A general criterion for synchronization of incrementally dissipative nonlinearly coupled agents

Whereas synchronization (consensus, agreement) in linear networks has been thoroughly studied in recent years up to certain exhaustiveness, reaching a synchrony among general nonlinear agents still remains a hard open problems. In this paper, we propose a general criteria of synchronization in undirected networks with nonlinear nodes, based on the idea of incremental dissipativity. We show that our result implies a number of existing synchronization criteria, employing incremental Lyapunov functions (such as e.g. the sum of squared deviations between the agents´ states). At the same time, unlike most of existing results of synchronization, our criterion is applicable not only to linearly coupling maps, but also to a wide class of nonlinear couplings, provided they are anti-symmetric and satisfy a quadratic constraint, which should be in correspondence with the incremental supply rate and the network topology. Our criterion thus allows to extend many existing synchronization criteria to the case of nonlinear couplings; moreover, it can be applied to some classes of agent, uncovered by previous results, among them are nonlinear agents in Lurie form and biochemical oscillators of Goodwin´s type.