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

Ping Zhou,Peng Zhu,Meihua Ke

doi:10.1155/2018/4192824

Ping Zhou, Peng Zhu + Show 1 more

Open Access

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

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Abstract

A 3D fractional-order nonlinear system with coexisting chaotic attractors is proposed in this paper. The necessary condition of the existence chaos is q≥0.8477. The fractional-order system exhibits chaotic attractors with the order as low as 2.5431. The largest Lyapunov exponent varying as fractional order q 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.

Highlights

Chaotic behaviors in nonlinear is a very interesting phenomenon
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
Same as the results reported by Zhou and Ke [6], there are coexisting “positive attractor” and “negative attractor” in fractional-order system (2)

Summary

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. More and more attention has been focused on the coexisting chaotic attractors in nonlinear chaotic systems. 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].

System Model and Basic Characteristics

Numerical Simulations Results

Conclusions