Conditions for impulsive synchronization of chaotic and hyperchaotic system

Experimental results show that chaotic and hyperchaotic systems can be synchronized by impulses sampled from one or two state variables. In this paper, we study the conditions under which chaotic and hyperchaotic systems can be synchronized by impulses sampled from a part of their state variables. By calculating the Lyapunov exponents of variational synchronization error systems, we show that this kind of impulsive synchronization can be applied to almost all hyperchaotic systems. We also study the selective synchronization of chaotic systems. In a selective synchronization scheme, the synchronizing signal is chosen in the time periods when the Lyapunov exponents of variational synchronization error systems are negative. Since only driving signals during the time periods when synchronization error can be reduced are applied to reduce the synchronization error, and no signal is applied during the time periods when synchronization error can be increased, selective synchronization scheme can be used to achieve synchronization even in the case when continuous synchronization schemes fail to work.