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