BIFURCATION ANALYSIS OF SYNCHRONIZED REGIONS IN COMPLEX DYNAMICAL NETWORKS

Abstract

This paper aims to investigate the bifurcation phenomena of synchronized regions in complex dynamical networks with varying node parameters and fixed inner-linking matrices. In particular, by using the unified chaotic system as network nodes, this paper further explores the bifurcation patterns of synchronized regions with synchronous states of both equilibrium points and attractors in complex dynamical networks based on two types of inner-linking matrices, respectively. Our results indicate that there does not exist any bifurcation phenomenon in the above synchronized regions for some specific inner-linking matrices. It means that the stability of network synchronous state will not change for some specific inner-linking matrices as the parameter of node dynamics changes. However, the above synchronized regions generate various bifurcation patterns for some inner-linking matrices and varying node parameters as follows: (i) The unbounded–empty set bifurcation mode; (ii) The bounded–empty set bifurcation mode; (iii) The single bounded–multiple bounded–single bounded–empty set bifurcation mode; (iv) The unbounded–multiple disconnected–empty set bifurcation mode. All these results tell us that the inner-linking matrices play a key role for determining the bifurcation patterns of synchronized regions.