Adaptive Modulus Hybrid Projective Combination Synchronization of Time-Delay Chaotic Systems with Uncertainty and Disturbance and its Application in Secure Communication

The present paper tries to study a new adaptive modulus hybrid projective combination synchronization (AMHPCS) with time-delay hyperchaotic (HC) systems containing uncertainty and disturbance. Time-delay, uncertainty, and disturbance are the three basic occurrences in nonlinear systems, which gravely affect the control and synchronization of chaotic systems. The controlling and parameter updating equations are designed through the adaptive controllers’ method and the Lyapunov function. This AMHPCS approach gives a systematic procedure for an automated arrangement of controllers in real-time, to perform or to sustain the wanted level of control system execution when the parameters of the chaotic systems are strange and/or vary in time. The time-delay complex HC Lu system and Lorenz systems have been considered as the master and slave systems respectively. This is the first problem in which modulus hybrid projective combination synchronization of time-delay HC with uncertainty and disturbance has been achieved for unknown parameters. Time-delay chaotic/HC exhibits multi-stability and is useful to construct more significant complex dynamics that improve the transmitted data security. Based on the AMHPCS, a secure communication scheme is described. The information message has been recovered accurately by using the chaotic signal masking method. Numerical simulations are also performed to establish the effectiveness of the proposed method, numerical simulations have been performed by using the Runge–Kutta method of the delay-differential equations. This ensures that the designed controllers and adaptive parameter laws are effective to secure and synchronize chaotic time-delay systems and our results demonstrate the novelty over the existing results.