Synchronization in coupled integer and fractional-order maps

Abstract

Coupled differential equations and coupled maps have been used to model numerous systems in science and engineering. The role of memory in these systems is modelled using fractional calculus. However, different parts of the system may respond to memory in a different manner. We study coupled system in which an integer order system is coupled to a fractional order α system bidirectionally or unidirectionally for various values of α. It is possible to analytically determine the stability of the fixed point for a unidirectionally coupled linear system. It is found to depend on the stability of the fractional system. The stability criterion extends to the nonlinear case as well. If we linearize the nonlinear map around the fixed point, the criterion for the linear case also holds for the stability of the fixed point of coupled nonlinear maps.