Complexity

Volume 2018, Article ID 4970152, 7 pages

https://doi.org/10.1155/2018/4970152

## An Integer-Order Memristive System with Two- to Four-Scroll Chaotic Attractors and Its Fractional-Order Version with a Coexisting Chaotic Attractor

^{1}Center of System Theory and Its Applications, Chongqing University of Posts and Telecommunications, Chongqing 400065, China^{2}Key Laboratory of Network Control and Intelligent Instrument of Ministry of Education, Chongqing University of Posts and Telecommunications, Chongqing 400065, China

Correspondence should be addressed to Ping Zhou; nc.ude.tpuqc@gnipuohz

Received 19 April 2018; Accepted 28 June 2018; Published 30 July 2018

Academic Editor: Viet-Thanh Pham

Copyright © 2018 Ping Zhou and Meihua Ke. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

First, based on a linear passive capacitor , a linear passive inductor , an active-charge-controlled memristor, and a fourth-degree polynomial function determined by charge, an integer-order memristive system is suggested. The proposed integer-order memristive system can generate two-scroll, three-scroll, and four-scroll chaotic attractors. The complex dynamics behaviors are investigated numerically. The Lyapunov exponent spectrum with respect to linear passive inductor and the two-scroll, three-scroll, and four-scroll chaotic attractors are yielded by numerical calculation. Second, based on the integer-order memristive chaotic system with a four-scroll attractor, a fractional-order version memristive system is suggested. The complex dynamics behaviors of its fractional-order version are studied numerically. The largest Lyapunov exponent spectrum with respect to fractional-order is yielded. The coexisting two kinds of three-scroll chaotic attractors and the coexisting three-scroll and four-scroll chaotic attractors can be found in its fractional-order version.

#### 1. Introduction

Chaos is an interesting phenomenon in nonlinear systems. High irregularity, unpredictability, and complexity are the typical characteristics of chaotic systems [1, 2]. These typical characteristics have great applications in the following fields: data encryption [3], secure communication [4–7], power grid protection [8, 9], and so on [10–16]. Therefore, more and more attentions have been attracted on the study of chaotic systems in the last few decades [17–20]. In 1971, Chua reported the fourth circuit element named memristor [21], and a solid-state implementation of a memristor has been successfully realized in Hewlett-Packard in 2008 [22]. After then, the applications of a memristor have caught many attentions in nonlinear science [23–28]. Meanwhile, chaotic and hyperchaotic attractors have been found in many memristor-based circuits [21, 23–26]. For example, Muthuswamy and Chua provided a memristor-based circuit with a single-scroll chaotic attractor [24], Bao et al*.* reported a memristor-based circuit with a double-scroll chaotic attractor [25], Teng et al. found a memristor-based circuit with double-scroll and four-scroll chaotic attractors [26], and so on [27, 28]. On the other hand, many real physical systems such as electromagnetic wave propagation, dielectric polarization, and heat conduction can be described by fractional-order differential equations [29, 30]. Meanwhile, chaotic phenomenon has been discussed in many fractional-order nonlinear systems such as the fractional-order electronic circuits [31], the fractional-order gyroscopes [32], the fractional-order chaotic brushless DC motor [12], the fractional-order microelectromechanical system [33], and the fractional-order neural networks [34, 35]. So, more attentions have been paid to research the chaotic behaviors of fractional-order nonlinear systems.

Motivated by the above considerations, first, based on a memristor-based chaotic circuit reported by Muthuswamy and Chua [24], Bao et al*.* [25], and Teng et al. [26], an integer-order memristive chaotic system with two-scroll, three-scroll, and four-scroll chaotic attractors is provided in this paper. It is noticed that there is only a single-scroll chaotic attractor in [24], only a double-scroll chaotic attractor in [25], and only double-scroll and four-scroll chaotic attractors in [26]. However, there are two-scroll, three-scroll, and four-scroll chaotic attractors in our memristive system. Meanwhile, the Lyapunov exponent spectrum, and phase diagram for our memristive chaotic system are obtained. Second, based on the proposed integer-order memristive chaotic system with a four-scroll chaotic attractor, a fractional-order version chaotic system is suggested. We find that the coexisting three-scroll and four-scroll chaotic attractors and coexisting two kinds of three-scroll chaotic attractors are emerged in the fractional-order version. To the best of our knowledge, this result is rarely reported.

The outline of this paper is organized as follows. In Section 2, the concept of a memristor and some memristor-based system are briefly reviewed. Based on the review, we present an integer-order memristive chaotic system with two-scroll, three-scroll, and four-scroll chaotic attractors and some basic dynamics behaviors are obtained. In Section 3, based on the integer-order memristive chaotic system with a four-scroll chaotic attractor, we present its fractional-order version and we find that there are coexisting chaotic attractors in its fractional-order system. In Section 4, the conclusion is given.

#### 2. An Integer-Order Memristive Chaotic System

The charge-controlled memristor [24, 26] is described by a nonlinear *I*-*V* characteristic as and . Here, , , and are the voltage, current, and charge associated to the device, respectively. is the memristance, and is the internal state function. In [24, 26], two schematics of the simplest memristor-based chaotic circuit with a linear passive inductor, linear passive capacitor, and a nonlinear active memristor have been reported. The state equations represent the current-voltage relation for the linear passive capacitor, and the inductor is described as
where denotes the voltage of the linear passive capacitor and denotes the current of the linear passive inductor .

In [24], the memristance is defined as , and the internal state function is defined as , where . The memristor-based circuit in [24] has a single-scroll chaotic attractor (for more details, see [24]), and its dynamics are described by

In [26], the memristance is chosen as and the internal state function is chosen as , where . The memristor-based circuit in [26] has double-scroll and four-scroll chaotic attractors (for more details, see [26]), and its dynamics are shown as

Now, based on [24, 26], an integer-order memristive system is suggested in our paper. The memristance is defined as , and the internal state function is defined as . So, the integer-order memristive chaotic system in this paper is suggested as where , , , , and .

The equilibrium points of system (4) can be calculated by

Obviously, only is the equilibrium point in system (4). The Jacobian matrix at this equilibrium point is and its eigenvalues are , , and . If , then . If , then . So, the equilibrium point in system (4) is unstable.

By numerical calculation, the Lyapunov exponent spectrum of integer-order memristive system (4) with respect to linear passive inductor can be obtained and is displayed in Figure 1.