Anti-synchronization of Identical Chaotic Systems Using Sliding Mode Control and an Application to Vaidyanathan–Madhavan Chaotic Systems

Abstract

Anti-synchronization is an important type of synchronization of a pair of chaotic systems called the master and slave systems. The anti-synchronization characterizes the asymptotic vanishing of the sum of the states of the master and slave systems. In other words, anti-synchronization of master and slave system is said to occur when the states of the synchronized systems have the same absolute values but opposite signs. Anti-synchronization has applications in science and engineering. This work derives a general result for the anti-synchronization of identical chaotic systems using sliding mode control. The main result has been proved using Lyapunov stability theory. Sliding mode control (SMC) is well-known as a robust approach and useful for controller design in systems with parameter uncertainties. Next, as an application of the main result, anti-synchronizing controller has been designed for Vaidyanathan–Madhavan chaotic systems (2013). The Lyapunov exponents of the Vaidyanathan–Madhavan chaotic system are found as \(L_1 = 3.2226, L_2 = 0\) and \(L_3 = -30.3406\) and the Lyapunov dimension of the novel chaotic system is found as \(D_L = 2.1095\). The maximal Lyapunov exponent of the Vaidyanathan–Madhavan chaotic system is \(L_1 = 3.2226\). As an application of the general result derived in this work, a sliding mode controller is derived for the anti-synchronization of the identical Vaidyanathan–Madhavan chaotic systems. MATLAB simulations have been provided to illustrate the qualitative properties of the novel 3-D chaotic system and the anti-synchronizer results for the identical novel 3-D chaotic systems.