Stability conditions for cluster synchronization in directed networks of diffusively coupled nonlinear systems

This paper investigates the stability issue of cluster synchronization manifold in networks of diffusively-coupled nonlinear system with directed topology. It is assumed that each cluster subdigraph is strongly connected and contains only cooperative interactions, but there may exist competitions among nodes of different clusters. In the case that the clusters meet the cluster-input-equivalence condition and each nonlinear system has a forward invariant set over which the system possesses a bounded Jacobian, some local stability conditions of cluster synchronization are derived. These conditions are expressed as an inequality of the matrix measure of the system's Jacobian matrix, the matrix measure of a reduced-order matrix about the digraph among the clusters, the coupling strength, and some eigenvalue conditions of the subdigraphs. Finally, the theoretical findings are validated by two numerical examples about coupled Lorenz-like system and coupled Hopfield neural network, respectively.