Synchronization of Fractional-Order Hyperchaotic Systems via Fractional-Order Controllers

Tianzeng Li,Yong Yang,Yu Wang

doi:10.1155/2014/408972

Abstract

In this paper, the synchronization of fractional-order chaotic systems is studied and a new fractional-order controller for hyperchaos synchronization is presented based on the Lyapunov stability theory. The proposed synchronized method can be applied to an arbitrary four-dimensional fractional hyperchaotic system. And we give the optimal value of control parameters to achieve synchronization of fractional hyperchaotic system. This approach is universal, simple, and theoretically rigorous. Numerical simulations of several fractional-order hyperchaotic systems demonstrate the universality and the effectiveness of the proposed method.

Highlights

Fractional calculus is a 300-year-old mathematical topic. It has a long history, it has not been used in physics and engineering for many years
Hyperchaotic system is usually defined as a chaotic system with more than one positive Lyapunov exponent, which implies that its dynamics are extended in several different directions simultaneously
According to the Lyapunov stability theorem of the fractional systems, the proposed method achieved to synchronize arbitrary 4D fractionalorder hyperchaotic systems

Summary

Introduction

Fractional calculus is a 300-year-old mathematical topic. it has a long history, it has not been used in physics and engineering for many years. A famous example of a continuous-time three-order system which exhibits chaos is the Lorenz system [11]. The order of this system can be defined as the sum of the orders of all involved derivatives. Hyperchaotic system is usually defined as a chaotic system with more than one positive Lyapunov exponent, which implies that its dynamics are extended in several different directions simultaneously. Most of the above synchronization methods concentrate on certain fractional systems, and there are few Discrete Dynamics in Nature and Society general methods that can be applied to control arbitrary fractional-order hyperchaotic systems. Numerical simulation results of synchronization of the fractional-order hyperchaotic Chen system, the fractional-order hyperchaotic Lorenz system, and the fractional-order hyperchaotic Lusystem demonstrate the effectiveness and the validity of the proposed method

Fractional Derivatives and Fractional Dynamic Systems

Synchronization of the Fractional-Order Hyperchaotic Systems

Simulation and Analysis

Conclusion