Chaos Synchronization Using Differential Equations and the Universal Approximation Theorem with Application to Secure Communication and Cryptography

In this paper, a new method has been presented for chaos synchronization using a nonlinear controller. In most so-far presented approaches, it is assumed that the mathematical models of the transmitter and reciever are completely the same. Due to the non-identical environmental circumstances in the transmitter and receiver and the influence of temperature on the chaotic system parameters, this assumption is not true. In this paper, a novel approach, in which uncertainties are modeled by a linear diffential equation with unknown constant coefficients, has been presented for estimation of these uncertainties . Since this function satisfies the conditions of the universal approximation theorem, it can estimate nonlinear functions with arbitrary small approximation error. However, since the coefficients are unknown, the parameters of these functions are unknown and should be estimated using the adaptation laws derived from the synchronization analysis. Simulation results verify the effectiveness of the proposed estimator. In comparison with other controllers such as fuzzy sliding mode controllers, the proposed controller response is faster. Moreover, its application in secure communications and cryptography has been studied, as well.