Characterizations and criteria for synchronization of heterogeneous networks to linear subspaces

In this paper, we investigate synchronization issues in heterogeneous complex networks. We start with a necessary condition for the existence of synchronous trajectories by analyzing the vector fields of isolated nodes. Then, by geometrical methods of differential equations, a closed invariant set is introduced to characterize the synchronous trajectories. Especially, the ω-limit sets of all the synchronous trajectories are contained in this closed invariant set. Afterward, assuming that the vector fields of all the isolated nodes are identical in a linear subspace, an equiv- alent network is found such that the equivalent network can be composed of two simpler networks with lower dimensions. Based on this composition, global synchronization criteria for the equivalent network are developed by combining the corresponding Lyapunov-like functions of quadratic form (and even beyond quadratic form) for these two simpler networks. Furthermore, we alternatively use different functions for different nodes to construct Lyapunov functions, arriving at an extended criterion which can be applicable for more networks. Finally, three examples are given to illustrate the validity and advantages of our theoretical results.