Anti-synchronization of Hyperchaotic Systems via Novel Sliding Mode Control and Its Application to Vaidyanathan Hyperjerk System

Abstract

Chaos in nonlinear dynamics occurs widely in physics, chemistry, biology, ecology, secure communications, cryptosystems and many scientific branches. Anti-synchronization of chaotic systems is an important research problem in chaos theory. Sliding mode control is an important method used to solve various problems in control systems engineering. In robust control systems, the sliding mode control is often adopted due to its inherent advantages of easy realization, fast response and good transient performance as well as insensitivity to parameter uncertainties and disturbance. In this work, we derive a novel sliding mode control method for the anti-synchronization of identical chaotic or hyperchaotic systems. The general result derived using novel sliding mode control method is proved using Lyapunov stability theory. As an application of the general result, the problem of anti-synchronization of identical Vaidyanathan hyperjerk hyperchaotic systems (2015) is studied and a new sliding mode controller is derived. The Lyapunov exponents of the Vaidyanathan hyperjerk system are obtained as \(L_1 = 0.1448\), \(L_2 = 0.0328\), \(L_3 = 0\) and \(L_4 = -1.1294\). Since the Vaidyanathan hyperjerk system has two positive Lyapunov exponents, it is hyperchaotic. Also, the Kaplan–Yorke dimension of the Vaidyanathan hyperjerk system is obtained as \(D_{KY} = 3.1573\). Numerical simulations using MATLAB have been shown to depict the phase portraits of the Vaidyanathan hyperjerk system and the sliding mode controller design for the anti-synchronization of identical Vaidyanathan hyperjerk systems.