Reconstruction and Synchronization of hyperchaotic Circuits via One State Variable

Many methods have been proposed to synchronize chaotic systems. The most widely used methods are the continuous synchronization schemes [Pecora & Carroll, 1990], where the driving signals are transmitted continuously to the driven systems. The other synchronization schemes are impulsive synchronization [Panas et al., 1998] and selective synchronization [Itoh et al., 2000]. In an impulsive synchronization scheme, only samples of state variables (or functions of state variables) called synchronization impulses are used to synchronize two chaotic systems. In a selective synchronization scheme, we select only those time periods of driving signals with strong synchronizing effect to the slave system and shut off the driving signals in some other time periods when they show strong desynchronizing effects. These three schemes were applied to several chaotic systems, and all three exhibited good performance [Panas et al., 1998; Itoh et al., 1999; Itoh et al., 2000; Itoh et al., 2001]. The stability of synchronization is closely connected to the values of Lyapunov components of variational systems, and so the conditions for stable impulsive synchronization in both chaotic and hyperchaotic systems are given in terms of Lyapunov exponents [Pecora & Carroll, 1990; Itoh et al., 1999; Itoh et al., 2000; Itoh et al., 2001]. According to Pyragas [Pyragas, 1993; Brucoli et al., 1999], the minimal number of controlled variables has to be equal to the number of positive Lyapunov exponents of the system. If the hyperchaotic systems have two positive Lyapunov exponents, then, at least two driving signals are needed to synchronize them by the continuous synchronization scheme. In the case of impulsive synchronization, the number of driving signals sequentially transmitted to the slave systems via time-division is greater than or equal to two [Itoh et al., 1999; Itoh et al., 2000;