as another structural feature of nonlinear but there are exceptions [38, 39]. systems [36]. In some cases, the perpetual points are useful to locate hidden attractors and to fnd coexisting attractors in 2.2. Conservation multistable systems [36, 37]. Defnition 1. Dynamical systems, whose Hamiltonian (i.e., Te main contribution of this paper is that a rare fve- conserved quantity) does not vary in time, are called conser- dimensional (5D) autonomous dynamical system is proposed vative systems; otherwise they are called dissipative systems and investigated. In comparison with the known chaotic [40]. systems, the system has the following four characteristics: (i) Te system with nonzero initial values has two non- Remark 2. Tere is diference between the conservative sys- hyperbolic fxed points. temandtheconservativemotion.Generally,theconservative motion is most ofen encountered in conservative systems, (ii) Te system is conservative, which can be theoretically especiallyinHamiltoniansystems,butitcanbealsofound verifed by the existence of a conserved quantity, in nonconservative systems, such as the Sprott A system; but cannot be confrmed by the trace of its Jacobian in other words, the conservative motion exists not only in matrix, which means that the fows generated by the conservative systems but also in nonconservative systems, system are compressible. which can be verifed by numerical techniques, such as LE (iii) Numerically, the symmetric LEs spectrum shows that spectra. the system has conservative chaos. By the defnition, if we can fnd a Hamiltonian for system (iv) Hyperchaotic motion can be found under specifc (2)andprovethetimederivativeoftheconservedquantity conditions in this conservative system. equals zero, system (2) is conservative. Te rest of this paper is organized as follows. In Section 2, we introduce a new 5D dynamical system and analyze its Teorem 3. One of the Hamiltonians of system (2) is �(�, 2 2 2 2 2 ̇ basic dynamics. In Section 3, the existence of perpetual point �, �, V, �) = (1/2)(� +� +� + V +� ),whichsatisfes�(�, of the new system is investigated. Section 4 illustrates the �, �, V,�)=0. dynamicalbehaviorsoftheproposedsystemwiththehelpof LEs and phase portraits. Te conclusion is presented in the Proof. Te linear frst-order PDE of system (2) can be last section. expressed as [41, 42] �� ̇ + �̇ + �̇ + V̇ + �̇ =0. (3) 2. System Model �� V �� In order to get the general solutions of (3), we choose (4) as Consider the following 5D continuous dynamical system: the associated system. 2 �=��+�̇ �� V �� = = = = . (4) �=−��̇ �� + �2 −�� V �� − �� −�V −�� Complexity 3

According to (4), for simplicity, we choose a particular Obviously, all FPs of system (2) are center FPs (nonhyperbolic solution of the following smooth positive-defnite standard FPs) since there exist one real root �1 =0and two pairs of quadratic form: purely imaginary roots �2,3 =±��and �4,5 =±��,which 1 can efectively exhibit that there are no asymptotically stable �(�,�,�,V,�)= (�2 +�2 +�2 + V2 +�2). 2 (5) equilibria or limit cycles in conservative systems in phase space. BasedontheTeorem3,wecanconcludethatsystem(2)is conservative. Teorem 5. Forsystem(2),therearetwocenterFPsunderany initial conditions (�0,�0,�0, V0,�0) with �0 =0̸ . In addition, we can also discuss the time evolution of volumes under a fow described by Liouville [43]. Proof. If the initial value of system (2) is (�0,�0,�0, V0,�0), we obtain Lemma 4. It is assumed that system (2) can generate a fow 5 ��(�,�,�,V,�,�).Let�� denote a domain in R and �� 1 2 2 2 2 2 0 � = (� +� +� + V +� ), � � (�,�,�,V,�,�) 0 0 0 0 0 0 (10) denote the evolution of �0 under the fow � . 2 If �(�) represents the volume of ��, system (2) is dissipative if ∇⋅�<0and conservative if ∇⋅�=0. which means that �(��,��,��, V�,��)=�0. Considering (��,��,��, V�,��)=(0,0,�,0,0),wehave According to Lemma 4, we might determine whether system (2) is dissipative or conservative by ∇⋅�.Wenotethat √ 2 2 2 2 2 �=± �0 +�0 +�0 + V0 +�0, (11) the divergence of fow of system (2) is ��̇ ��̇ ��̇ �V̇ ��̇ which shows that system (2) has two center FPs under ∇⋅�= + + + + = −�, (6) �� V �� any initial conditions except �0 =0.Actually,thesetwo center FPs are the intersections of �-axis and the hypersphere 2 2 2 2 2 2 which shows that it is difcult to directly discern whether �(�, �, �, V, �) = (1/2)(� +� +� + V +� ) = (1/2)� in ∇⋅� system (2) is dissipative or conservative since depends phase space. on the system variable �. Due to the existence of nonhyperbolic FPs, system (2) is 2.3. Fixed Points and Stability. In order to determine the fxed diferent from most published chaotic systems with hyper- points (FPs) and stability of system (2), it is necessary to bolic FP, such as the Lorenz and Lorenz-like systems, the calculate �=0̇ , �=0̇ , �=0̇ , V̇=0,and�=0̇ ;thenone Rossler¨ system, and Chua’s circuit [2, 16, 44, 45]. Chaotic obtains systems with nonhyperbolic FPs are rare, but they can be �� + �2 =0 found not only in conservative systems but also in dissipative systems. −��=0 In 1994, Sprott proposed the following three dissipative chaotic systems [15]: �V =0 (7) �� − �� = 0 �=��̇

−�V −��=0. �=�−�̇ (12) Solving (7) yields the FPs of system (2) as follows: �=1−�̇ 2, (��,��,��, V�,��) = (0,0,�,0,0),where�∈R is a parameter and denotes any point of the �-axis. Hence, the �-axis is the �=−�̇ line of FPs of system (2). Te Jacobian matrix of system (2) is �=�+�̇ (13) 0�0 02� �=��+3�̇ 2, −� 0 0 0 0 ( ) �=��̇ � (⋅) = ( V 00 � 0) . (8) 2 −� 0 −� 0 � �=�̇ −� (14) (−� 0 0 −� −�) �=1−4�.̇ For any FP of system (2), we can obtain the corresponding characteristic equation of (8) by calculating �(�) = |�� − �| = Obviously, there are two FPs in (12) and one FP in (13) and 0: (14). Tere exist zero real parts in the eigenvalues of these FPs; � (�) = |��−�| =�(�2 +�2)(�2 +�2)=0. thus these three systems belong to the kind of chaotic systems (9) with nonhyperbolic FPs. In 2005, Wei et al. presented the 4 Complexity following chaotic jerk system with one single nonhyperbolic where �,� = 1,2,...,� and the subscript “AP” denotes the FP [46, 47]: acceleration of these state variables at any point. Trough the �=�̇ comparison of (17) and (18), it is found that the solutions to (17) must be the solutions to (18), but there may exist other solutions to (18). PPs are these points where all the �=�̇ (15) acceleration of these state variables of the system (16) is zero �=ℎ(�,�,�),̇ but the velocities are not [36], which can be denoted by

2 2 �PP =(�AP −�AP ∩�FP). (19) where ℎ(�, �, �) = � + �1�+�2�+�3�+�4� +�5� + 2 �6� +�7�� + �8�� + �9�� (�, �� ∈ R, � = 1,2,...,9)is the Note that FPs and PPs are preserved under the trans- most general quadratic function. If �4 =0or �4 =0,�≠ formations of the invariance and time-reversibility of system 2 �1 /4�4, system (15) has only one nonhyperbolic FP. So far, (2) because these transformations are linear [49]. At frst, PP these chaotic systems with nonhyperbolic FP(s) are unusual canbeusedtodistinguishwhetheradynamicalsystemis relative to these chaotic system with hyperbolic FP(s). For a dissipative or conservative [36] and locate hidden attractors given set of initial values and system parameters, system (2) [37]. Based on the recent work, Jafari et al. pointed out the has two nonhyperbolic FPs, and moreover, the hyperchaotic limitation of PP for confrming conservation in dynamical motion can also be found in (2), which will be verifed in systems, which is supported by the Sprott A system [50], Section 4. and the insufciency of PP for locating hidden attractors in dynamical systems, which is also supported by three examples [51]. Terefore, the existence of PPs to distinguish 3. Analysis of Perpetual Point dissipative systems from conservative systems and locate Perpetual point (PP) as a new research topic in the feld hidden attractors is not applicable for all dynamical systems. of nonlinear dynamics has aroused the researchers’ interest. According to (7), the FPs of system (2) are Nazarimehr et al. gave the category of fows from the �PP viewpoint of FP and PP, which includes fows with FPs and 5 (20) PPs, with FPs but without any PPs, without any FPs but with ={(�,�,�,V,�)∈R |�=�=V =�=0,�=�}, PPs, and without any FPs and PPs [48]. where �∈R. Based on (18), we can get Generally, FPs are constant discrete points for ordinary diferential equations (ODEs), which are always discussed �̈ with the corresponding Jacobian matrix when we study the �̈ dynamical behaviors of nonlinear systems. Now, we consider ( ) the following generalized nonlinear autonomous dynamical (�̈ ) system: V̈

�1̇=�1 (�1,�2,...,��) (�̈) �̇=� (� ,� ,...,� ) 2 2 2 2 1 2 � −� �−2�V�−2�� (21) . (16) −�2�−��2 . ( ) ( 2 2 ) = ( �� + ��V −� �+V� ) �̇=� (� ,� ,...,� ), � � 1 2 � −�2V −��−��−�2V −��2 � �(⋅) 2 2 3 where � are system variables and � are smooth nonlinear (−� �+��(�+V) −��+� �−� ) functions, with � = 1,2,...,�. By calculating the FPs of the system (16), one obtains =0.

�FP Solving (21) yields (17) � � ={(�,�,�,V,�)∈R5 |�=�=V =�=0,� ={(�1,�2,...,��)∈R |�� (�1,�2,...,��)=0}. AP (22) Terefore, the point(s) with zero velocity of these state =�}, variables of the system (16) can be obtained. Similarly, we �∈R � =� can calculate the zero acceleration of these state variables of where parameter .Obviously, FP AP.Ten,we system (2) as follows: conclude �PP = (�AP −�AP ∩�FP) =0, (23) �AP which shows that system (2) belongs to the class of fows with { � �� (⋅) } (18) FPs but without any PPs. In this case, the conclusion obtained = (� ,� ,...,� )∈R� |�(⋅) ∑ � =0 , { 1 2 � � } by Prasad [36] is valid to determine the conservation of �=1 �� { } system (2) due to the inexistence of PP; that is, �PP =0. Complexity 5

1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 −0.2 −0.2

Lyapunov exponents Lyapunov −0.4

Lyapunov exponents Lyapunov −0.4 −0.6 −0.6 −0.8 −0.8 −1 −1 1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 a b

，％1 ，％4 ，％1 ，％4 ，％2 ，％5 ，％2 ，％5 ，％3 ，％3

Figure 1: Te dependence of fnite-time local LEs of the system (2) Figure 2: Te dependence of fnite-time local LEs of the system (2) on parameter �,with�=1. on parameter �,with�=1.

4. Dynamical Behaviors of the Conservative observed that the dynamics of system (2) switches among Hyperchaotic System hyperchaos, chaos, and quasiperiodic motion with increasing parameter �, and system (2) has a rather wide range of param- In this section, some numerical simulations for system (2) eter � in which the generated fows are chaotic. When �∈ are presented to investigate the dynamical behaviors of [1, 2.505], the largest LE is greater than zero and the second system (2). Generally, to analyze the dynamics of nonlinear largest LE is greater than or equal to zero, which means systems, there are many methods such as Lyapunov exponent system (2) is chaotic or hyperchaotic. When � ∈ (2.505, 3], spectrumandphaseportrait,whichcanbeusedtoanalyze all LEs equal zero, which implies system (2) undergoes system (2). Based on the fourth-order Runge-Kutta method, −9 aquasiperiodicmotion. we keep the absolute and relative error 10 and adopt Similarly, if the system parameter �=1and the initial val- variable step to solve system (2) in MATLAB. (� ,� ,� , V ,� ) = (1,1,1,1,1) Te dynamics of classical chaotic systems include peri- ues 0 0 0 0 0 , Figure 2 illustrates the � odic motion, quasiperiodic motion, and chaos and hyper- dependence of LEs of system (2) on parameter .Itisfound chaos, which can be confrmed by the calculation of Lya- that the curves refect the change of dynamics between hyper- punov exponents (LEs) [52]. If an autonomous dynamical chaos and quasiperiodic motion with increasing parameter system is chaotic, its dimension must be equal to or more �.When� ∈ [0.055, 3.75] ∪ [3.825, 4.265],thelargest than three and at least one positive LE is greater than zero. LEandthesecondlargestLEaregreaterthanzero,which If there is one positive LE, the system is chaotic; while there means system (2) is hyperchaotic. When � ∈ [0, 0.055) ∪ are two and more positive LEs, the system is hyperchaotic (3.75, 3.875) ∪ (4.265, 5],allLEsequalzero,whichimplies [53]. For a dynamical system with dissipative fows, the sum system (2) undergoes a quasiperiodic motion. of all LEs must be less than zero, while for a dynamical system Te LE spectra in Figures 1 and 2 are fnite-time local LE with conservative fows, the sum of all LEs equals zero [53]. In spectra [54–56], which are obtained by the Wolf algorithm in Sections 2 and 3, the theoretical analysis shows that system (2) MATLAB [52], with the time step of 0.1 second and the run- is conservative, but system (2) with conservative hyperchaos time of 1000 seconds. Note that, for transient chaos, the local needstobefurtherverifedbythesumofitsLEs. fnite-time LEs may be positive for a very long time, but the fnal values of LEs will be negative or zero. Remark 6. Generally, the dimension for integer-order dissipative dynamical systems with hyperchaos is greater than or Moreover, from Figures 1 and 2 we can see that the equal to FOUR, but the dimension for integer-order conser- nonzero LEs of system (2) are symmetric to zero LE, which vative systems with hyperchaos is at least FIVE because there can account for the conservative motion of the system exists linear dependence among of all system variables which (2) since the sum of all LEs is zero. Owing to the LEs arerefectedintheconservedquantity�(⋅) = constant. (LE1, LE2, LE3, LE4, LE5) = (0.836, 0.045, 0, −0.045, −0.836) when parameters �=�=1and the initial conditions (�0,�0, If the system parameter �=1and the initial values �0, V0,�0) = (1,1,1,1,1), the fows generated from system (�0,�0,�0, V0,�0) = (1,1,1,1,1),wecangettheLEspectra (2) are hyperchaotic. Te plots of the fow starting at (�0,�0, of system (2) by varying �, as shown in Figure 1. It can be �0, V0,�0) are shown in Figure 3. 6 Complexity

2.5 2.5

1.5 1.5

0.5 0.5 z w −0.5 −0.5

−1.5 −1.5

−2.5 −2.5 2.5 2.5 1.5 2.5 1.5 2.5 0.5 1.5 0.5 1.5 −0.5 0.5 0.5 y −0.5 y −0.5 −0.5 −1.5 −1.5 x −1.5 −1.5 x −2.5 −2.5 −2.5 −2.5 (a) (b)

2.5

1.5

0.5 w −0.5

−1.5 −2.5 2.5 1.5 2.5 0.5 1.5 0.5 v −0.5 −0.5 −1.5 −1.5 z −2.5 −2.5 (c)

Figure 3: Phase portraits of the system (2) with parameters �=1, �=1, and the initial conditions (�0,�0,�0, V0,�0) = (1,1,1,1,1):(a) �−�−�space; (b) �−�−�space; (c) �−V −�space.

Note that system (2) is conservative; as shown by the where � is the maximum value of � such that �� = LE1 +⋅⋅⋅+ existence of the invariant �(⋅), diferent initial conditions LEk ≥0. Obviously, the fractal dimension of system (2) is fve may lead to diferent dynamical behaviors. Terefore, system according to (24), which means system (2) is conservative. (2) is a multistable system. Visually, the hyperchaotic motion in Figure 3 forms a strange attractor, but in fact, there is no 5. Conclusion chaotic attractor in conservative systems [57]. Since the focus of this paper is investigating the hyperchaotic behavior of Tis paper proposed a new conservative system with two system (2), other dynamical behaviors including chaos and nonhyperbolic FPs. Te conservation of the system has been quasiperiodic motion are not studied. verifed theoretically, and the numerical results in MATLAB also showed there exist conservative hyperchaotic fows in the Remark 7. A 5D integer-order dissipative dynamical system proposed system. Compared with other chaotic systems, the has strange attractors with fractal dimension between two dynamics of the proposed hyperchaotic system are messier and fve, while a 5D integer-order conservative dynamical and more complex; hence the proposed hyperchaotic system system flls a 5D volume, and therefore, the fractal dimension may have a good potential application value in the feld of of system (2) is fve. Generally, the fractal dimension can information technology such as secure communication and � be expressed by the so-called Kaplan–Yorke dimension KY, encryption. which is always substituted by Lyapunov dimension and is defned as Conflicts of Interest ∑� � =� =�+ �=1 LE� , KY � � � (24) Te authors declare that there are no conficts of interest � � �LE�+1� regarding the publication of this paper. Complexity 7

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