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Dynamic Analysis, Circuit Implementation and Passive Control of a Novel Four-Dimensional Chaotic System with Multiscroll Attractor and Multiple Coexisting Attractors

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Dynamic Analysis, Circuit Implementation and Passive Control of a Novel Four-Dimensional Chaotic System with Multiscroll Attractor and Multiple Coexisting Attractors Pramana – J. Phys. (2018) 90:33 © Indian Academy of Sciences https://doi.org/10.1007/s12043-018-1525-1 Dynamic analysis, circuit implementation and passive control of a novel four-dimensional chaotic system with multiscroll attractor and multiple coexisting attractors BANG-CHENG LAI1,∗ and JIAN-JUN HE2 1Institute of Technology, East China Jiaotong University, Nanchang 330100, China 2College of Applied Science, Jiangxi University of Science and Technology, Ganzhou 341000, China ∗Corresponding author. E-mail: [email protected] MS received 25 August 2017; revised 3 October 2017; accepted 10 October 2017; published online 8 February 2018 Abstract. In this paper, we construct a novel 4D autonomous chaotic system with four cross-product nonlinear terms and five equilibria. The multiple coexisting attractors and the multiscroll attractor of the system are numer- ically investigated. Research results show that the system has various types of multiple attractors, including three strange attractors with a limit cycle, three limit cycles, two strange attractors with a pair of limit cycles, two coex- isting strange attractors. By using the passive control theory, a controller is designed for controlling the chaos of the system. Both analytical and numerical studies verify that the designed controller can suppress chaotic motion and stabilise the system at the origin. Moreover, an electronic circuit is presented for implementing the chaotic system. Keywords. Chaotic system; multiple attractors; multiscroll attractor; circuit implementation; passive control. PACS No. 05.45 1. Introduction pair of point attractors or limit cycles [7]. By using the polynomial function method, any number of coexist- The multiple coexisting attractors have been the focus ing chaotic attractors were constructed from the Sprott of research topics in nonlinear science in recent years. Bsystem[8]. Braga and Mello investigated the forma- Many academic papers have discussed the multiple tion mechanism of three types of attractors in a simple coexisting attractors in natural and artificial systems [1– chaotic system through rigorous mathematical verifica- 5]. The coexistence of multiple attractors usually means tion [9]. Tamba et al analysed an improved Colpitts that the system has multiple steady states. It provides oscillator that performs a period-1 limit cycle and a the system with the possibility of a variety of normal strange attractor for different initial conditions [10]. operating modes, which will benefit the system perfor- Pham et al put forward a no-equilibrium chaotic sys- mance. tem which coexists with a pair of strange attractors, or a In recent years, the multiple coexisting attractors in pair of limit cycles [11]. Sprott et al presented a special chaotic system has aroused great interest. The existing periodically-forced oscillator which generates an infi- research results indicated that some low-dimensional nite number of coexisting nested attractors, including nonlinear differential equations not only generate chaos, limit cycles, attracting tori and strange attractors [12]. but also generate multiple coexisting attractors. Li and Li et al proposed some variable-boostable chaotic flows Sprott discussed the multistability of Lorenz system with coexisting attractors, and found an interesting off- by computing its basin of attraction and bifurcation set boosting method for diagnosing the multistability diagram [6]. The butterfly attractor in Lorenz system of chaotic systems [13–15]. Moreover, the investiga- can be broken into two coexisting strange attractors tion of the multiple coexisting attractors has also been or two coexisting limit cycles for some special values reported in [16–22]. The study of the chaotic system of parameters. Lai and Chen proposed a new chaotic with multiscroll (or multiwing) strange attractor is also system with four coexisting strange attractors, four an interesting research topic. Previous studies showed coexisting limit cycles, two strange attractors with a that chaotic system with multiscroll attractor can be 33 Page 2 of 12 Pramana – J. Phys. (2018) 90:33 O(0, 0, 0, 0) used to secure communication and can greatly improve √ √ √ √ √ √ the encryption performance. Many effective methods A±(±r b, ±r a, ± ab, ac b/(m b + a)) √ √ √ √ √ √ have been found to construct chaotic system with multi- (± , ± , ± , /( − )), scroll attractor, such as trigonometric function method, B± s b s a ab ac b m b a piecewise-linear function method, coordinate transfor- where mation method, etc. [23–27]. √ √ √ Chaos control has become an important research topic r = cm b/(m b + a) since the famous OGY control method was put for- √ √ √ ward by Ott et al in 1990 [28]. The scientists started to s = cm b/(m b − a). believe that chaos is controllable, and many chaos con- trol methods were presented, including adaptive control, The stability of the equilibria can be determined by impulsive control, passive control, sliding mode con- computing their eigenvalues. It is easy to get that the , − , − , − trol, etc. [29–34]. The passive control has been applied eigenvalues of O are a b c m. Thus, O is an λ in many chaotic systems [35–38]. The studies showed unstable point. The eigenvalues ± of A± should meet that the controller designed on the basis of the passive the following equations: control theory can achieve the desired control effect. 4 3 2 2 λ+ + p1λ+ + p2λ+ + (p3 + 4abr )λ+ + p4 As an effective control method, the passive control has received widespread attention of the scholars. + 4abmr2 = 0, It is extremely interesting and significant to find and 4 3 2 λ + p λ + p λ + p λ− + p = 0, analyse multiple coexisting attractors in chaotic sys- − 1 − 2 − 3 4 tems, and consider their control problems. In this paper, where p1, p2, p3, p4 are given by we present a novel 4D autonomous chaotic system with = + + − , five equilibria and investigate the multiple coexisting p1 b c m a = ( + − ) + ( − )( − 2), attractors of the system. Research results show that p2 b c a m b a √c r 2 2 the system has different types of coexisting attractors, p3 = m(b −√a)(c − r ) − 2r ab, 2 including three coexisting chaotic attractors, three coex- p4 = 2ar ab. isting limit cycles and a chaotic attractor with two limit According to the Routh–Hurwitz criterion, we know that cycles. Numerical simulations are applied to analyse the A+ is stable as long as dynamical behaviours of the system. Based on the pas- ⎧ 2 sive control theory, a controller is proposed for driving ⎪ p1 > 0, p2 > 0, p3 > − 4abr , ⎨ 2 2 the system to the origin. In addition, we also study the p4 > − 4abmr , p1 p2 > p3 + 4abr circuit implementation of the chaotic system. ⎪ ( − − 2)( + 2) ⎩⎪ p1 p2 p3 4abr p3 4abr > 2( + 2) p1 p4 4abmr and A− is stable as long as 2. A novel 4D chaotic system p1 > 0, p2 > 0, p3 > 0, p4 > 0 The chaotic system considered in this paper is described > , > 2 + 2 . p1 p2 p3 p1 p2 p3 p3 p1 p4 by the following autonomous differential equations: ⎧ Similarly, B+ is stable if ⎪ x˙ = ax − yz ⎧ ⎨ ⎪ > , > , > − 2, y˙ =−by + xz ⎨⎪ q1 0 q2 0 q3 4abs (1) > − 2, > + 2 ⎩⎪ z˙ =−cz + xy + w q4 4abms q1q2 q3 4abs 2 ⎪ ( − + 2)( + 2) w˙ =−mw + y ⎩⎪ q1q2 q3 4abs q3 4abs > 2( + 2) q1 q4 4abms where x, y, z,ware the state variables, and a, b, c, m ∈ + and B− is stable if R are the parameters. System (1) is generated from the 3D chaotic system which was proposed by Liu and Chen q1 > 0, q2 > 0, q3 > 0, q4 > 0 [39] by introducing a variable w with w˙ =−mw + y2. > , > 2 + 2 q1q2 q3 q1q2q3 q3 q1 q4 If a < b + c + m, then system (1) is dissipative with its 2 divergence ∇V = ∂x˙/∂x +∂ y˙/∂y+∂z˙/∂z+∂w/∂w˙ = where q1 = p1, q2 = (b + c −√a)m + (b − a)(√c − s ), 2 2 2 a − b − c − m < 0. The equilibria of system (1) can be q3 = m(b − a)(c − s ) − 2s ab, p4 = 2as ab. obtained√ by letting x˙ =˙y =˙z =˙w = 0. Suppose that It is well known that the number and stability of the m > a/b, then system (1) will have five equilibria equilibria play a crucial role on the dynamic behaviour as follows: of the system. The existence of multiple equilibria is an Pramana – J. Phys. (2018) 90:33 Page 3 of 12 33 100 (a) 80 (a) 80 60 60 40 40 20 20 0 y −20 z 0 −40 −20 −60 −80 −40 −100 −60 −150 −100 −50 0 50 100 150 10 12 14 16 18 20 x (b) 100 (b) 5 80 0 60 40 −5 20 z 0 −10 −20 −15 −40 Lyapunov exponents −60 −20 −80 −150 −100 −50 0 50 100 150 −25 5 10 15 x b 100 (c) 5 80 (c) 60 0 40 −5 20 z 0 −10 −20 −15 −40 Lyapunov exponents −60 −20 −80 −100 −50 0 50 100 −25 y 5 10 15 b (d) 250 Figure 2. The bifurcation diagrams (a) and Lyapunov expo- 200 nents from initial values X0 (b), Y0 (c) with the parameter b ∈ (10, 20). 150 w 100 O(0, 0, 0, 0), 50 A±(±26.1128, ±11.6780, ±8.9443, ±17.0470), B±(±27.6159, ±12.3502, ±8.9443, ±19.0658). 0 −100 −50 0 50 100 y It can be verified that all the equilibria O, A±, B± are unstable. The Matlab simulation shows that sys- Figure 1. The four-scroll attractors of system (1) with tem (1) exists as a four-scroll chaotic attractor. The a = 4, b = 20, c = 36, m = 8: (a) x–y;(b) x–z;(c) y–z; phase portraits of system (1) plotted in figure 1 directly w (d) y– . illustrate the attractor with a = 4, b = 20, c = 36, m = 8. By using the Wolf method [40] with step size t = 0.01 and time interval t ∈[0, 500], the Lay- important factor in the generation of multiscroll attractor punov exponents of system (1) with given parameters and multiple coexisting attractors.
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