A Memristive Hyperchaotic Multiscroll Jerk System with Controllable Scroll Numbers

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

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 system. 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 complex 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 balance 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 Rössler [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

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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

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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 generates 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). (3) scroll, as shown in Fig. 1(b). Based on double-scroll m=1

(a) (b) Int. J. Bifurcation Chaos 2017.27. Downloaded from www.worldscientific.com by STATE UNIVERSITY OF NEW YORK @ BINGHAMTON on 07/05/17. For personal use only. (c) (d)

Fig. 1. The attractors of jerk system: (a) single-scroll, (b) double-scroll, (c) 7-scroll and (d) 8-scroll.

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According to the deﬁnition of memristor [Chua, 1971], the memristor is a two-terminal nonlinear element with variable resistance. Its nonlinear relationship between the voltage across a ﬂux-controlled memristor and the current into the memristor is given by i = W (ϕ) · v,˙ϕ = v,whereW (ϕ)is a memductance function. Therefore, we can gain the following state equation of the memristive jerk system:   y Cxx˙ = − W (ϕ)y  R1   z C y y ˙ = R 2 (5)   y z x F (x) Czz˙ = − − − +  R3 R4 R5 R6   ϕ˙ = y.

According to [Muthuswamy, 2010] and [Iu et al., 2011], we use a quadric nonlinearity to indi- cate memductance function: W (ϕ)=a +3bϕ2 (6) Fig. 2. Circuit implementation of the jerk system with a memristor. where a and b are two positive constants. We obtain the following dimensionless equations: Let A1 = A2 =0.5andN1 = N2 = 3, the sys-  tem (1) can generate 7-scroll and 8-scroll chaotic x˙ = y − ρW (w)y  attractors, as shown in Figs. 1(c) and 1(d). y˙ = z x y z If we use the variables , and to denote  (7) z˙ = −y − βz − x + F (x) voltages, then the state equations (4) can be easily  achieved with an analog circuit, as shown in Fig. 2. w˙ = y  ρ  y where is a positive parameter indicating the Cxx˙ =  R1 strength of the memristor [Li et al., 2015]. Int. J. Bifurcation Chaos 2017.27. Downloaded from www.worldscientific.com  β . F x F x  z Let =07, ( )= 1( ), we can get different Cyy˙ = (4) even-scroll attractors when we increase ρ from 0 to  R2  1, as shown in Table 1. Let β =0.7, F (x)=F2(x), by STATE UNIVERSITY OF NEW YORK @ BINGHAMTON on 07/05/17. For personal use only.   y z x F x  ( ) we can get different odd-scroll attractors when we Czz˙ = − − − + . ρ R3 R4 R5 R6 increase from0to2,asshowninTable1. According to Table 1, the scroll number of the Let τ = t·RC denote the physical time, where t memristive jerk system proposed in this paper is is the dimensionless time, C is a reference capacitor, controllable and is related to parameter ρ.When and R is a reference resistor, then the parameters F (x)=F1(x), ρ changes from 0 to 1, the mem- of jerk system can be considered as follows: Cx = ristive jerk system can generate 8-scroll, 6-scroll, Cy = Cz = C, R1 = R2 = R3 = R5 = R6 = R, 4-scroll or double-scroll chaotic attractors. Different R/R4 = β. values of ρ correspond to different chaotic attrac- To get a hyperchaotic memristive jerk system tors respectively, which are shown in Fig. 3. When with controllable number of scrolls, we introduce F (x)=F2(x), ρ changes from 0 to 2, the memristive a flux-controlled memristor, as illustrated by W jerk system can generate 7-scroll, 5-scroll, 3-scroll or in Fig. 2, to a three-dimensional multiscroll jerk single-scroll chaotic attractors. Different values of ρ, system. which are taken as 0.1, 0.21, 0.55, 2.0, correspond to

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A Memristive Hyperchaotic Multiscroll Jerk System

Table 1. The scroll number of memristive jerk system with diﬀerent ρ.

Parameter F (x) ρ The Scroll Number

β =0.7, A1,2 =0.5, N1,2 =3 F (x)=F1(x) ρ =0.02 8-scroll ρ =0.15 6-scroll ρ =0.34-scroll ρ =1.0 Double-scroll

F (x)=F2(x) ρ =0.17-scroll ρ =0.21 5-scroll ρ =0.55 3-scroll ρ =2.0 Single-scroll

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Fig. 3. Phase portraits of memristive jerk system with diﬀerent ρ when F (x)=F1(x): (a) ρ =0.02, (b) ρ =0.15, (c) ρ =0.3 and (d) ρ =1.0.

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(a) (b)

Fig. 4. Phase portraits of memristive jerk system with diﬀerent ρ when F (x)=F2(x): (a) ρ =0.1, (b) ρ =0.21, (c) ρ =0.55 and (d) ρ =2.0.

diﬀerent chaotic attractors respectively, which are It means that the volume of the chaotic attractor showninFig.4. decreases by a factor of e−β. Int. J. Bifurcation Chaos 2017.27. Downloaded from www.worldscientific.com 3. System Dynamics Analysis 3.2. Equilibrium point and stability analysis by STATE UNIVERSITY OF NEW YORK @ BINGHAMTON on 07/05/17. For personal use only. 3.1. Dissipativity When introducing a memristor into the jerk system, The dissipativity of system (7) can be described as the number and property of the equilibria will be ∂x ∂y ∂z ∂w greatly changed, which will be analyzed in detail as ∇V ˙ ˙ ˙ ˙ −β. = ∂x + ∂y + ∂z + ∂w = (8) follows. Letx ˙ =˙y =˙z =˙w = 0, we can get  β> y − ρW (w)y =0 When 0,thememristivejerksystemisobvi-   ously dissipative. This implies that asymptotic z =0 motion settles onto a chaotic attractor. Each vol-  (10) −y − βz − x + F (x)=0 ume containing the memristive jerk system tra-  jectory shrinks to zero at an exponential rate as y =0. t →∞.Moreover, When F (x)=F1(x), N1 =3,A1 =0.5, we can dV −β ﬁnd that eight saddle focal equilibria with index 2 of = e . (9) dt jerk system change into the corresponding equilibria

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A Memristive Hyperchaotic Multiscroll Jerk System

sets, Oi = {(x, y, z, w) | x = ±(k − 0.5),y= z = Table 2. Different values of the constant c corresponding to 0,w = c} (k =1, 2, 3, 4), where c can be any real the three nonzero characteristic roots. constant value. We can obtain the Jacobian matrix |c| λ1 λ2,3 Stability on Oi   0√ −0.8196 0.0598 ± 1.0462i Unstable focus 01− ρW (c)00 2 3/3 −0.70.0000 ± 1.0000i Hopf bifurcation   2 −0.3419 −0.1791 ± 0.9195i Stable saddle  0010 √ J  . 60 −0.3500 ± 0.9367i Critical Oi =   (11) −1 −1 −β 0 30.3344 −0.5172 ± 1.0384i Unstable saddle 40.7322 −0.7161 ± 1.2393i Unstable saddle 0100 Its characteristic equation is given by characteristic roots λj (j =1, 2, 3) of the equilib- 3 2 λ · [λ + βλ + λ − ρW (c) + 1] = 0 (12) ria sets Oi are shown in Table 2. F x F x N A . ρ c When ( )= 2( ), 2 =3, 2 =05, we can where and are variable parameters, and the find that seven saddle focal equilibria with index 2 coefficient of cubic polynomial equation shown in of jerk system change into the corresponding equi- the brackets of Eq. (12) are nonzero real con- libria sets, Oi = {(x, y, z, w) | x = ±k, y = z = stants. According to Routh–Hurwitz stable condi- 0,w = c} (k =0, 1, 2, 3), where c can be any real tion, when the relations of (13) are satisfied, all the constant value. The corresponding stability analysis roots of the cubic polynomial equation possess a of the equilibrium sets is consistent with the previ- negative real part. It means that the equilibria sets ous analysis. Oi are stable. −ρW (c)+1> 0 (13) 3.3. Lyapunov exponents and β − (−ρW (c)+1)> 0. bifurcation diagrams ρ c We can get the relationship between and , In order to further explore the nonlinear dynamics   of the memristive jerk system, we investigate the  − β − aρ − aρ  1 < |c| < 1 Lyapunov exponents and bifurcation diagrams. The   3bρ 3bρ Matlab simulation results are described in Figs. 5   and 6. By comparing Fig. 5 with Fig. 6, we can find − β − aρ that the Lyapunov exponents and the bifurcation 1 (14) c1 =  3bρ diagrams match very well. Both of them can show   that the memristive jerk system can generate com-   1 − aρ plex dynamic behaviors. Int. J. Bifurcation Chaos 2017.27. Downloaded from www.worldscientific.com c2 = . F x F x 3bρ When ( )= 1( ), at the beginning, the memristive strength ρ is 0, and chaotic dynamics c < |c| c2 or |c|

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(a) (b)

Fig. 5. First three Lyapunov exponents by adjusting ρ:(a)F (x)=F1(x)and(b)F (x)=F2(x).

When F (x)=F2(x), at the beginning, the 3.4. Coexisting of multiple memristive strength ρ is 0, and chaotic dynamics is attractors relying on the also initiated from the original jerk system, which memristor’s initial condition shows common chaotic behavior. The Lyapunov Coexisting of multiple attractors imply two or more exponents are 0.12, 0.00, 0.00, and −0.82, respectively. With the strength ρ increasing from 0, chaos attractors in a system with the same parame- transits to hyperchaos at ρ ≈ 0.10. A typical hyper- ter but diﬀerent initial conditions. Coexistence of chaos is at ρ ≈ 0.83 and the Lyapunov exponents multiple attractors means that dynamical charac- are 0.08, 0.02, 0, and −0.75, respectively. When teristics of nonlinear systems are sensitive to the the strength ρ keeps increasing from 1.06, periodic changes of initial condition. Under initial conditions orbits begin to emerge until ρ ≈ 1.39 except in of the memristive jerk system (7) changing from some special interval. A typical periodic orbit can be (0, 0.1, 0.1, 0.1) to (0, 0.1, 0.1, 9.5) as Table 3, there observed at ρ ≈ 1.13 and the Lyapunov exponents appear new forms of chaotic attractors as shown in are 0, −0.04, −0.04, and −0.7, respectively. Fur- Figs. 7(a)–7(f). Coexistence of multiple attractors ther simulations show that the hyperchaotic attrac- with a variable number of scrolls can be observed tor appears again at ρ ≈ 1.91 and disappears at in the memristive jerk system (7). It is obvious ρ ≈ 2.25. from Fig. 7(a) that the memristive jerk system (7) Int. J. Bifurcation Chaos 2017.27. Downloaded from www.worldscientific.com by STATE UNIVERSITY OF NEW YORK @ BINGHAMTON on 07/05/17. For personal use only.

(a) (b)

Fig. 6. Bifurcation diagram by adjusting ρ:(a)F (x)=F1(x)and(b)F (x)=F2(x).

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A Memristive Hyperchaotic Multiscroll Jerk System

Table 3. Coexisting attractors for various initial conditions of memristor.

Parameter F (x) ρ Initial Conditions Coexisting Attractors

β =0.7, A1,2 =0.5, N1,2 =3 F (x)=F1(x) ρ =0.055 (0, 0.1, 0.1, 0.1) Fig. 7(a) β =0.7, A1,2 =0.5, N1,2 =3 F (x)=F1(x) ρ =0.055 (0, 0.1, 0.1, 2) Fig. 7(b) β =0.7, A1,2 =0.5, N1,2 =3 F (x)=F1(x) ρ =0.055 (0, 0.1, 0.1, 3) Fig. 7(c) β =0.7, A1,2 =0.5, N1,2 =3 F (x)=F1(x) ρ =0.055 (0, 0.1, 0.1, 4) Fig. 7(d) β =0.7, A1,2 =0.5, N1,2 =3 F (x)=F1(x) ρ =0.055 (0, 0.1, 0.1, 4.7) Fig. 7(e) β =0.7, A1,2 =0.5, N1,2 =3 F (x)=F1(x) ρ =0.055 (0, 0.1, 0.1, 9.5) Fig. 7(f)

(a) 1 (b) 1

0.5 0.5 y y 0 0

-0.5 -0.5

-1 -1 -5 -4 -3 -2 -1 0 1 2 3 4 5 -6 -5 -4 -3 -2 -1 0 1 2 3 4 x x

0.5 0.5 y y 0 0

-0.5 -0.5

-1 -1 -6 -5 -4 -3 -2 -1 0 1 2 3 4 -6 -5 -4 -3 -2 -1 0 1 2 Int. J. Bifurcation Chaos 2017.27. Downloaded from www.worldscientific.com x x

1 1 by STATE UNIVERSITY OF NEW YORK @ BINGHAMTON on 07/05/17. For personal use only. (e) (f)

0.5 0.5 y y 0 0

-0.5 -0.5

-1 -1 -0.5 0 0.5 1 1.5 -0.3 -0.2 -0.1 0 0.1 0.2 x x

Fig. 7. Phase portraits of coexisting attractors in the x–y plane: (a) (0, 0.1, 0.1, 0.1), (b) (0, 0.1, 0.1, 2), (c) (0, 0.1, 0.1, 3), (d) (0, 0.1, 0.1, 4), (e) (0, 0.1, 0.1, 4.7) and (f) (0, 0.1, 0.1, 9.5).

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has 8-scroll chaotic attractors within x varying in we use operational amplifier TL082CP and ana- the interval [−4, 4] for initial (x, y, z, w)=(0, 0.1, log multiplier AD633JN, whose supply voltages are 0.1, 0.1). It has 7-scroll chaotic attractors in taken as VEE = ±15 V. The whole circuit shown Fig. 7(b) within x varying in the interval [−4, 3] in Fig. 8 consists of four modules. Each module for initial (x, y, z, w)=(0, 0.1, 0.1, 2). If the initial is equivalent to realize a dimensionless equation in condition of the memristive jerk are set as (x, y, Eqs. (7). In the first module, two amplifiers U1, z,w)=(0, 0.1, 0.1, 9.5), the x–y phase diagram is U2 and two multipliers G1, G2 are used to realize described in Fig. 7(f), and the memristor jerk sys- the memristive circuit. In the second module, the tem (7) enters the state of torus. amplifier U3 is only used to implement the function of integration. In the third module, the usage of two amplifiers U4 and U5 is to realize the func- 4. Circuit Implementation of the tion of integration and inversion. In the last mod- Memristive Jerk System ule, the amplifier U6 is used to realize the function In order to further validate the memristive jerk sys- of integration and addition. Its negative terminal tem with controllable number of scrolls, we con- is connected to a staircase wave circuit F (x). The struct two analog circuits with classical components amplifier U7 is used to implement the function of as shown in Figs. 8 and 9. In the circuit design, inversion. The circuits of nonlinear function F (x) Int. J. Bifurcation Chaos 2017.27. Downloaded from www.worldscientific.com by STATE UNIVERSITY OF NEW YORK @ BINGHAMTON on 07/05/17. For personal use only.

Fig. 8. The circuit of memristive jerk system.

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Fig. 9. The circuit of nonlinear function F (x). Int. J. Bifurcation Chaos 2017.27. Downloaded from www.worldscientific.com by STATE UNIVERSITY OF NEW YORK @ BINGHAMTON on 07/05/17. For personal use only.

Fig. 10. The circuit experimental prototype of the memristive jerk system.

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including F1(x)andF2(x) are described in Fig. 9. By comparing Eqs. (16) with Eqs. (5), we can get The experimental prototype is shown in Fig. 10. the values of the capacitors and resistors of the cir- Let vx = x, vy = y, vz = z, vw = w,wherevx, cuit in Fig. 2 as follows: vy, vz and vw are the voltages across capacitors. By   setting a time scale factor RC on the dimensionless R1 = R2 = R3 = R5 = R6 = R7  time and adding a multiplication factor 0.1/V on   = R8 = R9 = R10 = R all multiplications with AD633JN, we can rewrite  R (17) Eqs. (7) as follows: R4 =   β    vy 0.1  C · v˙x = − · 10ρ C C C C.  R V x = y = z =    . v Let us take R = 100 kΩ and C =3.3 nF. According  · 0 1 · b · v · v a · y  30 w w + R to the parameters of the memristive jerk system,  V  i.e. β =0.7, we have R1 = R2 = R3 = R5 = R6 = vz R R R R R C · v˙y = 100 kΩ, 4 = 143 kΩ, 7 = 8 = 9 = 10 =  R C C C .  100 kΩ and x = y = z =33nF.Inthememris-  C C R R R  vy βvz vx F (vx) tor, if we take w = , w = , c = 100 kΩ C · v˙z = − − − + V .  R R R R and 0 =01 V, then it is not hard to see that  R R/ ρ V R /R a R /R b  d = 10 , 0 c a = , c b =30. Accord-  vy a . C · v˙w = . ing to the above discussion, when =01V and R b =0.05, we can easily get the values of the two (16) resistors: Ra = 100 kΩ, Rb =66.7kΩ.

(a) (b) Int. J. Bifurcation Chaos 2017.27. Downloaded from www.worldscientific.com

by STATE UNIVERSITY OF NEW YORK @ BINGHAMTON on 07/05/17. For personal use only. (c) (d)

Fig. 11. The circuit experimental results with diﬀerent Rd when F (x)=F1(x): (a) Rd = 500 kΩ, (b) Rd =66.7kΩ, (c) Rd =33.3kΩand(d) Rd =10kΩ.

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A Memristive Hyperchaotic Multiscroll Jerk System

(a) (b)

Fig. 12. The circuit experimental results with diﬀerent Rd when F (x)=F2(x): (a) Rd = 100 kΩ, (b) Rd =47.6kΩ, (c) Rd =18.2kΩand(d) Rd =5kΩ.

Obviously, we can see that system parameter ρ scroll attractors. Although the modiﬁcation keeps is corresponding to circuit parameter Rd.Byonly the dissipativity of the original system, it brings

Int. J. Bifurcation Chaos 2017.27. Downloaded from www.worldscientific.com changing the value of resistance Rd, we can easily the new system abundant complex dynamics, such control the scroll number of the memristive jerk sys- as limit cycles, chaos, and even hyperchaos. Fur- tem. The experimental results are shown in Figs. 11 thermore, coexisting attractors are observed in the by STATE UNIVERSITY OF NEW YORK @ BINGHAMTON on 07/05/17. For personal use only. and 12. We observe that the circuit experimental memristive jerk system. The dynamics of the pro- results of the memristive jerk system are consistent posed chaotic system is investigated, including the with the numerical simulation results. stability of the equilibria, the numerical simulation of bifurcation and Lyapunov exponents. Theoretical analysis, numerical simulation and circuit experi- 5. Conclusions mental results have confirmed the effectiveness of In this paper, we have presented a new hyper- this approach. chaotic memristive jerk system, which is modified from the jerk system. The main feature of this system is that we can get different scroll attrac- Acknowledgments tors by varying the strength of the memristor in This work is supported by the National Natural Sci- this system, which is a significant difference from ence Foundation of China (No. 61571185), the Nat- other chaotic systems reported before, where we ural Science Foundation of Hunan Province, China must change the circuit structure to obtain different (No. 2016JJ2030) and the Open Fund Project

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C. H. Wang et al.

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