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A Decomposition Approach for Synchronization of Heterogeneous Complex Networks

In this paper, the synchronization problem of complex networks with linearly diffusively coupled nonidentical nodes is investigated. Starting with the boundedness condition of network trajectories, we introduce an invariant set such that it contains all limit points of ultimately synchronous trajectories. Then, we develop a decomposition technique for the heterogeneous network. With this decomposition, the synchronization of the network can be investigated by the convergence of one decomposed network and the synchronization of the other decomposed homogeneous-like network. Moreover, for a particular case that the invariant set is a linear subspace, conditional synchronization analysis is provided to reduce the coupling complexity between the two decomposed networks. It is noted that our decomposition technique is quite simple yet general: by this technique, the synchronization of various heterogeneous complex networks can be transformed into the stability of nonlinear systems and synchronization of homogeneous-like complex networks. Finally, we present several numerical examples to demonstrate the effectiveness of the theoretical results. In particular, we use an example to show that our theoretical procedure is also feasible for some heterogeneous networks with a general invariant submanifold instead of linear subspace.