Chaotic Circuit Based on two Memristors

Yongbin Yu1, Yongtuo Li1, Ju Jin2, and Wenshu Zhang1

1 School of Computer Science and Engineering, School of Information and Software Engineering,2 School of Life Science and Technology,

University of Electronic Science and Technology of China, Chengdu, ChinaEmail: ybyu@uestc.edu.cn

Abstract—Taking advantage of memristor property, this paper presentschaotic circuit based on two memristors. By numerical simulation, a chaot-ic attractor is observed in the memristive circuit. According to chaoticevidence of Lyapunov exponent and bifurcation, chaotic behavior of thememristive circuit is illustrated.

I. Introduction

TRADITIONALLY, there are only three fundamental circuit

elements of capacitors, resistors, and inductors, discovered

respectively in 1745, 1827, and 1831[1]. In 1971, the memris-

tor was postulated as the fourth circuit element by L.O.Chua[2],

which extended to resistance switching memories[3]. In 2008,

HP labs successfully manufactured the prototypical device of

memristor[4], and established characterization of memristive

switches to enable stateful logic operations[5]. Ever since the

physical fabrication, the reemergence of the missing memristor

as a fundamental circuit element has kicked open new vistas in

multiple frontiers ranging from chaos theory to nanoelectronic-

s and neuromorphic systems. The works of Chua et al. have

brought the study of memristor to the forefront, proving that

chaos can happen frequently in the nonlinear memristive circuit-

s, and memristor-based chaotic circuits can hold very promising

applications especially in secure communication. Current pa-

pers in this area strive to use memristor properties to make chaos

generator. By the use of the second-order properties of mem-

ristor, the memristor connected to a power source can exhibit

chaos[6]. One of the first memristor-based chaotic circuits has

been proposed by Itoh and Chua, which derived memristor os-

cillators from Chuas oscillators by replacing Chuas diodes with

memristor[7]. At the same way, memristive circuits can achieve

chaos by adapting a modified Chua’s circuit[8], [9]. Memris-

tor based chaotic circuits can be built by analog components

and observed experimental chaos[10]. The simplest chaotic cir-

cuit with a linear passive inductor, a linear passive capacitor,

and a nonlinear active memristor is presented[11]. However,

these proposed chaotic circuits do not consider two memristors

or consider it less in the nonlinear electronics. Therefore, this

paper studies the chaotic circuit based on two memristors, and

its rich dynamical behaviors.

This paper is organized as follows. First, the circuit model

based on two memristor is designed. Then, numerical evidence

of chaos such as Lyapunov exponent, bifurcation, and so on,

is presented. Finally, the paper concludes with a remark to the

memristive circuit.

II. Memristive Circuit

The memristor defined as following is a passive two-terminal

electronic element which imposes a relationship between the

charge q and the flux φ. If the constitutive relation can be writ-

ten as φ = φ(q), it is called a charge-controlled memristor, that

is memristance M(q). On the other hand, it is called a flux-

controlled memristor, that is memductance W(φ), if the con-

stitutive relation can be written as q = q(φ). In this paper, we

design memristive circuit using two flux-controlled memristors.

v = M(q)i, M(q) �dφ(q)

dq

i =W(φ)v, W(φ) �dq(φ)

dφ

(1)

According to circuit laws, a general class of memristive systems

can be designed by using the three traditional basic circuit ele-

ments and memristor. Without loss of generality, we can utilize

two memristors, two capacitors, two resistors, and one induc-

tor to design the memristive circuit. Figure 1 shows the model

graphically, where M1 and M2 are two flux-controlled mem-

ristors, two capacitors C1 and C2, one inductor L, two resistorsrand R. Applying circuit laws to memristor, we can obtain a rela-

tion between the two fundamental circuit variables of the charge

and flux. As for capacitor, inductor and resistor, we can use the

voltage and the current to get the relation. When we apply the

Kirchhoff’s circuit laws to the memristive circuit, we can obtain

the following equations.

Fig.1. Memristive circuit

dφ1

dt= v3

dφ2

dt=v4 − v3

RW2 + 1

dv3dt=

1

C1

[(−W1)v3 − W2

RW2 + 1(v3 − v4)]

dv4dt=

1

C2

[W2

RW2 + 1(v3 − v4)+ i5]

di5dt= − 1

Lv4 − r

Li5

(2)

The equations (2) define the relation among the four fundamen-

tal circuit variables of voltage, current, charge, and flux. Let

492978-1-4799-3051-7/13/$31.00 ©2013 IEEE

x = φ1, y = φ2, z = v3, m = v4, n = i5, α = 1C1

, β = 1L , γ = r

L ,

g = R, C2 = 1. When q = q(φ) = φ+ φ3 is designed, memduc-

tance W(φ) is defined as W(φ) � dq(φ)dφ = 1 + 3φ2, and W1 and

W2 in equations (2) are respectively defined as W1 = 1 + 3x2,

W2 = 1+3y2. So, the equations in 2 can be transformed into the

following form.

x = z

y =m− z

g(1+ 3y2)+ 1

z = α[(−1− 3x2)z− 1+ 3y2

g(1+ 3y2)+ 1(z−m)]

m =1+ 3y2

g(1+ 3y2)+ 1(z−m)+ n

n = −mβ− nγ

(3)

The equations (3) are a set of five first-order differential systems,

which describe the dynamics of the memristive circuit shown in

Fig.1.

III. Numerical Evidence of Chaos

Lyapunov exponents, which characterize the rate of separa-

tion of infinitesimally close trajectories in state space, provide

empirical evidence of chaotic behavior. A positive Lyapunov

exponent implies an expanding direction in phase space. Fur-

thermore, the negative sum of Lyapunov exponents indicates

contracting volumes in phase space. These two seemingly con-

tradictory properties are characteristic of chaos in a dynamical

system. On the other hand, a bifurcation is a period doubling,

quadrupling, and so on, that accompanies the onset of chaos in a

dynamical system. So, Lyapunov exponent and bifurcation are

studied to indicate chaotic behavior of the memristive circuit in

this section.

A. Chaotic Attractor

Let α = 8, β2 = 10, γ = 0, g = −12, and initial condition-

s [x, y, z,m, n] = [0, 0, 0, 0.08, 0], chaotic attractor is shown in

Fig.2. In the dynamical memristive circuit, the shape structure

of strange attractor has three scrolls, which is different from the

double-scroll Chua’s attractors.

−50

5

−500

50−1000

0

1000

y(t)z(t)

m(t)

−50

5

−50

5−1000

0

1000

x(t)y(t)

m(t)

−4 −2 0 2 4−40

−20

0

20

40

y(t)

z(t)

−4 −2 0 2 4−4

−2

0

2

4

x(t)

y(t)

Fig.2. Memristive circuit

By computing the values of Lyapunov exponents, the biggest

Lyapunov exponents is positive, and the sum of Lyapunov ex-

ponents is negative. These values of Lyapunov exponents are

indications of chaotic behavior in the memristive circuit.

B. Bifurcation

Bifurcation theory has intensively investigated the topic that

bears on chaotic dynamics. In the memristive circuit, bifurca-

tion is a qualitative change in its dynamics produced by varying

parameters. The bifurcation parameter from Eq.3 is g. When the

circuit parameter makes change of g > 0, plots of bifurcation is

shown in Fig.3. On the other hand, plots of bifurcation corre-

sponding to g < 0 is presented in Fig.4 by numerical simulation.

These figures illustrate the route to chaos, and characterize the

existence of chaotic attractors.

−6 −5.5 −5 −4.5 −4 −3.5 −30

0.5

1

1.5

2

2.5

3x 104

Fig.3. Plots of bifurcation when g > 0

0 2 4 6 8 10−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

Fig.4. Plots of bifurcation when g < 0

C. Lyapunov Exponent Spectrum

By computer simulation, we plot the Lyapunov exponents

spectrum of the memristive circuit. Figure 5 shows result from

the memristive circuit. When the circuit parameter g is choosed,

a plot of the Lyapunov exponent obtained by making simulation

is shown in Fig.5. The number of Lyapunov exponents is same

as the number of dimensions of the memristive circuit. So for

a five-dimensional system described by the equations (3), we

will have five Lyapunov exponents including L1,L2,L3,L4, and

L5. In Fig.5, there exists a positive Lyapunov exponent, and

the sum of Lyapunov exponents is negative. So, these proper-

ties of Lyapunov exponents are to verify chaotic behavior in the

memristive circuit.

0 20 40 60 80 100−5

−4

−3

−2

−1

0

1

2

3

L1L2L3L4L5

L3

L5

L4 L1,L2

Fig.5. Lyapunov exponent spectrum

493

IV. Concluding Remarks

In this paper, chaotic circuit based on two memristors is p-

resented. According to the numerical simulation of Lyapunov

exponent and bifurcation, chaotic behavior of the memristive

circuit is studied. An interesting future direction would be to

further investigate the dynamics and stability analysis of mem-

ristive circuit.

This work is supported by the National Natural Science Foun-

dation of China (NSFC Grant No.61174025 and No.70932005)

and the Fundamental Research Funds for the Central Universi-

ties (Program No.ZYGX2010J073).

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