Bifurcation Analysis of Synchronized Regions in Complex Dynamical Networks with Coupling Delay

According to the master stability function (MSF) framework, synchronized regions play an extremely important role in network synchronization. On these grounds, this paper casts sight on network synchronous state stability via studying the bifurcation (or transition) problem of network synchronized regions with varying nodal dynamics, and the effects of time delay on the bifurcation of synchronized regions. Theoretical and numerical investigations show that in complex networks with coupling delay, there exist rich bifurcation behaviors of synchronized regions. The coupling delay can not only enlarge or narrow synchronized regions, but also change bifurcation points. More importantly, a very small delay can result in the conversion of an unbounded or empty synchronized region into a bounded one, implying that coupling delay can enhance or suppress synchronization in complex dynamical networks. These results will further strengthen our understanding for synchronous state stability in complex dynamical networks.