Prediction of Partially Synchronous Regimes of Delay-Coupled Nonlinear Oscillators

Abstract We present an approach which allows to accurately predict both the occurrence and type of partially synchronous regimes of delay-coupled non-linear oscillators. Unlike the conventional approach, we build on an analysis of the stability properties of the synchronized equilibrium in the (coupling gain, delay) parameter space. As partially synchronous regimes are closely related to the presence of invariant manifolds, we first present necessary and sufficient conditions for the existence of forward invariant sets. Next, from the existence of these invariant sets and from the characterization of solutions in the unstable manifold of the synchronized equilibrium, we predict which (gain, delay) parameters may result in fully/partially synchronous behavior. We illustrate the approach for a network of delay coupled Hindmarsh-Rose neurons.