Partial synchronization manifolds for linearly time-delay coupled systems

Sometimes a network of dynamical systems shows a form of incomplete synchronization characterized by synchronization of some but not all of its systems. This type of incomplete synchronization is called partial 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 systems that interact via linear coupling functions subject to time-delays. For such networks a number of necessary and sufficient conditions for the existence of partial synchronization manifolds are presented. A simple, illustrative example demonstrates the application of our results.