Topological Symmetry And Existence of Partial Synchronization

We study the relationship between the partial synchronous (PaS) state and the coupling structure in general dynamical systems. By the exact proof, we find the sufficient and necessary condition of the existence of PaS state for the coupling structure. Our result shows that the symmetry of the coupling structure is not the equivalent condition which is supposed before but only the sufficient condition. Furthermore, for the existence of the PaS state, the general structure is the equal-degree random.