Generalized projective synchronization of Vaidyanathan Chaotic system via active and adaptive control

Abstract

Generalized projective synchronization (GPS) of chaotic systems generalizes known types of synchronization schemes such as complete synchronization (CS), anti-synchronization (AS), hybrid synchronization (HS), projective synchronization (PS), etc. In this work, we have designed active and adaptive controllers for the generalized projective synchronization (GPS) of identical Vaidyanathan chaotic systems (2014). Vaidyanathan system is an eight-term chaotic system with three quadratic nonlinearities. The Lyapunov exponents of the Vaidyanathan chaotic system are obtained as L1 = 6.5294, L2 = 0 and L3 = −26.4696. Since themaximal Lyapunov exponent of the Vaidyanathan system is L1 = 6.5294, the system exhibits highly chaotic behaviour. The Kaplan–Yorke dimension of the Vaidyanathan chaotic system is obtained as DKY = 2.2467. The main GPS results in this work have been established using Lyapunov stability theory. MATLAB plots have been depicted to illustrate the phase portraits of the Vaidyanathan chaotic system and also the GPS results for Vaidyanathan chaotic systems using active and adaptive controllers.