Stability switches and Hopf bifurcations in a pair of delay-coupled oscillators

In this paper, we consider a pair of delay-coupled limit-cycle oscillators. Regarding the arithmetical average of two delays as a parameter, we investigate the effect of time delays on its dynamics. We show that there exist stability switches for time delays under certain conditions, which do not occur for the corresponding coupled system without time delays. A similar result has been reported for the same delay by Ramana Reddy et al. (Physica D, 129 [1999]), but in the present paper we give more detailed and specific conditions determining the amplitude death for different delays. On the other hand, we also investigate Hopf bifurcations induced by time delays using the normal form theory and center manifold reduction. In the region where the stability switches may occur, we not only specifically determine the direction of Hopf bifurcations but also show that the bifurcating periodic solutions are orbitally asymptotically stable. Numerical simulation results are also given to support the theoretical predictions.