Vaidyanathan S and Azar AT (2015) Anti-Synchronization of Identical Chaotic Systems using Sliding Mode Control and an Application to Vaidyanathan-Madhavan Chaotic Systems. In: A.T Azar, Q. Zhu, (eds.), Advances and Applications in Sliding Mode Control systems, Studies in Computational Intelligence book Series, Vol. 576, pp. 527-547, Springer-Verlag GmbH Berlin/Heidelberg.
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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 L1 = 3.2226,L2 = 0 and L3 = − 30.3406 and the Lyapunov dimension of the novel chaotic system is found as DL = 2.1095. The maximal Lyapunov exponent of the Vaidyanathan-Madhavan chaotic system is L1 = 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.