Implementation of a New Memristor-Based Multiscroll Hyperchaotic System

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Implementation of a new memristor-based multiscroll hyperchaotic system

Article in Pramana · February 2017 DOI: 10.1007/s12043-016-1342-3

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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 proposed. 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 stability 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 memristor 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 circuit 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 memductance 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. When f > 10f0, the pinched hysteresis loop ⎩⎪ z z y 7 z 6 contracts to a single-valued function. Here we cannot ϕ˙ = vx. directly measure current in the multisim simulation. So we convert the current i into the corresponding Further, we obtain ⎧ voltage vp by using AD844. The converting resistor C v˙ =v /R + v /R ⎪ x x x 1 y 2 Rp = 10 k and the converting voltage vp = i · Rp. ⎪ ⎪ 2 So we can conclude that the proposed memrsitor model ⎪ 1 g1g2 2 g1g3g4 4 ⎨ − vx − ϕ + ϕ satisfies the pinched hysteresis characteristic of the R R (R C )2 R (R C )4 a b 0 0 c 0 0 . ⎪ ˙ = + − (13) memrsitor [1,2,21]. ⎪Cyvy vx/R3 vy/R4 vz/R5 ⎪ ⎪C v˙ =v /R − v /R ⎩⎪ z z y 7 z 6 3. Chaotic system ϕ˙ =vx. In order to further study the effectiveness of the pro- Let τ = t ·RC denotes the physical time, where t is the posed memristor model, this section introduces the use dimensionless time, R = 100 k is a reference resistor

Figure 3. The memristor i–v curves for (a) f = f0,(b) f = 4f0 and (c) f = 10f0. 34 Page 4 of 7 Pramana – J. Phys. (2017) 88: 34

10 1

5 0.5

x 0 y 0

-5 -0.5

-10 -1 -4 -2 0 2 4 -4 -2 0 2 4 w w (a) (b)

1 0.5 0.5

z 0 z 0 -0.5

-0.5 -1 1 5 0 -1 0 -4 -2 0 2 4 -1 y -5 w w Figure 4. Circuit implementation of the 4D chaotic system (c) (d) with a memristor. Figure 5. Three-scroll chaotic attractor: (a) w–x plane, (b) w–y plane, (c) w–z plane and (d) w–y–z plane. and C=100 nF is a reference capacitor. Equations (13) can be written as follows: attractor generated by means of MATLAB simulation ⎧ v R is illustrated in figure 5. We can see a three-scroll ⎪ Cxdvx = vxR + y ⎪ chaotic attractor which is different from the attractors ⎪ Cdt R1 R2 ⎪ generated by other memristor-based chaotic systems. ⎪ R Rg g ϕ 2 ⎪ − αv − 1 2 ⎪ x ⎪ αRa αRb R0C0 3.2 Lyapunov exponents and bifurcation diagram ⎪ ⎪ 2 4 ⎪ Rg g3g4 ϕ ⎨ + 1 , The Lyapunov exponents and bifurcation diagram are αRc R0C0 (14) effective approaches for determining whether a system ⎪ ⎪ C dv v R v R v R is chaotic. The rate of separation can be different for ⎪ y y = x + y − z ⎪ , different orientations of the initial separation vector, ⎪ Cdt R3 R4 R5 ⎪ and hence the number of Lyapunov exponents is equal ⎪ Czdvz vy vz ⎪ = − , to the number of dimensions in phase space. For a four- ⎪ Cdt R7 R6 ⎪ dimensional autonomous continuous time system, this ⎩⎪ d ϕ system will have four Lyapunov exponents. A positive = vx. dt RC Lyapunov exponent implies an expanding direction in The parameters of the memristive chaotic circuit can phase space. However, if the sum of Lyapunov expo- be considered as follows: Cx = Cy = Cz = C0 = C, nents is negative, then this system has contracting R/R1 = R/R2 = α, R/R3 = β, R/R4 = γ , R/R7 = volumes in phase space. These two seemingly contra- δ, R/R5 = R/R6 = R/R0 = 1. Let x = vx, y = vy, dictory properties are indications of chaotic behaviour z = vz, w = ϕ/R0C0, then eqs. (14) are equivalent to in a dynamical system. eq. (11). The function W(w) in eq. (11) represents the So we provide the corresponding Lyapunov exponent dimensionless function of the memristor as follows: spectrum of system (11), which is shown in figure 6.As 2 the fourth Lyapunov exponent is too large, we only R Rg g Rg g3g4 W(w) = − 1 2 w2 + 1 w4, (15) show the first three Lyapunov exponents. In the numer- αRa αRb αRc ical simulation, only α varies, while the other param- 2 = = = where R/αRa = 1.8, Rg1g2/αRb = 1.45, Rg g3g4/ eters are fixed at β 0.067, γ 1.033 and δ 1 = αRc = 0.3. 2.1. Notice that for α 7.5, the corresponding Lya- Typical α, β, γ and δ values are: α = 7.5, β = 0.067, punov exponents are: LE1 = 0.06, LE2 = 0.03, γ = 1.033 and δ = 2.1. Choosing the initial conditions LE3 = 0, LE4 =−12. There are two positive Lya- (x0, y0, z0, w0) = (0, 0, 0.01, 0), system (11) has been punov exponents and the sum of the Lyapunov expo- simulated using the above parameters. The chaotic nents is negative, indicating hyperchaotic behaviour. Pramana – J. Phys. (2017) 88: 34 Page 5 of 7 34

0.15 Table 1. Different values of d corresponding to the four characteristic roots. 0.1 dλ1 λ2,3 λ4 0.05 0 −6.0669 0.0499 ± 0.9848i 0 0 0.1 −5.9592 0.0504 ± 0.9839i 0 Lyapunov exponents 0.2 −5.6391 0.0518 ± 0.9807i 0 -0.05 3 4 5 6 7 8 0.5 −3.5152 0.0632 ± 0.9443i 0 α 1 2.8386 −0.0903 ± 1.0750i 0 Figure 6. Lyapunov exponent spectrum. 2 1.8270 −0.1470 ± 1.0628i 0 3 −90.3805 0.0192 ± 1.0301i 0 1

According to table 1, the equilibria set contain a 0.5 number of saddle focal equilibria with index-1 or 2,

X which provides the possibility to generate chaos. 0 4. The chaotic circuit implementation

-0.5 3 4 5 6 7 8 α Modular design method is applied to obtain a chaotic circuit and a chaotic attractor is obtained by multi- Figure 7. Bifurcation diagram. sim simulation. The amplifier TL082 is used in this paper, whose power supply voltage is VCC = 15 V, Figure 7 shows the bifurcation diagram of system (11) VEE =−15 V. The whole circuit shown in figure 8 with the same parameters. consists of four modules. Each module can implement a dimensionless equation in eq. (11). In the first mod- 3.3 Equilibrium points and stability analysis ule, the two amplifiers U1 and U2 are used to realize the function of integration and inversion, respectively. Let x˙ =˙y =˙z =˙w = 0, then we can get the In the second module also, the two amplifiers U3 and equilibrium set of chaotic system (11). U4 are used to realize the function of integration and inversion. In the third module, amplifier U5 is only = = P0 (x0,y0,z0,w0) (0, 0, 0,d). (16) used to implement the function of integration. In the Equation (16) is an equilibrium set of this system. We last module, the variable x will be reversed and applied can see that d is uncertain but constant. We can write to the input of the amplifier U6, realizing the process the Jacobian matrix of this system at the equilibrium of integration, and then the output from U6 would be set P as follows: applied to multiplication, amplification and inversion 0 ⎡ ⎤ operation circuit in order to obtain the variable w2 α(1 − W(d)) α 00 and 0.1w4. The whole circuit of the chaotic system ⎢ ⎥ βγ−10 is shown in figure 8 and the chaotic attractor is shown J(p ) = ⎢ ⎥ . (17) 0 ⎣ 0 δ −10⎦ in figure 9. 1000 The descriptions of U1, U3, U5, U6 in figure 8 are as follows: The system characteristic equation can be written as ⎧ ⎪ 1 1 g2 2 3 + 2 + + = ⎪ vx =− vx − vy − v vx λ(λ a2λ a1λ a0) 0, (18) ⎪ w ⎪ R1C1 R2C1 R4C1 ⎪ g g where ⎪ + 1 4 4 ⎧ ⎪ vwvx dt ⎨ a = α(−1 + W(d)) (δ − γ ) − αβ ⎪ R3C1 0 ⎨ a = α(−1 + W(d)) (1 − γ ) +δ − γ − αβ. (19) 1 1 1 ⎩ 1 vy =− − vx − vy + vz dt. (20) a = α(−1 + W(d)) + 1 − γ ⎪ R C R C R C 2 ⎪ 5 2 6 2 7 2 ⎪ 1 1 When α = 7.5, β = 0.067, γ = 1.033 and δ = 2.1, ⎪ v =− − v + v dt ⎪ z y z we can choose a series of typical values of the constant ⎪ R8C3 R9C3 ⎪ d, and the corresponding characteristic roots are shown ⎩⎪ 1 vw =− − vx dt in table 1. R10C4 34 Page 6 of 7 Pramana – J. Phys. (2017) 88: 34

So we set R1 = 16.7k, R2 = 13.3k, R3 = 444 , R4 = 920 , R5 = 1587.3k, R6 = 96.4k, R8 = 47.6k, R7 = R9 = R10 = 100 k, C1 = C2 = −1 C3 = C4 = 100 nF and g1 = g2 = g3 = g4 = 0.1V , denoting the coefficient factor of the multiplier. The circuit including amplifiers U2, U4 and U8 realizes reverse operation. The arithmetic unit U7 is used to implement amplification. So we can set R11 = R12 = R13 = R14 = R15 = R17 = R18 = 1k and R16 = 10 k. Comparing the attractor shown in figure 9 with that in figure 5, we can conclude that the results of the numerical simulation agree with the results of the circuit experiment.

5. Conclusion

This paper proposes a new memristor-based multiscroll hyperchaotic system, in which we can observe Figure 8. The three-scroll chaotic system circuit. the complex chaotic phenomenon. Higher-order memductance are used in this chaotic system to increase complexity of the chaotic system. So we can generate a three-scroll chaotic attractor. Modular design method is applied to obtain a chaotic circuit, which is easy to implement. The dynamics of the proposed chaotic system is investigated, including the stability of the equilibrium points, the numerical simulations of bifur- (a) (b) cation and Lyapunov exponents. Theoretical analysis, numerical simulation and circuit simulation results have conﬁrmed the effectiveness of this approach.

Acknowledgements This work is supported by the National Natural Science Foundation of China (No. 61571185), the Natural Science Foundation of Hunan Province, China (No. (c) (d) 2016JJ2030) and the Open Fund Project of Key Lab- Figure 9. Chaotic attractor obtained by circuit simulation. oratory in Hunan Universities (No. 15K027). (a) w–x plane, (b) w–y plane, (c) w–z plane and (d) time domain waveform of w. References ⎧Its differential equations: ⎪ 1 1 [1] L O Chua, IEEE Trans. Circuit Theory CT-18(5), 507 (1971) ⎪ v˙ =− v + v ⎪ x R C x R C y [2] D B Strukov, Nature 453, 80 (2008) ⎪ 1 1 2 1 [3] M Itoh, Int. J. Bifurcat. Chaos 18(11), 3183 (2008) ⎪ g2 g1g4 ⎪ + v2 v − v4 v [4] B Muthuswamy, IETE Technical Rev. 26(6), 415 (2009) ⎪ w x w x ⎨⎪ R4C1 R3C1 [5] B Muthuswamy, Int. J. Bifurcat. Chaos 20(5), 1335 (2010) 1 1 1 v˙ = v + v − v [6] B C Bao, Chin. Phys. Lett. 27(7), 070504 (2010) ⎪ y x y z . (21) ⎪ R5C2 R6C2 R7C2 [7] B C Bao, Chin. Phys. B 20(12), 120502 (2011) ⎪ [8] B C Bao, Chin. Phys. B 19(3), 030510 (2010) ⎪ ˙ = 1 − 1 ⎪ vz vy vz [9] B C Bao, Electron. Lett. 46(3), 228 (2010) ⎪ R C R C ⎪ 8 3 9 3 [10] B C Bao, Acta Phys. Sinica 59(6), 3785 (2010) ⎩⎪ 1 v˙w = vx [11] B C Bao, Acta Phys. Sinica 60(12), 120502 (2011) R10C4 [12] B C Bao, Int. J. Bifurcat. Chaos 21(9), 2629 (2011) Pramana – J. Phys. (2017) 88: 34 Page 7 of 7 34

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