Synchronization and bifurcation in Limit Cycle oscillators with Delayed Couplings

In this paper, a system of three globally coupled limit cycle oscillators with a linear time-delayed coupling are investigated. Considering the delay as a parameter, we also study the effect of time delay on the dynamics. Next, Hopf bifurcations induced by time delays using the normal form theory and center manifold reduction are obtained. Based on the symmetric Hopf bifurcation theorem, we investigate stable phase-locking and unstable waves. Then later, the directions of Hopf bifurcations are determined in some region, where stability switches may occur. The results show that the bifurcating periodic solutions are orbitally asymptotically stable. Numerical simulations are applied to verify the theoretical predictions.