Synchronization on coupled dynamical networks

In this paper, partial synchronization (PaS) in networks of coupled chaotic oscillator systems and synchronization in sparsely coupled spatiotemporal systems are explored. For the PaS, we reveal that the existence of PaS patterns depends on the symmetry property of the network topology, while the emergence of the PaS pattern depends crucially on the stability of the corresponding solution. An analytical criterion in judging the stability of PaS state on a given network are proposed in terms of a comparison between the Lyapunov exponent spectrum of the PaS manifold and that of the transversal manifold. The competition and selections of the PaS patterns induced by the presence of multiple topological symmetries of the network are studied in terms of the criterion. The phase diagram in distinguishing the synchronous and the asynchronous states is given. The criterion in judging PaS is further applied to the study of synchronization of two sparsely coupled spatiotemporal chaotic systems. Different synchronization regimes are distinguished. The present study reveals the intrinsic collective bifurcation of coupled dynamical systems prior to the emergence of global synchronization.