1 Introduction

Memristor, the fourth circuit element, was postulated in 1971 by L.O.Chua^[1], which represented a relationship between charge and flux.In memristive system, the nonlinear characteristics of memristor is indispensable.It could bring about the chaos.In 2008, researchers of HP labs fabricated the first memristor^[2].Memristors' potential applications span multitudinous fields ranging from modeling neural networks^[3] to applying nano-scale's nonvolatile memories^[4].Since memristor is a basic circuit element, the applications of circuit based on memristor are also a research hot spot^[5], especially in chaotic circuits, for example, Dasgupta et al.^[6] put forward a chaotic circuit based on memristor which was applied in image encryption.

Recently, many physical systems have been found to display fractional-order dynamics, and researches about system with fractional-order have attracted increasing attention.Fractional calculus involves in many fields, for example, in physics^[7], electrical circuit theory^[8], control systems^[9], signal processing ^[10-11], and chemical mixing^[12].Particularly, it plays a great significant role in chaos theory, which means that chaotic phenomena are existent in fractional order dynamic systems.In 1995, Chua system with fractional-order^[13] was firstly presented by Hartley, and he pointed out that when the system's total order was less than 3, the system would exhibit chaotic behavior.Then Lorenz system with fractional-order was studied by Griagorenko et al.^[14], and its chaotic dynamics was also analyzed.In 2010, Petrá š provided memristor-based Chua's circuit^[15] with fractional order.It has also other fractional-order counterparts proposed in Refs.[16-18]. Moreover, the chaotic circuit system with fractional order has also achieved great progress.For instance, Shaik et al.^[19] realized an experimental circuit system which was a simple modulus nonlinearity jerk equation, and they also used the function of a variable control parameters to study the chaotic behavior.Meanwhile, several systems with fractional-order derived from the corresponding integer-order system have been proposed.

In the paper, a quadratic memristor function is used in the chaotic circuit using the flux-controlled memristor^[20] to replace the Chua's circuit's diode.Recalling that applications of traditional Chua's system may be multitudinous, while the analyses of this fractional-order memeristor Chua's system are few.In particular, to demonstrate the chaotic dynamics, the theoretical analysis of its dynamics is described in detail.Meanwhile we study this fractional-order system's complex chaotic behavior by bifurcation diagrams with different bifurcation parameters and the largest Lyapunov exponent diagram.Fraction-order system can generate complex chaotic behavior with lower order than the integer-order system.

The structure of this paper is as follows.In the second section, we first discuss the mathematical preliminaries of fractional derivative and memristor.The dynamic equations of the fractional-order memristive system are given in the third section.Then in the fourth section, we analyze the stability of equilibrium point about the memristive system with fractional-order. In addition, we discuss phase portraits, bifurcation diagram and largest Lyapunov exponent diagram under different order q about the system in Section 5.This paper concludes with a summary of this work in Section 6.

2 Background 2.1 Fractional Calculus

When it comes to fractional calculus, there are many ways to define derivatives presented in Refs.[21-22].In this paper, we use Caputo^[23] to define the fractional derivative:

$ {D_{0,t}}^qw\left( t \right) \buildrel \Delta \over = \frac{1}{{\mathit\Gamma \left( {m - q} \right)}}\int_0^t {\frac{{{w^{(m)}}\left( \tau \right)}}{{{{\left( {t - \tau } \right)}^{q + 1 - m}}}}} {\rm{d}}t $

(1)

where Γ(q) is the gamma function, m-1 < q≤m and m∈ N.It is worthy to note that the Caputo formula (1) is somewhat similar to the Davison-Essex formula reported in Ref.[24].

2.2 Memristor Model

Memristor is a two-terminal element, which is expressed as a function, and the function describes the relationship between the flux (φ) around the terminals and the charge (Q) that passes through the element.We use a flux controlled memristor in this paper, and express it by a function of its incremental memductance W(φ) :

$ W\left( \phi \right) \buildrel \Delta \over = \frac{{{\rm{d}}Q\left( \phi \right)}}{{{\rm{d}}\phi }} $

(2)

The relationship between the voltage v(t) and current i(t) through the memristor can be written as:

$ i\left( t \right) \buildrel \Delta \over = \frac{{{\rm{d}}Q}}{{{\rm{d}}\phi }} = \frac{{{\rm{d}}Q}}{{{\rm{d}}t}}\frac{{{\rm{d}}t}}{{{\rm{d}}\phi }} = W\left( {\phi \left( t \right)} \right)v\left( t \right) $

(3)

In Eq.(3), we simplified $\frac{{{\rm{d}}Q}}{{{\rm{d}}t}}$ by using the chain rule and then used the definition of W(φ) from Eq.(2). The memristor is just a resistor when W(φ)=W(∫v(t))=constan t.The memristive circuit model is shown in Fig. 1.

3 Memristive Chaotic System Based on Chua's Circuit 3.1 Integer-Order Case

The integer-order memristive circuit described in this paper is an autonomous Chua's circuit system presented in Ref.[20], and it simply uses a flux-controlled memristor^[25] instead of Chua's diode.

In Fig. 2, four fundamental circuit elements including resistor R, capacitor C, inductor L, and memristor M, constitute the memristive dynamical circuit.By utilizing KVL around the loop, the dynamics of the circuit based on memristor^[25] are described as:

$ \left\{ \begin{array}{l} \frac{{{\rm{d}}\phi }}{{{\rm{d}}t}} = {v_1}\left( t \right)\\ \frac{{{\rm{d}}{v_1}\left( t \right)}}{{{\rm{d}}t}} = \frac{1}{{{C_1}}}(\frac{{{v_2}\left( t \right)-{v_1}\left( t \right)}}{R}-i\left( t \right))\\ \frac{{{\rm{d}}{v_2}\left( t \right)}}{{{\rm{d}}t}} = \frac{1}{{{C_2}}}(\frac{{{v_1}\left( t \right)-{v_2}\left( t \right)}}{R} - {i_L}\left( t \right))\\ \frac{{{\rm{d}}{i_L}\left( t \right)}}{{{\rm{d}}t}} = \frac{{{v_2}\left( t \right)}}{L} \end{array} \right. $

(4)

And i(t) is defined by:

$ i\left( t \right) = W\left( {\phi \left( t \right)} \right){v_1}\left( t \right) = \frac{{{\rm{d}}Q}}{{{\rm{d}}\phi }}{v_1}\left( t \right) $

(5)

A cubic nonlinear function was chosen to define the Q(φ) :

$ Q\left( \phi \right) = \alpha \phi + \beta {\phi ^3} $

(6)

It is clear that the model of memristor with smooth nonlinearities would be more easy to be fabricated.And chaos can be occurred in chua's circuit based on a cubic nonlinear memristor^[26].Thus, the function of memductance W(φ) is described as:

$ W\left( \phi \right) = \frac{{{\rm{d}}Q}}{{{\rm{d}}\phi }} = \alpha + 3\beta {\phi ^2} $

(7)

Let α=-0.769×10^-3 and β=0.029×10^-3 in Eq.(6), which is similar to value in Ref.[27].From the results in Ref.[20], one can point out that, according to simulations in Fig. 3, the system can cause chaotic behavior.

3.2 Fractional-Order Case

The fractional equations^[28] can be easily deduced from the corresponding integer-order system (4) by making the change as:

$ w \buildrel \Delta \over = \phi, x \buildrel \Delta \over = {v_1}\left( t \right), y \buildrel \Delta \over = {v_2}\left( t \right), z \buildrel \Delta \over = {i_{\rm{L}}}\left( t \right) $

Then:

$ \left\{ \begin{array}{l} {D^q}w = x\\ {D^q}x = \frac{1}{{{C_1}}}\left( {\frac{{y-x}}{R}-W\left( x \right)x} \right)\\ {D^q}y = \frac{1}{{{C_2}}}\left( {\frac{{x-y}}{R} - z} \right)\\ {D^q}z = \frac{y}{L} \end{array} \right. $

(8)

The initial time t₀=0 and order q∈[0, -1]. Current and voltage variables are rescaled as:

$ w \buildrel \Delta \over = \phi, x \buildrel \Delta \over = \frac{{{v_1}}}{\delta }, y \buildrel \Delta \over = \frac{{{v_2}}}{\delta }, z \buildrel \Delta \over = \frac{{{i_{\rm{L}}}}}{\delta } $

By simplifying the equations, we get:

$ \left\{ \begin{array}{l} {D^q}w = \delta x\\ {D^q}x = \frac{1}{{{C_1}}}\left( {\frac{{y-x}}{R}-W\left( w \right)x} \right)\\ {D^q}y = \frac{1}{{{C_2}}}\left( {\frac{{x-y}}{R} - z} \right)\\ {D^q}z = \frac{y}{L} \end{array} \right. $

By using $\delta \buildrel \Delta \over =-\frac{1}{\zeta }$, the result can be written as:

$ \left\{ \begin{array}{l} {D^q}w =-\frac{x}{\zeta }\\ {D^q}x = \frac{1}{{{C_1}}}\left( {\frac{{y-x}}{R}-W\left( w \right)x} \right)\\ {D^q}y = \frac{1}{{{C_2}}}\left( {\frac{{x - y}}{R} - z} \right)\\ {D^q}z = \frac{y}{L} \end{array} \right. $

(9)

4 Stability Analysis of Memristive Chaotic System with Fractional-Order

We discuss the stability analysis about equilibrium point of system (9), and consider a necessary condition which makes the system remain chaos.The vector X =[w x y z]^T, then general nonlinear form of system (9) was given:

$ {D^q}\mathit{\boldsymbol{X}} = f\left( \mathit{\boldsymbol{X}} \right) $

(10)

By solving the following equation, we can calculate the equilibrium point of the system (10) as:

$ {D^q}\mathit{\boldsymbol{x}} = 0 \Leftrightarrow f({\mathit{\boldsymbol{x}}_{{\rm{eq}}}}) = 0 $

Obviously, system (9) has one equilibrium point which is located at x_eq=[0 0 0 0]^T.Based on Lyapunov's indirect method^[29], we can investigate the stability property through considering stability of nonlinear system (9) as the equilibrium point of the linear counterpart.Then the system (10) is linearized as:

$ {D^q}\mathit{\boldsymbol{x}} = \mathit{\boldsymbol{Jx}},(0 < q < 1) $

(11)

where

$ \mathit{\boldsymbol{J}} = {(\frac{{\partial f}}{{\partial x}})_x}_{_{{\rm{eq}}}} = 0 $

is the Jacobian matrix at the equilibrium point.

Now, the following theorem is proposed to explain the stability of linear fractional system (11).Those theorems are stated and proved in Refs.[29-31].

Theorem 1^[28]   If all the eigenvalues λ of the fractional-order system (11)'s Jacobian matrix J satisfy:

$ q\frac{\pi }{2} < \left| {{\rm{arg}}\left( \lambda \right)} \right| $

The linear system (11) is asymptotically stable.

Then, according to the extension of the Lyapunov's direct method, the Theorem 2 is given to better describe the relationship between the stability of the linear fractional system (11) and nonlinear fractional system (10).

Theorem 2^[28]   If the fractional linear system (11) is asymptotically stable at the equilibrium point, then the corresponding fractional nonlinear system (10) is asymptotically stable.

Corollary 1^[29]   If at least one eigenvalue λ of the Jacobian matrix J at the equilibrium point x_eq=0 satisfies $q\frac{\pi }{2} > \left| {{\rm{arg}}\left( \lambda \right)} \right|$ then the fractional nonlinear system (10) is unstable.

The stable and unstable regions are shown in Fig. 4.The dark region is the stable region with $\left| {{\rm{arg}}\left( \lambda \right)} \right| > q\frac{\pi }{2}$ while the light region is the unstable region with $\left| {{\rm{arg}}\left( \lambda \right)} \right| < q\frac{\pi }{2}$.

According to Ref.[32]:if the fractional system (9) is supposed to remain chaotic, a necessary condition is in the unstable region, there is at least one eigenvalue λ.

Based on above-mentioned stability results, fractional-order memristive system (9) has a unique equilibrium point x_eq=0 which is readily shown, and have four eigenvalues:

$ \left\{ \begin{array}{l} {\lambda _1} = 0\\ {\lambda _2} = 3.286{\rm{ }}7\\ {\lambda _3} =-0.783{\rm{ }}0 + 2.343{\rm{ }}4{\rm{i}}\\ {\lambda _4} =-0.783{\rm{ }}0-2.343{\rm{ }}4{\rm{i}} \end{array} \right. $

Here the eigenvalues λ₂ in the unstable region.In Fig. 5, it shows the distribution of Jacobian matrix J's eigenvalue.

5 Numerical Simulations

The parameters of system (9) are set as those in Ref.[20]: ζ=8 200 Ω·47×10^-9 nF, L=18 mH, C₂=68 nF, C₂=6.8 nF.In order to obtain the chaotic attractor, R is set as 2 000 Ω.We perform the simulation and select the initial conditions as:

$ \left[{w\left( 0 \right), x\left( 0 \right), y\left( 0 \right), z\left( 0 \right)} \right] = \left[{0, 0.11, 0.11, 0} \right] $

Fig. 6 and Fig. 7 show the time domain curve and phase plots of a chaotic attractor found with q=0.97.

Since any system containing at least one positive Lyapunov exponent is defined to be chaotic, we considered the Largest Lyapunov Exponent (LLE) λ_max at first.Then the important problem is how to measure the LLE.The more popular algorithm for calculating LLE is Wolf and Jacobian algorithms^[33].However, for calculating LLE of a fractional order system, Jacobian algorithm is not applicable.Therefore, in this paper, Wolf algorithm^[34] is chosen, and the diagram of LLE is plotted in Fig. 8.From the figure, we can see when q>0.93, the system has positive Lyapunov exponents.Namely, the system generates chaos when q>0.93, which is consistent with the result in bifurcation diagram shown in Fig. 9.

We simulate the system with R=2 000, and 0.8≤q≤1, the increment of q is 0.000 5.Bifurcation diagram with bifurcation parameter q is shown in Fig. 9.From the bifurcation diagram, we can see when 0.8 < q < 0.93, a limit cycle which attracts nearby solutions appears, and when q>0.93, the system losses its stability and bifurcation is observed, the corresponding phase diagrams are shown in Fig. 10 and Fig. 11.From Fig. 7 and Fig. 11, we can see a trend that the chaotic behavior in system will be strengthened as increase of q. Bifurcation diagram with q as bifurcation parameter verifies the correctness of these results in Fig. 8.

Now, when we choose q=0.97, the system can be calculated numerically with R∈[1000, 2400].From the bifurcation diagram with bifurcation parameter R in Fig. 12, we can see when R < 1 800 and R>2 100 the system is in a relatively steady state, when R is in the range of [1800,2100], the system exhibits nonlinear or chaotic-like behavior.In order to demonstrate the chaotic-like behavior, we calculate the Lyapunov exponents with different value of R, and the results are listed in Table 1. Considering numerical error in the calculation process, we analyze Lyapunov exponent which is less than 0.01 to be equivalent to zero.When R is set as 1 500 Ω and 2 300 Ω, the system remains stable and trajectories converge as shown in Fig. 13.

6 Conclusions

In this article, dynamic behaviors of memristive system with fractional-order are studied via stability analysis and numerical simulations.The phase diagrams, bifurcation diagrams and LLE show that the fractional-order memristive system exhibits chaos when the system parameters exceed the certain threshold.LLE is calculated by the Wolf algorithm, which was readily used in preview research.The simulation results show the effectiveness of the algorithm, and good agreement with theoretical analysis.