A new approach to study the coexistence of some synchronization types between chaotic maps with different dimensions

Abstract

When synchronizing chaotic systems, an interesting phenomenon that may occur is the coexistence of several synchronization types. Referring to discrete-time chaotic (hyperchaotic) systems, this paper presents a new approach to rigorously study the coexistence of some synchronization types between maps with different dimensions. By exploiting a Lyapunov-based approach, the paper first analyzes the coexistence of full state hybrid projective synchronization and generalized synchronization when the drive system is a three-dimensional map and the response system is a two-dimensional map. Successively, the coexistence of three different synchronization types is illustrated, i.e., inverse projective synchronization, inverse generalized synchronization and Q–S synchronization are proved to coexist between two-dimensional drive systems and three-dimensional response systems. Finally, by exploiting a pole placement technique, the coexistence of antiphase synchronization and dislocated full state hybrid projective synchronization is proved to be achieved between three-dimensional drive system maps and two-dimensional response system maps. Since synchronization is achieved in finite time, this last case represents an example of dead beat coexistence of some synchronization types in maps with different dimensions. Several numerical examples of coexistence of synchronization types are illustrated, with the aim to show the effectiveness of the novel approach developed herein.