Characterization and Computation of Partial Synchronization Manifolds for Diffusive Delay-Coupled Systems

Abstract

Sometimes a network of dynamical systems shows a form of incomplete synchronization, characterized by synchronization of some but not all of its component systems. This type of incomplete synchronization is called partial synchronization or cluster synchronization. Partial synchronization is associated with the existence of partial synchronization manifolds, which are linear invariant subspaces of C, the state space of the network of systems. We focus on partial synchronization manifolds in networks of identical systems, characterized by linear diffusive coupling described by a weighted graph, and allowing for time-delay in the coupling. We present equivalent existence criteria for partial synchronization manifolds in terms of invariant spaces, the block-structure of a reordered adjacency matrix, and the solvability of a Sylvester equation. We emphasize decomposable networks, according to the rational dependency structure of the coupling weights, and according to the delay values, respectively. It is obvious that if the existence conditions for partial synchronization manifolds are satisfied for all subnetworks simultaneously, they hold for the original network, yet the converse result is not always true, as we shall illustrate with an example. Furthermore, as our main results, we show that if the decomposition is according to the weights and the basis weights are rationally independent numbers, or if the decomposition is according to different delay values, then finding a partial synchronization manifold for the original network is equivalent to finding common partial synchronization manifolds for the subnetworks, i.e., restricting to the analysis of the subnetworks does not impose any conservatism, which simplifies the analysis significantly. For the case of decomposable networks according to the weights, with rationally independent basis weights, we provide a fourth existence criterion for partial synchronization manifolds in terms of a balanced coloring of an associated multigraph. In addition, we briefly describe publicly available software for detecting partial synchronization manifolds. Our equivalent existence criteria, which depend on the network and delay structure but not on the dynamics of the systems at the nodes, are sufficient for the presence of a partial synchronization manifold. We show that under a mild assumption on the systems at the nodes, namely, left-invertibility, these conditions are necessary as well. In all criteria it turns out that the distinction between noninvasive and invasive delayed coupling is important, i.e., whether or not a coupling term between two systems vanishes whenever the latter are synchronized.