See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/312519067 Implementation of a new memristor-based multiscroll hyperchaotic system Article in Pramana · February 2017 DOI: 10.1007/s12043-016-1342-3 CITATIONS READS 6 50 3 authors, including: Chunhua Wang Ling Zhou Hunan University Hunan University 167 PUBLICATIONS 792 CITATIONS 9 PUBLICATIONS 62 CITATIONS SEE PROFILE SEE PROFILE Some of the authors of this publication are also working on these related projects: memristor chaotic system View project RF CMOS receiver View project All content following this page was uploaded by Chunhua Wang on 09 October 2017. The user has requested enhancement of the downloaded file. Pramana – J. Phys. (2017) 88: 34 c Indian Academy of Sciences DOI 10.1007/s12043-016-1342-3 Implementation of a new memristor-based multiscroll hyperchaotic system CHUNHUA WANG, HU XIA∗ and LING ZHOU College of Computer Science and Electronic Engineering, Hunan University, Changsha, 410082, China ∗Corresponding author. E-mail: [email protected] MS received 14 December 2015; revised 23 June 2016; accepted 29 July 2016; published online 17 January 2017 Abstract. In this paper, a new type of flux-controlled memristor model with fifth-order flux polynomials is presented. An equivalent circuit which realizes the action of higher-order flux-controlled memristor is also pro- posed. We use the memristor model to establish a memristor-based four-dimensional (4D) chaotic system, which can generate three-scroll chaotic attractor. By adjusting the system parameters, the proposed chaotic system performs hyperchaos. Phase portraits, Lyapunov exponents, bifurcation diagram, equilibrium points and stabil- ity analysis have been used to research the basic dynamics of this chaotic system. The consistency of circuit implementation and numerical simulation verifies the effectiveness of the system design. Keywords. Memristor; hyperchaos; three-scroll chaotic attractor; circuit implementation. PACS Nos 05.40.Jc; 05.45.Pq 1. Introduction So far, memristor-based chaotic systems are lim- ited to single-scroll and double-scroll, and so chaotic Leon Chua proposed the concept of memristor as dynamics was not complicated. Memristor-based mul- the fourth circuit element in 1971 [1]. It represents a tiscroll chaotic systems have been rarely found. In relationship between the flux and the charge. Memristor 2014, Li [19] presented a memristor-based three-scroll is a two-terminal element with variable resistance called chaotic system. However, Li [19] used the additional memristance which depends on how much electric ordinary nonlinear function to generate a three-scroll charge has been passed through it in a particular attractor. So the circuit is more complicated and the direction. In 2008, researchers in Hewlett–Packard system is not a memristor-based multiscroll chaotic announced that a solid-state implementation of mem- system in the real sense. In 2014, Teng [20] used mem- ristor has been successfully fabricated [2]. Since then, ristor to achieve multiscroll chaotic attractor, without memristor-based chaotic systems gained a lot of atten- the additional ordinary nonlinear function. A fifth-order tion. Researchers mainly focussed on using a memris- generalized charge-controlled memristor is presented tor to substitute Chua’s diode of Chua’s circuit [3–12], in [20] and multiscroll chaotic system is implemented. building various chaotic circuits based on memristor However, only two-scroll and four-scroll chaotic sys- and then completing the analysis of dynamics. In addi- tems are obtained in [20], and then odd-scroll could not tion, Li [13] proposed a memristor oscillator based on be achieved. Moreover, charge-controlled memristor a twin-T network. Through coupling, Corinto [14] real- was used to implement chaotic system in [20], and then ized a chaotic oscillator containing three flux-controlled charge-controlled memristor emulator is complicated memristor. Based on a nonlinear model of HP TiO2 and has difficulty in circuit implementation. Further- memristor, two different memristor-based chaotic cir- more, memristor and memristor-based chaotic system cuits are constructed [15,16]. In the past few years, in [20] are realized using mathematical simulation many researchers devoted themselves to introducing only, without circuit implementation. the memristor to Lü, Chen and Lorenz chaotic systems Based on the above theory, this paper proposes a new [17,18], and then nonlinear systems are further studied type of flux-controlled memristor model with fifth- using topological horseshoe theory. order flux polynomials, and the memristor model is 1 34 Page 2 of 7 Pramana – J. Phys. (2017) 88: 34 used in a four-dimensional (4D) chaotic system, which In this paper, we assume that the definition of mem- can generate chaotic phenomenon. We use higher- ristor is a smooth and nonlinear function, which is degree polynomial memductance function to increase defined as complexity of the chaos. So the chaotic system can = 5 + 3 + + generate a three-scroll chaotic attractor. The memris- q(ϕ) aϕ bϕ cϕ d. (3) tor model is simulated by circuit simulation. On this From eq. (3), memductance W(ϕ) is given by basis, memristor-based chaotic system is also imple- 4 2 mented by circuit simulation. In this paper, we propose W(ϕ) = 5aϕ + 3bϕ + c. (4) the memristor model to establish a memristor-based When 5a = 2227 S/Wb4,3b =−1.087 S/Wb2, four-dimensional multiscroll attractor, while Teng [20] c = 0.135 mS, then W(ϕ)can be given by modelled three-dimensional multiscroll attractor. The rest of this paper is organized as follows: In W(ϕ) = (2227 S/Wb4)ϕ4 − (1.087 S/Wb2)ϕ2 §2, the memristor theory is presented and then a new + (0.135 mS). (5) memristor emulator circuit is proposed. In §3,the chaotic system equations and dynamical characteristics The relationship curve between ϕ and W(ϕ) is shown are described. In §4, the chaotic circuit and multisim in figure 1. simulation results are presented. In §5, the conclusion is presented. 2.2 Memristor emulator circuit 2. Memristor system On the basis of memristor theory, we propose a cir- cuit to emulate the behaviour of the memristor. The 2.1 Memristor model schematic is depicted in figure 2. The schematic is composed of two operational Memristor is a new passive two-terminal circuit ele- amplifiers, four multipliers, one capacitor and four ment in which there is a functional relationship resistances. The first-level operational amplifier U1 is between charge q and magnetic flux ϕ. The memristor used to avoid load effect. The second-level operational is governed by the following relation: amplifier U2, resistance R and capacitor C together = 0 0 v M(q)i form the integral circuit, used to implement integral = , (1) i W(ϕ)v function: where the two nonlinear functions M(q) and W(ϕ), 1 t 1 called the memristance and memductance, respec- va(t) =− v(τ) dτ =− ϕ(t). (6) R0C0 −∞ R0C0 ⎧tively, are defined by ⎪ dq(ϕ) According to the input-to-output function of G1–G4 ⎨⎪ W(ϕ) = dϕ (AD633), vb(t) and vd (t) can be calculated using the . (2) ⎪ dϕ(q) equations ⎩⎪ M(q) = 2 dq vb(t) =−g1g2[va(t)] v(t) , (7) = 2 [ ]4 2500 vd (t) g1g3g4 va(t) v(t). (8) As shown in eqs (7)and(8), coefficients g1, g2, g3 and 2000 g4 are the scaling factors of multipliers G1, G2, G3 and 1500 )/S ϕ W( 1000 500 0 -1 -0.5 0 0.5 1 ϕ/Wb Figure 1. The relationship curve between ϕ and W(ϕ). Figure 2. Schematic for realizing a memristor. Pramana – J. Phys. (2017) 88: 34 Page 3 of 7 34 G4. Therefore, the current going through the emulator of memristors within fourth-order differential equa- can be rewritten as tions aiming at the generation of chaotic attractors. v(t) v (t) v (t) i(t) = + b + d . (9) R R R a b c 3.1 System equations and phase portraits By combining eqs (6)–(9), the flux-controlled mem- ductance can be calculated as The proposed dimensionless chaotic system is as 2 1 g g g g3g4 follows: W ϕ = − 1 2 ϕ2 + 1 ϕ4. ⎧ ( ) 2 4 (10) Ra Rb(R0C0) Rc(R0C0) ⎪ dx ⎪ = α(y + x − W(w)x), Let Ra = 7.4 k, Rb = 92 , Rc = 4.4 , R0 = ⎪ dt ⎪ 100 k, C0 = 100 nF and g1 = g2 = g3 = g4 = ⎪ dy −1 ⎨ = βx + γy − z, 0.1 V . We obtain the flux-controlled memductance dt = (11) corresponding to eq. (5). Here t0 R0C0 is the time ⎪ dz ⎪ = δy − z, that determines the pace of the integration. t0 and ⎪ dt ⎪ f0 provide the natural time and frequency scales for ⎪ dw ⎩ = x, the memristive circuit (the preceding parameters imply dt t0 ∼0.01 s and f0 ∼100 Hz). In order to validate that the memristor model which where x, y, z and w are the states of the chaotic system we propose in this article have real memristor char- and α, β, γ and δ are the parameters. If vx, vy, vz indi- acteristics, we have done a multisim simulation to cate voltages and ϕ indicates the magnetic flux, then obtain diverse i–v characteristics by choosing appro- eq. (12) can be easily achieved with an analog circuit, priate functional forms of vi(t). Figure 3 shows the as shown in figure 4. Therefore, the state equation of i–v curves when v (t) = v sin(2πf t) for f = f , the memristive chaotic circuit is i 0 0 ⎧ f = 4f and f = 10f .Whenf ≤ 10f , a pinched ˙ = + − 0 0 0 ⎪ Cxvx vx/R1 vy/R2 vxW(ϕ), hysteresis loop is found in the i–v curve. By increasing ⎨⎪ Cyv˙y = vx/R3 + vy/R4 − vz/R5, the frequency, the pinched hysteresis loop gradually (12) ⎪ C v˙ = v /R − v /R , contracts.