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Infinitely Many Coexisting Attractors in No-Equilibrium Chaotic System

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Infinitely Many Coexisting Attractors in No-Equilibrium Chaotic System Hindawi Complexity Volume 2020, Article ID 8175639, 17 pages https://doi.org/10.1155/2020/8175639 Research Article Infinitely Many Coexisting Attractors in No-Equilibrium Chaotic System Qiang Lai ,1 Paul Didier Kamdem Kuate ,2 Huiqin Pei ,1 and Hilaire Fotsin 2 1School of Electrical and Automation Engineering, East China Jiaotong University, Nanchang 330013, China 2Laboratory of Condensed Matter, Electronics and Signal Processing Department of Physics, University of Dschang, P.O. Box 067, Dschang, Cameroon Correspondence should be addressed to Qiang Lai; [email protected] Received 17 January 2020; Revised 18 February 2020; Accepted 26 February 2020; Published 28 March 2020 Guest Editor: Chun-Lai Li Copyright © 2020 Qiang Lai et al. *is 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. *is paper proposes a new no-equilibrium chaotic system that has the ability to yield infinitely many coexisting hidden attractors. Dynamic behaviors of the system with respect to the parameters and initial conditions are numerically studied. It shows that the system has chaotic, quasiperiodic, and periodic motions for different parameters and coexists with a large number of hidden attractors for different initial conditions. *e circuit and microcontroller implementations of the system are given for illustrating its physical meaning. Also, the synchronization conditions of the system are established based on the adaptive control method. 1. Introduction system may generate hidden attractor. Also, a lot of pre- vious studies have shown that chaotic systems with mul- Encouraging progress has been made on chaos in the past tiple unstable equilibria usually have richer dynamic few decades. An important change is to recognize the great behaviors and are more likely to produce coexisting application potential of chaos in engineering. Nowadays, attractors. *erefore, many scholars tend to construct chaos generation has become an important research issue chaotic systems and distinguish their dynamic properties arousing constant concern. Inspired by the well-known by configuring different quantities and types of equilibria. Lorenz system [1], many different chaotic systems have Chaotic systems with no equilibrium, one stable equilib- been created [2–7]. *ere are two interesting directions in rium, one unstable node, two saddle foci, circular equi- generating new chaotic systems. One is to discover chaos in libria, a line equilibria, and other types of equilibria have nonlinear systems with very simple mathematical models. been reported [16–19]. *e most representative work was made by Sprott who Recently, the study of coexisting attractors and hidden established nineteen polynomial chaotic systems with ei- attractors in chaotic systems has aroused great enthusiasm ther five terms and two nonlinear terms or six terms and among scholars. Li et al. studied the coexisting attractors in one nonlinear term [8]. *e other is to construct chaotic Lorenz systems by some numerical experiments and pro- systems with special strange attractors including butterfly posed the conditional symmetry method to yield any attractor, multiscroll attractor, multiwing attractor, hidden number of coexisting attractors [20, 21]. Kengne et al. put attractor, and coexisting attractors [9–14]. *e number and forward some simple jerk systems with coexisting attractors type of equilibria play a decisive role in the dynamic [22]. Lai et al. introduced some effective methods to con- properties of chaotic system to some extent. A classic ar- struct chaotic systems with infinitely many coexisting gument is that a dynamic system with one saddle focus strange attractors [23, 24]. Li et al. designed a programmable connected by homoclinic orbit or two saddle foci con- chaotic circuit with infinitely many chaotic attractors and nected by heteroclinic orbit generates horseshoe chaos [15]. established the coexistence of multiple attractors by simu- It is generally recognized that no equilibrium chaotic lation methods [25]. Bao et al. found that memristor-based 2 Complexity chaotic systems can generate different types of coexisting 2. New Chaotic System attractors [26, 27]. Li et al. studied the coexisting attractors in memristor-based chaotic circuit in depth and Looking back, an augmented Sprott B system is described by considered its application in image encryption [28, 29]. the following differential equations [41]: Wei et al. studied the nonstationary chimeras in Hind- x_ � a x − x ; >8 1 2 1 � marsh–Rose neuronal network and found that the net- > <> x_ � x x + x ; work coexists with periodic and chaotic states [30]. *e 2 1 3 4 (1) > history of studying the hidden attractors in dynamical > x_3 � b − x1x2; :> systems without equilibria dates back to the well-known x_ � − cx ; Sommerfeld effect, arising from feedback in the energy 4 2 exchange between vibrating systems [31]. *e electro- where x1, x2, x3, and x4 are state variables and a, b, and c are mechanical and drilling systems without equilibria have parameters. *e previous study has shown that system (1) the ability to yield multiple hidden attractors [32, 33]. It is has no equilibrium and coexists with two symmetric strange very interesting to generate as many hidden attractors as attractors. Replacing the nonlinear term x1x3 of system (1) possible in chaotic system via some periodic functions with kx1 sin(x3) (k > 0 is a real number), then the following which can continuously replicate the attractors in phase new chaotic system is established: space. *e attractors produced by this method usually x_ � a x − x ; >8 1 2 1 � have the same properties. Another interesting question is > how many nontrivial attractors can be coexist in phase <> x_ � kx sinx � + x ; 2 1 3 4 (2) space of nonlinear system, which corresponds to a chaotic > > x_3 � b − x1x2; generalization [34] of the second part of Hilbert’s 16th :> problem. Recently, the study of hidden attractors in x_4 � − cx2: chaotic systems has received increasing concern. Wei et al. System (2) and system (1) have some similar properties generated an extended Rikitake system with hidden including the following: (i) they have no equilibrium because attractors [35] and investigated the hidden hyperchaos of there exists no point that satisfies x_ � x_ � x_ � x_ � 0; (ii) five-dimensional self-exciting homopolar disc dynamo via 1 2 3 4 they are dissipative with the divergence ∇V � zx_ /zx + analytical and numerical approaches [36]. Pham et al. did 1 1 zx_ /zx + zx_ /zx + zx_ /zx � − a < 0 for a > 0; (iii) they many important works on hidden attractors and first 2 2 3 3 4 4 have the symmetry under the transformation (x ; x ; showed different families of hidden attractors in chaotic 1 2 x ; x ) ⟶ (− x ; − x ; x ; − x ); and (iv) they have hidden systems [37, 38]. Danca et al. discovered the hidden 3 4 1 2 3 4 attractors whose basins of attraction do not intersect with attractors of the classic Rabinovich–Fabrikant system small neighborhoods of any equilibria. However, the re- [39]. Hidden attractor that refers to attractor whose basin markable difference between them is that system (2) can of attraction does not intersect with small neighborhoods generate infinitely many coexisting hidden attractors while of equilibria [40] embodies some mysterious unknown system (1) only generates no more than three attractors for dynamic behaviors of the system. *e phenomenon of given parameters. Also, to our best knowledge, system (2) coexisting attractors that corresponds to the generation of has never been reported before. *us, system (2) can be multiple attractors with independent basins of attractions classified as a new chaotic system. *e following section will implies the strong influence of initial conditions on the give an intuitive observation of the dynamic properties of final state of system. Both hidden attractors and coexisting system (2). attractors are interesting nonlinear dynamics worthy of further study. *e construction of simple no-equilibrium chaotic system with an infinite number of hidden 3. Coexisting Hidden Attractors attractors is an interesting and challenging thing. It will be of great significance to reveal the complex behaviors of Here, we will investigate the influence of the parameters and simple systems. However, most existing systems can initial conditions on the dynamic behaviors of system (2) by generate a limited number of attractors. So this paper aims simulation experiments, emphasizing the existence of infi- to present a new autonomous chaotic system with infi- nitely many hidden chaotic and periodic attractors in system nitely many coexisting hidden attractors from the per- (2). We apply the well-known fourth-fifth-order Run- spective of building complex behavior of simple system. ge–Kutta method to numerically solved system (2) on Necessary theoretical and experimental researches are Matlab platform. All the simulation results are obtained by given to illustrate the dynamic properties of the proposed fixing the step size Δt � 0:01 and time region t ∈ [0; 300]. system. *e circuit and microcontroller implementations *e Lyapunov exponents are calculated along the trajecto- and synchronization control of the system are studied. *e ries of system (2) by using the Wolf method [42]. We also paper is organized as follows. Section 2 derives the model can further consider the Lyapunov exponents by using the of the new system. Section 3 presents the coexisting method proposed in literature [43]. Here, the Lyapunov hidden attractors of the system. Section 4 implements the exponents are used to distinguish the chaotic, periodic, and system for physically illustrating its dynamics. Section 5 stable states of system (2). So, we use the conventional considers its synchronization problem. Section 6 sum- method to calculate them. *ere is no doubt that a system marizes the conclusions. with the positive largest Lyapunov exponent cannot be Complexity 3 judged to be chaotic directly. It also requires that the system behaviors for suitable sets of parameters and initial condi- is bounded. However, it is difficult to quantitatively establish tions.
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