The Generation, Analysis, and Circuit Implementation of a New Memristor Based Chaotic System

Abstract

We present a new memristor based chaotic circuit, which is generated by replacing the nonlinear resistor in Chua’s circuit with a flux-controlled memristor and a negative conductance. The dynamical behaviors are verified not only by computer simulations but also by Lyapunov exponents, bifurcation analysis, Poincaré mapping, power spectrum analysis, and laboratory experiments.

1. Introduction

The memristor was postulated in 1971 by circuit theorist Chua as a missing nonlinear two-terminal electrical component relating electric charge and magnetic flux linkage [1]. In 2008, a team at HP Labs introduced a model for a memristance function based on thin films of titanium dioxide [2]. From then on, memristor has attracted worldwide interest due to its promising applications in nanoelectronic memories, computer logic, and neuromorphic computer architectures [3].

Chaos has great potential application in many engineering fields, such as image encryption and secure communications. As a nonlinear circuit element, the memristor is very suitable for the design of chaotic circuits and the generation of chaotic systems. Some memristor based chaotic circuits were presented by replacing the Chua diodes with memristors characterized by monotone-increasing piecewise-linear function [4–6]. Several smooth flux-controlled memristors characterized by cubic or piecewise-quadratic nonlinearities were developed in [7–9]. Moreover, attempts have been made on the generation of memristor based hyperchaotic circuits [10, 11].

In this paper, a memristor based chaotic circuit is presented by replacing the nonlinear resistor in Chua’s circuit with a flux-controlled memristor which is characterized by a smooth cubic nonlinear resistor. The rest of this paper is organized as follows. The new chaotic circuit with memristor and the corresponding mathematical model are presented in Section 2. The dynamical behaviors of the new system are analyzed in Section 3. Circuit for the implementation of this chaotic circuit is proposed in Section 4. Finally, some conclusions are drawn in Section 5.

2. Chaotic Circuit with Memristor

The two nonlinear functions 𝑀(𝑞) and 𝑊(𝜙), called the memristance and memductance [6], respectively, are defined by 𝑀𝑞=𝑑𝜙𝑞,𝑊𝜙=𝜙𝑑𝑞𝑑𝑞.𝑑𝜙 (1)(2)

The memristor used in this paper is a flux-controlled memristor, and we assume that the flux-controlled memristor is characterized by a smooth continuous cubic monotone-increasing nonlinearity as follows: 𝑞𝜙=𝑎₀+𝑎₁𝜙+𝑎₂𝜙³, (3) where 𝑎₁<0,  𝑎₂>0.

In view of (2), the memductance is given by 𝑊𝜙=𝑑𝑞𝜙𝑑𝜙=𝑎₁+3𝑎₂𝜙². (4) Thus, by replacing Chua’s diode in Chua’s circuit with an active two-terminal memristive circuit consisting of a negative conductance and a flux-controlled memristor characterized by (3), a smooth memristor oscillator can be designed.

The memristor based chaotic circuit is shown in Figure 1. This circuit consists of four basic circuit elements and has only one negative element in addition to the nonlinearity.

Figure 1 

Chua’s circuit with a flux-controlled memristor.

Applying Kirchhoff’s voltage and current laws and component’s current-voltage relationship to the circuit in Figure 1, we can obtain a system of differential equations governing the dynamical behavior of the circuit in Figure 1, described by 𝐶₁𝑑𝑣₁=𝑣𝑑𝑡²−𝑣₁𝑅+𝐺𝑣₁−𝑊𝜙𝑣₁,𝐶₂𝑑𝑣₂=𝑣𝑑𝑡¹−𝑣₂𝑅𝐿−𝑖,𝑑𝑖𝑑𝑡=𝑣₂+𝑟𝑖,𝑑𝜙𝑑𝑡=𝑣₁, (5) where the 𝜙-𝑞 characteristic curve of the flux-controlled memristor is described by (3) and (4).

Equation (5) can be transformed into the dimensionless form as 𝑑𝑥(𝑤)𝑥,𝑑𝑡=𝛼𝑦−𝑥+𝛾𝑥−𝑊𝑑𝑦𝑑𝑡=𝑥−𝑦+𝑧,𝑑𝑧𝑑𝑡=−𝛽𝑦+𝑘𝑧,𝑑𝑤𝑑𝑡=𝑥 (6) by letting 𝑥=𝑣₁,  𝑦=𝑣₂,  𝑧=−𝑖,  𝑤=𝜃,  𝛼=1/𝐶₁,  𝛽=1/𝐿,  𝛾=𝐺,  𝑘=𝑟/𝐿,  𝑅=1,  𝐶₂=1 and defining the nonlinear functions 𝑞(𝑤) and 𝑊(𝑤) as 𝑞(𝑤)=𝑎₀+𝑎₁𝑤+𝑎₂𝑤³,𝑊(𝑤)=𝑑𝑞(𝑤)𝑑𝑤=𝑎₁+3𝑎₂𝑤². (7)

Take 𝛼=4,  𝛽=12,  𝑘=0.688,  𝛾=1.5,  𝑎₁=−0.67×10⁻³,  𝑎₂=−0.02×10⁻³, and 𝑘=0.688. For initial conditions (1,1,1,1), the system (6) is chaotic with Lyapunov exponents 𝜆₁=0.21383,  𝜆₂=0,  𝜆₃=−0.0002, and 𝜆₄=−1.5259,   and the Lyapunov dimension is 𝑑_𝐿=3.14; the chaotic attractors are shown in Figure 2.

(a) 𝑥-𝑦

(b) 𝑥-𝑧

(c) 𝑦-𝑧

(d) 𝑥-𝑤

(e) 𝑦-𝑤

(f) 𝑧-𝑤

(g) 𝑥-𝑦-𝑧

(h) 𝑥-𝑦-𝑤

Figure 2 

Phase portraits of the chaotic system (6).

3. Numerical Analysis

Similar to dynamical analysis of general chaotic circuits, by using the conventional dynamical analysis methods, such as bifurcation diagram, Lyapunov exponent spectra, the dynamical behaviors of the chaotic system (6) are studied when the system parameters are varied.

3.1. Bifurcation Analysis

Holding the values of 𝛼=4,  𝛽=12,  𝛾=1.5,  𝑎₁=−0.67×10⁻³,  𝑎₂=−0.02×10⁻³ fixed and changing the value of 𝑘, the spectra of Lyapunov exponents on the (𝑘-𝜆) plane are obtained as shown in Figure 3, and the corresponding bifurcation diagram is given in Figure 4.

Figure 3 

The Lyapunov exponents in the (𝑘-𝜆) plane.

Figure 4 

Bifurcation diagram for increasing 𝑘.

It can be seen from Figures 4 and 5 that the bifurcation diagram is consistent well with the Lyapunov exponent spectrum. From Figure 5, we observe that system (6) undergoes the following routes as 𝑘 increases.(1)When 0.2<𝑘≤0.321,  𝜆₁=0,  𝜆₂,𝜆₃,𝜆₄<0, system (6) is periodic (Figure 5(a)).(2)When 0.322<𝑘≤0.508,  𝜆₁=𝜆₂=0,  𝜆₃,𝜆₄<0, system (6) is quasi-periodic (Figure 5(b)).(3)When 0.509<𝑘≤0.631, 𝜆₁>0,  𝜆₂=0,  𝜆₃,𝜆₄<0, system (6) is chaotic (Figure 5(c)).(4)When 0.632<𝑘≤0.638,  𝜆₁=𝜆₂=0,  𝜆₃,𝜆₄<0, system (6) is quasi-periodic (Figure 5(d)).(5)When 0.639<𝑘≤0.66,𝜆₁>0,  𝜆₂=0,  𝜆₃,𝜆₄<0, system (6) is chaotic (Figure 5(e)).(6)When 0.661<𝑘≤0.682,  𝜆₁=𝜆₂=0,  𝜆₃,𝜆₄<0, system (6) is quasi-periodic (Figure 5(f)).(7)When 0.683<𝑘≤0.7,𝜆₁>0,  𝜆₂=0,  𝜆₃,𝜆₄<0, system (6) is chaotic (Figure 5(g)).

Figure 5 

Phase portraits of system (6) in (i) 𝑦-𝑤 plane and (ii) 𝑧-𝑤 plane with different 𝑘. (a) Period orbit with 𝑘=0.321; (b) quasi-period orbit with 𝑘=0.486; (c) chaotic attractor with 𝑘=0.569; (d) quasi-period orbit with 𝑘=0.635; (e) chaotic attractor with 𝑘=0.659; (f) quasi-period orbit with 𝑘=0.675; (g) chaotic attractor with 𝑘=0.688.

3.2. Poincaré Maps and Power Spectrum Analysis

Poincaré mapping is shown in Figure 6, and system parameters are selected as 𝛼=4,  𝛽=12, 𝑎₁=−0.67×10⁻³,  𝑎₂=0.02×10⁻³,   and 0.2<𝑘≤0.7.

(a)

(b)

(c)

Figure 6 

Poincaré mapping of system (6) with different 𝑘: (a) period orbit with 𝑘=0.321; (b) quasi-period orbit with 𝑘=0.486; (c) chaotic attractor with 𝑘=0.688.

The power spectrum of system (6) with different 𝑘 is displayed in Figure 7.

(a)

(b)

(c)

Figure 7 

Power spectrum of system (6) with different 𝑘: (a) period orbit with 𝑘=0.321; (b) quasi-period orbit with 𝑘=0.486; (c) chaotic attractor with 𝑘=0.688.

4. Circuit Design for Chaotic System

In this section, we present experimental confirmation of the above numerical results through Multisim modeling. The circuit is designed as in Figure 8.

Figure 8 

The chaotic circuit.

The two-terminal memristor consists of five Op-Amps TL084, two analog multipliers AD633, thirteen resistors, and three capacitances. The parameter values of the circuit are 𝑅₁,𝑅₂,  𝑅₇,  𝑅₈,  𝑅₁₁,  𝑅₁₂=1kΩ; 𝑅₃=12.5kΩ; 𝑅₄=2.5kΩ; 𝑅₅=36kΩ; 𝑅₆=1.5kΩ; 𝑅₉=3.1kΩ; 𝑅₁₀=0.5kΩ; 𝐶₁=10nF; 𝐶₂=100nF,  𝐶₃=47nF. The power supplies are 𝑉_𝑝=+15 V and 𝑉_𝑛=−15 V. For different parameter R13, the attractors are observed as in Figure 9.

Figure 9 

Phase portraits of system (6) with different 𝑅13, measured from the experimental setup: (a) period orbit, 𝑅13=0.305kΩ; (b) quasi-period orbit, 𝑅13=0.329kΩ; (c) chaotic attractor, 𝑅13=0.527kΩ; (d) quasi-period orbit, 𝑅13=0.612kΩ; (e) chaotic attractor, 𝑅13=0.639 kΩ; (f) quasi-period orbit, 𝑅13=0.667 kΩ; (g) chaotic attractor, 𝑅13=0.689kΩ.

5. Conclusions

In this paper, we present a memristor chaotic circuit, which is derived from Chua’s oscillator by replacing Chua’s diode with a charge-controlled memristor. This chaotic circuit uses only four basic circuit elements and has only one negative element in addition to the nonlinearity. The resulting chaotic system is demonstrated by computer simulations and verified by Lyapunov exponents, bifurcation, poincaré mapping, and power spectrum analysis.

Acknowledgment

This research was supported by the National Natural Science Foundation of China (Grant no. 60971022 and no. 61004078).