Nonlinear synchronization on connected undirected networks

In this paper, we give sufficient conditions to have a complete synchronization of oscillators in connected undirected networks. The oscillators we are considering are not necessarily identical and the synchronization terms can be nonlinear. Many results in the literature deal with sufficient conditions insuring complete synchronization. This is a difficult problem since such conditions require to take into account the individual dynamics of the oscillators and also the graph topology. In this paper, we extend one of these results, the connection graph stability method, to nonlinear coupling functions and we also give an existence condition of trajectories of the oscillators. The sufficient conditions for synchronization presented in this paper are based on the study of a Lyapunov function and on the use of pseudometrics which enable us to link network dynamics and graph theory. These results are applied to a network of Chua’s oscillators and lead to an explicit condition insuring the complete synchronization of the oscillators.