Effect of the coupling matrix with a weight parameter on synchronization pattern in a globally coupled network

In this paper, effect of the coupling matrix with a weight parameter on synchronization pattern in a globally coupled network is investigated. On the basis of matrix theory, the threshold values of the coupling strength and the weight parameter for cluster synchronization have been obtained by utilizing the attractiveness criteria of the invariant synchronization manifold. It shows that cluster synchronization bifurcation comes forth, which concept is first put forward. That is to say, via changing the weight parameter and the coupling strength, the purpose of controlling the number of clusters is achieved, which provides a new idea for control the number of clusters in a network. Numerical simulations are given to demonstrate the theoretical results. In addition, the theoretical results and the numerical simulations also show that full synchronization may not be realized even if the network is globally coupled when there are some negative couplings.