Specific features of generalized synchronization in unidirectionally and mutually coupled mappings and flows: Method of phase tubes

A concept of generalized synchronization in flow systems and discrete mappings is corrected and completed. It is rigorously demonstrated that the state vectors of interacting systems in the course of generalized synchronization must be considered as interrelated with the aid of a functional rather than a functional relation (as it is commonly accepted). An approach based on the analysis of the tubes of trajectories in phase space is proposed to determine the threshold of the generalized synchronization and study such type of synchronous behavior in the systems with discrete and continuous time. We conclude that the concept of weak and strong generalized synchronization must also be reconsidered.