Novel Second Order Sliding Mode Control Design for the Anti-synchronization of Chaotic Systems with an Application to a Novel Four-Wing Chaotic System

Abstract

Chaos in nonlinear dynamics occurs widely in physics, chemistry, biology, ecology, secure communications, cryptosystems and many scientific branches. 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. This work derives a new result for the anti-synchronization of identical chaotic systems via novel second order sliding mode control method. The main control result is established by Lyapunov stability theory. As an application of the general result, the problem of anti-synchronization of novel four-wing chaotic systems is studied and a new sliding mode controller is derived. The Lyapunov exponents of the novel four-wing chaotic system are obtained as \(L_1 = 0.8312\), \(L_2 = 0\) and \(L_3 = -27.4625\). The Kaplan-Yorke dimension of the novel chaotic system is obtained as \(D_{KY} = 2.0303\). We show that the novel four-wing chaotic system has five unstable equilibrium points. We also show that the novel four-wing chaotic system has rotation symmetry about the \(x_3\)-axis. Numerical simulations using MATLAB have been shown to depict the phase portraits of the novel four-wing chaotic system and the global anti-synchronization of the novel four-wing chaotic systems.