Connecting Orbits from Synchronous Periodic Solutions to Phase-Locked Periodic Solutions in a Delay Differential System

Abstract

We consider a system of delay differential equations modelling the excitatory interaction of two identical neurons. Assuming the delay is sufficiently large, we show that the closure of the forward extension W5 of a 5-dimensional leading unstable manifold of the trivial solution contains a phase-locked periodic orbit and a synchronized periodic orbit and we classify the dynamics of the semiflow restricted to W5. We also obtain the precise information about the Floquet multipliers of the synchronized periodic orbit, which enables us to establish the existence of heteroclinic orbits from the synchronized periodic orbit to the phase-locked orbit.