June 20, 2017 15:16 WSPC/S0218-1274 1750091 International Journal of Bifurcation and Chaos, Vol. 27, No. 6 (2017) 1750091 (15 pages) c World Scientific Publishing Company DOI: 10.1142/S0218127417500912 A Memristive Hyperchaotic Multiscroll Jerk System with Controllable Scroll Numbers Chunhua Wang, Hu Xia∗ and Ling Zhou College of Computer Science and Electronic Engineering, Hunan University, Changsha 410082, P. R. China ∗[email protected] Received November 24, 2016; Revised March 26, 2017 A memristor is the fourth circuit element, which has wide applications in chaos generation. In this paper, a four-dimensional hyperchaotic jerk system based on a memristor is proposed, where the scroll number of the memristive jerk system is controllable. The new system is constructed by introducing one extra flux-controlled memristor into three-dimensional multiscroll jerk sys- tem. We can get different scroll attractors by varying the strength of memristor in this system without changing the circuit structure. Such a method for controlling the scroll number without changing the circuit structure is very important in designing the modern circuits and systems. The new memristive jerk system can exhibit a hyperchaotic attractor, which has more com- plex dynamic behavior. Furthermore, coexisting attractors are observed in the system. Phase portraits, dissipativity, equilibria, Lyapunov exponents and bifurcation diagrams are analyzed. Finally, the circuit implementation is carried out to verify the new system. Keywords: Hyperchaos; multiscroll attractor; memristor; circuit implementation. 1. Introduction they belong to common chaotic systems. On the In recent years, the generation of chaotic attrac- other hand, switch is used to control nonlinear func- tors has become a hot topic in the investigation of tion, which decides the number of saddle focal bal- ance index of 2, so as to adjust the scroll number Int. J. Bifurcation Chaos 2017.27. Downloaded from www.worldscientific.com chaos theory and application. Especially, multiscroll attractors have more complex dynamics than single- [Peng et al., 2014]. The deficiency of this method is scroll and double-scroll chaotic attractors, and the that gaining different scroll attractors will change by STATE UNIVERSITY OF NEW YORK @ BINGHAMTON on 07/05/17. For personal use only. generation of multiscroll chaotic system has been the circuit structure. Moreover, switch is not con- intensively researched (see e.g. [Suykens & Vande- ducive to integration. walle, 1991; Tang et al., 2011; Wang & Liu, 2008; Hyperchaos, which was first introduced by Zhang, 2009; Sanchez-Lopez et al., 2010; Wang R¨ossler [1979], is defined as a chaotic attractor with et al., 2013; Peng et al., 2014]). The multiscroll at least two or more positive Lyapunov exponents. chaotic system can be constructed by expanding Compared with common chaos with only one pos- the number of saddle focal equilibria with index 2. itive Lyapunov exponent, hyperchaos can exhibit However, on the one hand, all the systems (see e.g. multidirectional expansion which leads to more [Suykens & Vandewalle, 1991; Tang et al., 2011; complex and richer dynamical behaviors. Therefore, Wang & Liu, 2008; Zhang, 2009; Sanchez-Lopez hyperchaotic systems have been intensively stud- et al., 2010; Wang et al., 2013; Peng et al., 2014]) ied and are often considered better than common have only one positive Lyapunov exponent, and chaotic systems in the engineering fields, such as ∗ Author for correspondence 1750091-1 June 20, 2017 15:16 WSPC/S0218-1274 1750091 C. H. Wang et al. cryptosystem (see e.g. [Grassi & Mascolo, 1999; into Wien-bridge oscillator, a kind of memristive Wang & Zhang, 2016]), secure communication [Wu Wien-bridge chaotic oscillator was proposed [Yu et al., 2014], neural network (see e.g. [Zhang & Shen, et al., 2014]. A new charge-controlled memristor- 2014; Zhang et al., 2015]), and laser design [Zunino based simplest chaotic circuit was deduced, and the et al., 2011]. circuit has only three basic elements [Chen et al., Hyperchaotic multiscroll attractors which have 2015]. Recently, a new method of generating hyper- greater research value were recently proposed (see chaotic attractor has been proposed. By replacing e.g. [Ahmad, 2006; Liu et al., 2012]). Compared the resistor of Chua’s circuit with a memristor, a with common multiscroll chaotic system, hyper- four-dimensional hyperchaotic memristive system is chaotic multiscroll attractor systems have more obtained [Yang et al., 2014]. However, it can only complex and richer dynamical behaviors. A hyper- generate double-scroll attractor. A new memristor- chaotic multiscroll attractor is constructed by based multiscroll hyperchaotic system is designed, adding multiple breakpoints to nonlinear section various coexisting attractors and hidden coexisting of the system [Ahmad, 2006]. However, the hyper- attractors are observed in this system [Yuan et al., chaotic multiscroll attractors proposed in the liter- 2016]. By utilizing a memristor to substitute a cou- ature [Ahmad, 2006] were generated on the basis pling resistor in the realization circuit of a three- of an existing hyperchaotic system. This method dimensional chaotic system, a novel memristive is not suitable to obtain hyperchaotic multiscroll hyperchaotic system with coexisting infinitely many attractors based on common chaotic systems. By hidden attractors is presented [Bao et al., 2017]. A introducing time delay to the control function, a new type of flux-controlled memristor model with hyperchaotic attractor can be obtained in an exist- five-order flux polynomials is presented, and the ing multiscroll chaotic attractor from Chen system memristor model is used to establish a memristor- [Liu et al., 2012]. Nevertheless, the feedback con- based four-dimensional (4D) chaotic system, which trol function possesses four different parameters. can generate three-scroll chaotic attractor [Wang The method creates trouble during the selection of et al., 2017]. parameters. In this paper, we propose a method to obtain The concept of memristor as the fourth cir- hyperchaotic multiscroll attractor based on a mem- cuit element was first proposed by Leon Chua in ristor. By introducing a flux-controlled memristor 1971 [Chua, 1971]. It represents the relationship to three-dimensional multiscroll jerk system, we between charge and flux. Until 2008, researchers can obtain a hyperchaotic memristive system with in Hewlett-Packard successfully fabricated a solid controllable number of scrolls. Compared with the state implementation of memristor [Strukov et al., methods (see e.g. [Ahmad, 2006; Liu et al., 2012]), 2008]. Since then, memristive chaotic systems have the method proposed in this paper is relatively gained a lot of attention and success (see e.g. [Bao easy to realize, and it can be easily extended to other three-dimensional chaotic systems. Compared Int. J. Bifurcation Chaos 2017.27. Downloaded from www.worldscientific.com et al., 2010a, 2010b, 2010c; Muthuswamy, 2010; Bao et al., 2011; Iu et al., 2011; Wang, 2012; Bus- with [Yang et al., 2014], this present paper can carino, 2012; Li & Zeng, 2013; Yu et al., 2014; achieve four-dimensional hyperchaotic multiscroll by STATE UNIVERSITY OF NEW YORK @ BINGHAMTON on 07/05/17. For personal use only. Chen et al., 2015; Yang et al., 2014]). Researchers attractor, whereas [Yang et al., 2014] realizes four- mainly focused on using a quadratic or cubic dimensional hyperchaotic double-scroll attractor. nonlinear flux-controlled memristor to substitute Compared with common multiscroll chaotic system Chua’s diode of Chua’s circuit (see e.g. [Bao et al., with adjustable scroll [Peng et al., 2014], the mem- 2010a, 2010b, 2010c; Muthuswamy, 2010; Bao et al., ristive jerk system just varies the strength of the 2011]), building types of memristor-based Chua’s memristor to get different scroll attractors, with- chaotic circuits and then completing the analysis of out changing the circuit structure. Moreover, by dynamics. Based on a nonlinear model of HP TiO2 varying the strength of memristor, we can also find memristor, two different memristor-based chaotic rich and complex dynamics, such as limit cycles, circuits are constructed (see e.g. [Wang, 2012; Bus- chaos, and even hyperchaos. Furthermore, coexist- carino, 2012]). In addition, Li et al. proposed a ing attractors are observed in the memristive jerk memristor oscillator based on a twin-T network system. The rest of this paper is organized as fol- [Li & Zeng, 2013]. Through introducing a gen- lows. In Sec. 2, the memristive jerk system and eralized memristor and a LC absorbing network its phase portraits are described. In Sec. 3, the 1750091-2 June 20, 2017 15:16 WSPC/S0218-1274 1750091 A Memristive Hyperchaotic Multiscroll Jerk System dynamical characteristics of memristive jerk system jerk system, we can get multiscroll jerk system by are analyzed. In Sec. 4, circuit implementation of expanding the number of saddle focal equilibria the memristive jerk system with controllable num- with index 2 in the x direction. Let f(x)=F (x)−x, ber of scrolls is presented. In Sec. 5, some conclu- according to the symmetry of nonlinear function, sions are finally drawn. we can expand saddle focal balance with index 2 on both sides of the origin. The nonlinear function generating even-scroll is expressed as follows: 2. The Jerk System Based on aMemristor N1 F1(x)=A1 sgn(x)+A1 sgn(x − 2nA1) The following jerk system proposed by J. C. Sprott n=1 (see e.g. [Sprott, 2000a, 2000b]) is considered N1 x y ˙ = + A1 sgn(x +2mA1). (2) y˙ = z (1) m=1 z˙ = −y − βz + f(x) Nonlinear function generating odd-scroll is expressed as follows: where x, y and z are the state variables, β (β = N 0.45 ∼ 0.7) are system parameter and f(x)is 2 F x A x − n − A nonlinear function which decides the scroll num- 2( )= 2 sgn( (2 1) 2) n=1 ber. When f(x)=|x|−1, the jerk system gen- erates single-scroll, as shown in Fig. 1(a). When N2 f(x)=sgn(x)−x, the jerk system generates double- + A2 sgn(x +(2m − 1)A2).