Inphase and Antiphase Synchronization in a Delay-Coupled System With Applications to a Delay-Coupled FitzHugh–Nagumo System

A time delay is inevitable in the coupled system and is an essential property of the coupling, which cannot be neglected in many realistic coupled systems. In this paper, we first study the existence of a Hopf bifurcation induced by coupling time delay and then investigate the influence of coupling time delay on the patterns of Hopf-bifurcating periodic oscillations. How the coupling time delay leads to complex scenarios of synchronized inphase or antiphase oscillations is analytically investigated. As an example, we study the delay-coupled FitzHugh-Nagumo system. We find conditional stability, absolute stability, and stability switches of the steady state provoked by the coupling time delay. Then we investigate the inphase and antiphase synchronized periodic solutions induced by delay, and determine the direction and stability of these bifurcating periodic orbits by employing the center manifold reduction and normal form theory. We find that in the region where stability switches occur, there exist synchronization transitions, i.e., synchronized dynamics can be switched from inphase (antiphase) to antiphase (inphase) and back to inphase (antiphase) and so on just by progressive increase of the coupling time delay.