Synchronization of Chua’s Oscillators with the Third State as the Driving Signal

Although research concerning the synchronization of chaotic systems is still at an early stage, the literature on the subject is growing at a fast pace, as documented by the surveys of Ogorzalek [1993] and Hasler [1997], and the bibliography by Chen [1997]. In most of the synchronization schemes, the driving signal can be taken as one of the state variables of the drive system. The driving signal is used to synchronize the response to the drive in the sense that the trajectory of the response tracks the trajectory of the drive. The synchronization stability is proven by numerically computing the conditional Lyapunov exponents of the response system [Pecora & Carroll, 1991] or finding a Lyapunov function [Wu & Chua, 1994]. A widely used system for generating a chaotic carrier signal is Chua’s oscillator [Chua et al., 1993a] described by Eq. (1). Many authors have considered the synchronization of two Chua’s oscillators through linear feedback (coupling) [Chua et al., 1993b; Wu & Chua, 1994; Chua et al., 1996]. Wu and Chua proved that synchronization can be attained with x1, the first state of the drive system as the driving signal when the feedback gain is large enough. However, the single nonlinearity in Chua’s oscillator is a piecewise linear function which depends only on x1. If x1 is taken as the driving signal, one could find out in which linear region is the drive system at any time; this in turn would enable an unauthorized receiver to estimate system parameters with conventional linear short term system identification methods. In the computer simulations, we can see that synchronization can be achieved with x2, the second state of the drive system as the driving signal when the feedback gain is large enough. However, we were not able to prove this conjecture rigorously [Wu & Chua, 1994]. Schweizer et al. [1995] show that, if we can get x2 and its derivative, ẋ2 simultaneously at the response, then we can develop an error feedback synchronization scheme such that a global synchronization proof is possible. However, a large error can potentially occur when one calculates the derivative of x2. To the best of our knowledge, no one have used x3, the third state as the driving signal. In this work, we consider the linear feedback method for the synchronization of two Chua’s oscillators with x3 as the driving signal. The feedback vector depends on a free parameter. We prove that one can achieve globally asymptotic synchronization when the parameter is large enough.