Complexity

Volume 2018, Article ID 4192824, 7 pages

https://doi.org/10.1155/2018/4192824

## A Fractional-Order System with Coexisting Chaotic Attractors and Control Chaos via a Single State Variable Linear Controller

^{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 23 March 2018; Accepted 16 April 2018; Published 25 June 2018

Academic Editor: Viet-Thanh Pham

Copyright © 2018 Ping Zhou et al. 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

A 3D fractional-order nonlinear system with coexisting chaotic attractors is proposed in this paper. The necessary condition of the existence chaos is . The fractional-order system exhibits chaotic attractors with the order as low as 2.5431. The largest Lyapunov exponent varying as fractional order is given. Furthermore, there are the coexisting “positive attractor” and “negative attractor” in this fractional-order chaotic system, and the necessary condition for “positive attractor” and “negative attractor” is obtained. Meanwhile, a control scheme for the stabilization of the unstable equilibrium is suggested via a single state variable linear controller. Numerical results show that the control scheme is valid.

#### 1. Introduction

Chaotic behaviors in nonlinear is a very interesting phenomenon. The high irregularity, unpredictability, and complexity in chaotic systems [1, 2] have been widely used in the field of engineering and technology such as secure communications, image steganography, authenticated encryption, motor control, and power system protection. Recently, coexisting chaotic attractors have been found in chaotic systems [3–9]. For example, the coexisting chaotic attractors in a 3D no-equilibrium system were reported by Pham et al. [3], the coexisting multiple attractors in Hopfield neural network were found by Bao et al. [4], the coexisting chaotic attractors in a hyperchaotic hyperjerk system were given by Wang et al. [5], the coexisting “positive attractor” and “negative attractor” in a 3D autonomous continuous chaotic system were found by Zhou and Ke [6], and so on [7–9]. Therefore, more and more attention has been focused on the coexisting chaotic attractors in nonlinear chaotic systems.

On the other hand, the fractional-order differential equations [10–12] can be accurately described in the real-world physical systems such as viscoelasticity, dielectric polarization, electrode-electrolyte polarization, electromagnetic waves, heat conduction, diffusion-wave, and superdiffusion. Chaotic behaviors have been found in many real-world physical fractional-order nonlinear systems, for example, the fractional-order chaotic brushless DC motor [13], the fractional-order electronic circuits [14], the fractional-order microelectromechanical system [15], and the fractional-order gyroscopes [16]. Therefore, more and more attention has been paid to the chaotic behaviors in fractional-order nonlinear systems.

Motivated by the above discussions, based on a 3D autonomous continuous chaotic system reported by Zhou and Ke [6], we suggested a 3D autonomous continuous fractional-order system. We have shown that the chaotic system reported by Zhou and Ke [6] can be extended to its fractional-order version where the coexisting “positive attractor” and “negative attractor” can be observed. We obtained that the fractional-order system with the order as low as 2.5431 exhibits chaotic attractors. Moreover, we obtained the largest Lyapunov exponent varying as fractional order. Finally, for the stabilization of the unstable equilibrium, one control scheme is proposed via a single state variable linear controller.

The outline of this paper is organized as follows. In Section 2, based on a 3D autonomous continuous chaotic system reported by Zhou and Ke [6], the fractional-order version nonlinear system is given, and some basic dynamical properties of this fractional-order version nonlinear system are obtained including the necessary condition of the existence chaos, the largest Lyapunov exponent varying as fractional order, and the coexisting “positive attractor” and “negative attractor.” In Section 3, by a single state variable, stabilization of the unstable equilibrium points of the fractional-order chaotic system is discussed. Finally, the conclusions are given in Section 4.

#### 2. System Model and Basic Characteristics

In this paper, the Caputo definition of the fractional derivative will be used in next. The Caputo definition of the fractional derivative is described aswhere is the Caputo operator, is the first integer which is not less than , and is the -order derivative in usual sense of .

Next, based on the 3D autonomous continuous chaotic system reported by Zhou and Ke [6], a fractional-order system is addressed aswhere fractional order . Fractional-order system (2) has five equilibrium points. They are , , , , and , respectively.

Now, we can obtain the eigenvalues of the five equilibrium points. The eigenvalues of equilibrium point are (−6, −1, 4). Thus, equilibrium point is an unstable saddle point of index one. The eigenvalues of equilibrium points and are (−5.3957, *j*, *j*). Thus, equilibrium points and are unstable saddle points of index two. The eigenvalues of equilibrium points and are (−6.7558, *j*, *j*). Thus, equilibrium points and are unstable saddle points of index two.

Tavazoei and Haeri [17] have obtained that a necessary condition for a fractional-order nonlinear system to exist chaotic iswhere is the eigenvalues of saddle equilibrium point of index two in fractional-order nonlinear system.

Now, we can obtain the necessary condition of the existence chaos in fractional-order system (2). According to (3), we have the following:Thus, the necessary condition of existence chaos in fractional-order system (2) is . This result indicates that fractional-order system (2) with the order as low as 2.5431 can exhibit chaotic attractors.

In this paper, the improved version of Adams-Bashforth-Moulton [18] (denoted by IVABM) numerical algorithm is used to deal with fractional-order system (2). The IVABM numerical algorithm will be introduced next. Now, consider the fractional-order systemwith initial condition . Let , and . By IVABM numerical algorithm, system (5) can be discretized as follows:where

The error of this IVABM numerical algorithm is

We can yield the largest Lyapunov exponent varying as via IVABM numerical algorithm, which is shown in Figure 1, where and the initial condition is (−5, 2, 5).