Synchronization Between Adaptively Coupled Systems With Discrete and Distributed Time-Delays

This paper investigates complete synchronization of unidirectionally and adaptively coupled systems with discrete and distributed time delays. Instead of the conventional hypothesis of a uniform Lipschitz condition on the system´s vector fields, only a local Lipschitz condition is adopted. It is proved that the local complete synchronization can be achieved through a unidirectional and adaptive coupling, and that the global complete synchronization can be realized when the nonlinear degree of the vector fields is smaller than some derived critical value. The results are illustrated in some representative models with time delays. Also considered is complete synchronization with an exponential convergence rate on the adaptively coupled time-delayed systems with vector fields that are one-sided uniformly Lipschitz (systems of this type can admit time-varying discrete and distributed delays). All the results can be further generalized to various types of synchronization between bidirectionally coupled time-delayed systems or among delayed kinetic systems of a complex network.