Bipartite synchronization in a network of nonlinear systems: A contraction approach

This paper studies the bipartite synchronization in a network of nonlinear systems with collaborative and antagonistic interactions. Under the assumption that the signed graph is structurally balanced and the considered domain does not contain the origin, we use contraction theory to obtain some sufficient conditions such that the network admits a bipartite synchronization solution. These conditions are described by coupling matrices and the contractivity of lower-dimensional dynamic systems. In particular, if the nonlinear system satisfies a one-sided Lipschitz condition and the coupling matrices are identical, we also obtain some sufficient conditions about the second smallest eigenvalue of signed graph for the bipartite synchronization. Some numerical examples are presented to illustrate the effectiveness of the obtained results.