Hopf bifurcation and spatio-temporal patterns in delay-coupled van der Pol oscillators

In this paper, the dynamics of a pair of van der Pol oscillators with delayed velocity coupling is studied by taking the time delay as a bifurcation parameter. We first investigate the stability of the zero equilibrium and the existence of Hopf bifurcations induced by delay, and then study the direction and stability of the Hopf bifurcations. Then by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups, we investigate the spatio-temporal patterns of Hopf bifurcating periodic oscillations. We find that there are different in-phase and anti-phase patterns as the coupling time delay is increased. The analytical theory is supported by numerical simulations, which show good agreement with the theory.