Anti-synchronization of Two Different Hyperchaotic Systems via Active and Adaptive Control

Since the idea of synchronizing chaotic systems was introduced by Pecora and Carroll[1] in 1990, chaos synchronization has received increasing attention due to its theoretical challenge and its great potential applications in secure communication,chemical reaction and biological systems[2]. Recently several different types of synchronizations have been proposed in the literature, for example, generalized synchronization[3–6], phase synchronization[7, 8], lag synchronization[9–11], anti-synchronization [12–14] and so on .The generalized synchronization implies the establishment of functional relations between drive and response systems. Phase synchronization is characterized in that the phase difference between two chaotic systems are locked within 2π, while their amplitudes remain chaotic and uncorrelated. Lag synchronization is described as the coincidence of two chaotic trajectories with a constant time lag. The state vectors of synchronized systems have the same absolute values but opposite signs, i.e. the sum of the output signals of two systems can converge to zero, called the anti-synchronization(AS) method. A chaotic system is extremely sensitive to tiny variations of parameters. But in practical situation, some systems parameters cannot be exactly known in priori, the effect of these uncertainties will destroy the synchronization and even break it. Therefore, it is important and necessary to study the synchronization in such systems with unknown parameters. In this paper, we apply active and adaptive control theory to anti-synchronize two different hyperchaotic systems. We demonstrate the technique capability on the anti-synchronization of hyperchaotic Lü and hyperchaotic Chen systems.