Pramana J. Phys. (7) 88: 3 DOI.7/s3-6-3-3 c Indian Academy of Sciences Implementation of a ne memristor-based multiscroll hyperchaotic system CHUNHUA WANG, HU XIA and LING ZHOU College of Computer Science and Electronic Engineering, Hunan University, Changsha, 8, China Corresponding author. E-mail: 8537@qq.com MS received December 5; revised 3 June 6; accepted 9 July 6; published online 7 January 7 Abstract. In this paper, a ne type of flux-controlled memristor model ith fifth-order flux polynomials is presented. An equivalent circuit hich realizes the action of higher-order flux-controlled memristor is also proposed. We use the memristor model to establish a memristor-based four-dimensional (D) chaotic system, hich 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. Keyords. PACS Nos Memristor; hyperchaos; three-scroll chaotic attractor; circuit implementation. 5..Jc; 5.5.Pq. Introduction Leon Chua proposed the concept of memristor as the fourth circuit element in 97 [. It represents a relationship beteen the flux and the charge. Memristor is a to-terminal element ith variable resistance called memristance hich depends on ho much electric charge has been passed through it in a particular direction. In 8, researchers in Helett Packard announced that a solid-state implementation of memristor has been successfully fabricated [. Since then, memristor-based chaotic systems gained a lot of attention. Researchers mainly focussed on using a memristor to substitute Chua s diode of Chua s circuit [3, building various chaotic circuits based on memristor and then completing the analysis of dynamics. In addition, Li [3 proposed a memristor oscillator based on a tin-t netork. Through coupling, Corinto [ realized a chaotic oscillator containing three flux-controlled memristor. Based on a nonlinear model of HP TiO memristor, to different memristor-based chaotic circuits are constructed [5,6. In the past fe years, many researchers devoted themselves to introducing the memristor to Lü, Chen and Lorenz chaotic systems [7,8, and then nonlinear systems are further studied using topological horseshoe theory. So far, memristor-based chaotic systems are limited to single-scroll and double-scroll, and so chaotic dynamics as not complicated. Memristor-based multiscroll chaotic systems have been rarely found. In, Li [9 presented a memristor-based three-scroll chaotic system. Hoever, Li [9 used the additional ordinary nonlinear function to generate a three-scroll attractor. So the circuit is more complicated and the system is not a memristor-based multiscroll chaotic system in the real sense. In, Teng [ used memristor to achieve multiscroll chaotic attractor, ithout the additional ordinary nonlinear function. A fifth-order generalized charge-controlled memristor is presented in [ and multiscroll chaotic system is implemented. Hoever, only to-scroll and four-scroll chaotic systems are obtained in [, and then odd-scroll could not be achieved. Moreover, charge-controlled memristor as used to implement chaotic system in [, and then charge-controlled memristor emulator is complicated and has difficulty in circuit implementation. Furthermore, memristor and memristor-based chaotic system in [ are realized using mathematical simulation only, ithout circuit implementation. Based on the above theory, this paper proposes a ne type of flux-controlled memristor model ith fifthorder flux polynomials, and the memristor model is
3 Page of 7 Pramana J. Phys. (7) 88: 3 used in a four-dimensional (D) chaotic system, hich can generate chaotic phenomenon. We use higherdegree polynomial memductance function to increase complexity of the chaos. So the chaotic system can generate a three-scroll chaotic attractor. The memristor model is simulated by circuit simulation. On this basis, memristor-based chaotic system is also implemented by circuit simulation. In this paper, e propose the memristor model to establish a memristor-based four-dimensional multiscroll attractor, hile Teng [ modelled three-dimensional multiscroll attractor. The rest of this paper is organized as follos: In, the memristor theory is presented and then a ne memristor emulator circuit is proposed. In 3, the chaotic system equations and dynamical characteristics are described. In, the chaotic circuit and multisim simulation results are presented. In 5, the conclusion is presented.. Memristor system. Memristor model Memristor is a ne passive to-terminal circuit element in hich there is a functional relationship beteen charge q and magnetic flux ϕ. The memristor is governed by the folloing relation: { v = M(q)i i = W(ϕ)v, () here the to nonlinear functions M(q) and W(ϕ), called the memristance and memductance, respectively, are defined by W(ϕ) = dq(ϕ) dϕ M(q) = dϕ(q). () dq 5 In this paper, e assume that the definition of memristor is a smooth and nonlinear function, hich is defined as q(ϕ) = aϕ 5 + bϕ 3 + cϕ + d. (3) From eq. (3), memductance W(ϕ) is given by W(ϕ) = 5aϕ + 3bϕ + c. () When 5a = 7 S/Wb, 3b =.87 S/Wb, c =.35 ms, then W(ϕ) can be given by W(ϕ) = (7 S/Wb )ϕ (.87 S/Wb )ϕ + (.35 ms). (5) The relationship curve beteen ϕ and W(ϕ) is shon in figure.. Memristor emulator circuit On the basis of memristor theory, e propose a circuit to emulate the behaviour of the memristor. The schematic is depicted in figure. The schematic is composed of to operational amplifiers, four multipliers, one capacitor and four resistances. The first-level operational amplifier U is used to avoid load effect. The second-level operational amplifier U, resistance R and capacitor C together form the integral circuit, used to implement integral function: v a (t) = t v(τ) dτ = ϕ(t). (6) R C R C According to the input-to-output function of G G (AD633), v b (t) and v d (t) can be calculated using the equations v b (t) =g g [v a (t) v(t), (7) v d (t) = g g 3g [v a (t) v(t). (8) As shon in eqs (7) and (8), coefficients g, g, g 3 and g are the scaling factors of multipliers G, G, G3 and W(ϕ)/S 5 5 - -.5.5 ϕ/wb Figure. The relationship curve beteen ϕ and W(ϕ). Figure. Schematic for realizing a memristor.
Pramana J. Phys. (7) 88: 3 Page 3 of 7 3 G. Therefore, the current going through the emulator can be reritten as i(t) = v(t) + v b(t) + v d(t). (9) R a R b R c By combining eqs (6) (9), the flux-controlled memductance can be calculated as W(ϕ) = g g R a R b (R C ) ϕ + g g 3g R c (R C ) ϕ. () Let R a = 7. k, R b = 9, R c =., R = k, C = nf and g = g = g 3 = g =. V. We obtain the flux-controlled memductance corresponding to eq. (5). Here t = R C is the time that determines the pace of the integration. t and f provide the natural time and frequency scales for the memristive circuit (the preceding parameters imply t. s and f Hz). In order to validate that the memristor model hich e propose in this article have real memristor characteristics, e have done a multisim simulation to obtain diverse i v characteristics by choosing appropriate functional forms of v i (t). Figure 3 shos the i v curves hen v i (t) = v sin(πf t) for f = f, f = f and f = f.whenf f, a pinched hysteresis loop is found in the i v curve. By increasing the frequency, the pinched hysteresis loop gradually contracts. When f > f, the pinched hysteresis loop contracts to a single-valued function. Here e cannot directly measure current in the multisim simulation. So e convert the current i into the corresponding voltage v p by using AD8. The converting resistor R p = k and the converting voltage v p = i R p. So e can conclude that the proposed memrsitor model satisfies the pinched hysteresis characteristic of the memrsitor [,,. 3. Chaotic system In order to further study the effectiveness of the proposed memristor model, this section introduces the use of memristors ithin fourth-order differential equations aiming at the generation of chaotic attractors. 3. System equations and phase portraits The proposed dimensionless chaotic system is as follos: dx = α(y + x W()x), dy = βx + γy z, () dz = δy z, d = x, here x, y, z and are the states of the chaotic system and α, β, γ and δ are the parameters. If v x, v y, v z indicate voltages and ϕ indicates the magnetic flux, then eq. () can be easily achieved ith an analog circuit, as shon in figure. Therefore, the state equation of the memristive chaotic circuit is C x v x = v x /R + v y /R v x W(ϕ), C y v y = v x /R 3 + v y /R v z /R 5, () C z v z = v y /R 7 v z /R 6, ϕ = v x. Further, e obtain C x v x =v x /R + v y /R [ v x g g R a R b (R C ) ϕ + g g 3g R c (R C ) ϕ C y v y =v x /R 3 + v y /R v z /R 5 C z v z =v y /R 7 v z /R 6 ϕ =v x.. (3) Let τ = t RC denotes the physical time, here t is the dimensionless time, R = k is a reference resistor Figure 3. The memristor i v curves for (a) f = f,(b) f = f and (c) f = f.
Page of 7 Pramana J. Phys. (7) 88: 3 5.5 y x 3-5 - - -.5 - - - - (a) (b).5 z z.5 -.5 -.5-5 - - Figure. Circuit implementation of the D chaotic system ith a memristor. and C= nf is a reference capacitor. Equations (3) can be ritten as follos: C dv vx R v y R x x = + C R R R ϕ Rg g αvx αra αrb R C Rg g3 g ϕ +, αrc R C () Cy dvy v x R vy R vz R = +, C R3 R R5 vy Cz dvz vz =, C R7 R6 ϕ d = vx. RC The parameters of the memristive chaotic circuit can be considered as follos: Cx = Cy = Cz = C = C, R/R = R/R = α, R/R3 = β, R/R = γ, R/R7 = δ, R/R5 = R/R6 = R/R =. Let x = vx, y = vy, z = vz, = ϕ/r C, then eqs. () are equivalent to eq. (). The function W () in eq. () represents the dimensionless function of the memristor as follos: W () = Rg g Rg g3 g R +, αra αrb αrc (5) here R/αRa =.8, Rg g /αrb =.5, Rg g3 g / αrc =.3. Typical α, β, γ and δ values are: α = 7.5, β =.67, γ =.33 and δ =.. Choosing the initial conditions (x, y, z, ) = (,,., ), system () has been simulated using the above parameters. The chaotic - (c) y - -5 (d) Figure 5. Three-scroll chaotic attractor: (a) x plane, (b) y plane, (c) z plane and (d) y z plane. attractor generated by means of MATLAB simulation is illustrated in figure 5. We can see a three-scroll chaotic attractor hich is different from the attractors generated by other memristor-based chaotic systems. 3. Lyapunov exponents and bifurcation diagram The Lyapunov exponents and bifurcation diagram are effective approaches for determining hether a system is chaotic. The rate of separation can be different for different orientations of the initial separation vector, and hence the number of Lyapunov exponents is equal to the number of dimensions in phase space. For a fourdimensional autonomous continuous time system, this system ill have four Lyapunov exponents. A positive Lyapunov exponent implies an expanding direction in phase space. Hoever, if the sum of Lyapunov exponents is negative, then this system has contracting volumes in phase space. These to seemingly contradictory properties are indications of chaotic behaviour in a dynamical system. So e provide the corresponding Lyapunov exponent spectrum of system (), hich is shon in figure 6. As the fourth Lyapunov exponent is too large, e only sho the first three Lyapunov exponents. In the numerical simulation, only α varies, hile the other parameters are fixed at β =.67, γ =.33 and δ =.. Notice that for α = 7.5, the corresponding Lyapunov exponents are: LE =.6, LE =.3, LE3 =, LE =. There are to positive Lyapunov exponents and the sum of the Lyapunov exponents is negative, indicating hyperchaotic behaviour.
Pramana J. Phys. (7) 88: 3 Page 5 of 7 3 Lyapunov exponents.5..5 -.5 3 5 6 7 8 α Figure 6. Lyapunov exponent spectrum. X.5 -.5 3 5 6 7 8 α Figure 7. Bifurcation diagram. Figure 7 shos the bifurcation diagram of system () ith the same parameters. 3.3 Equilibrium points and stability analysis Let ẋ = ẏ = ż = ẇ =, then e can get the equilibrium set of chaotic system (). P = (x,y,z, ) = (,,,d). (6) Equation (6) is an equilibrium set of this system. We can see that d is uncertain but constant. We can rite the Jacobian matrix of this system at the equilibrium set P as follos: α( W(d)) α J(p ) = β γ δ. (7) The system characteristic equation can be ritten as λ(λ 3 + a λ + a λ + a ) =, (8) here a = α( + W(d)) (δ γ ) αβ a = α( + W(d)) ( γ ) +δ γ αβ. (9) a = α( + W(d)) + γ When α = 7.5, β =.67, γ =.33 and δ =., e can choose a series of typical values of the constant d, and the corresponding characteristic roots are shon in table. Table. Different values of d corresponding to the four characteristic roots. d λ λ,3 λ 6.669.99 ±.988i. 5.959.5 ±.9839i. 5.639.58 ±.987i.5 3.55.63 ±.93i.8386.93 ±.75i.87.7 ±.68i 3 9.385.9 ±.3i According to table, the equilibria set contain a number of saddle focal equilibria ith index- or, hich provides the possibility to generate chaos.. The chaotic circuit implementation Modular design method is applied to obtain a chaotic circuit and a chaotic attractor is obtained by multisim simulation. The amplifier TL8 is used in this paper, hose poer supply voltage is V CC = 5 V, V EE =5 V. The hole circuit shon in figure 8 consists of four modules. Each module can implement a dimensionless equation in eq. (). In the first module, the to amplifiers U and U are used to realize the function of integration and inversion, respectively. In the second module also, the to amplifiers U3 and U are used to realize the function of integration and inversion. In the third module, amplifier U5 is only used to implement the function of integration. In the last module, the variable x ill be reversed and applied to the input of the amplifier U6, realizing the process of integration, and then the output from U6 ould be applied to multiplication, amplification and inversion operation circuit in order to obtain the variable and.. The hole circuit of the chaotic system is shon in figure 8 and the chaotic attractor is shon in figure 9. The descriptions of U, U3, U5, U6 in figure 8 are as follos: v x = [ R C v x + g g R 3 C v v x v y g v R C R C v x v y = [ R 5 C v x R 6 C v y + v z = [ R 8 C 3 v y + v = [ R C v x R 9 C 3 v z v z. () R 7 C
3 Page 6 of 7 Pramana J. Phys. (7) 88: 3 So e set R = 6.7 k, R = 3.3 k, R 3 =, R = 9, R 5 = 587.3 k, R 6 = 96. k, R 8 = 7.6k, R 7 = R 9 = R = k, C = C = C 3 = C = nf and g = g = g 3 = g =.V, denoting the coefficient factor of the multiplier. The circuit including amplifiers U, U and U8 realizes reverse operation. The arithmetic unit U7 is used to implement amplification. So e can set R = R = R 3 = R = R 5 = R 7 = R 8 = k and R 6 = k. Comparing the attractor shon in figure 9 ith that in figure 5, e can conclude that the results of the numerical simulation agree ith the results of the circuit experiment. 5. Conclusion Figure 8. The three-scroll chaotic system circuit. (a) (b) This paper proposes a ne memristor-based multiscroll hyperchaotic system, in hich e can observe the complex chaotic phenomenon. Higher-order memductance are used in this chaotic system to increase complexity of the chaotic system. So e can generate a three-scroll chaotic attractor. Modular design method is applied to obtain a chaotic circuit, hich is easy to implement. The dynamics of the proposed chaotic system is investigated, including the stability of the equilibrium points, the numerical simulations of bifurcation and Lyapunov exponents. Theoretical analysis, numerical simulation and circuit simulation results have confirmed the effectiveness of this approach. Acknoledgements (c) Figure 9. Chaotic attractor obtained by circuit simulation. (a) x plane, (b) y plane, (c) z plane and (d) time domain aveform of. Its differential equations: v x = v x + v y R C R C + g v R C v x g g v R 3 C v x v y = v x + v y v z. () R 5 C R 6 C R 7 C v z = v y v z R 8 C 3 R 9 C 3 v = v x R C (d) This ork is supported by the National Natural Science Foundation of China (No. 65785), the Natural Science Foundation of Hunan Province, China (No. 6JJ3) and the Open Fund Project of Key Laboratory in Hunan Universities (No. 5K7). References [ L O Chua, IEEE Trans. Circuit Theory CT-8(5), 57 (97) [ D B Strukov, Nature 53, 8 (8) [3 M Itoh, Int. J. Bifurcat. Chaos 8(), 383 (8) [ B Muthusamy, IETE Technical Rev. 6(6), 5 (9) [5 B Muthusamy, Int. J. Bifurcat. Chaos (5), 335 () [6 B C Bao, Chin. Phys. Lett. 7(7), 75 () [7 B C Bao, Chin. Phys. B (), 5 () [8 B C Bao, Chin. Phys. B 9(3), 35 () [9 B C Bao, Electron. Lett. 6(3), 8 () [ B C Bao, Acta Phys. Sinica 59(6), 3785 () [ B C Bao, Acta Phys. Sinica 6(), 5 () [ B C Bao, Int. J. Bifurcat. Chaos (9), 69 ()
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