Stability and bifurcation analysis in the delay-coupled van der Pol oscillators

In this paper, the dynamics of a system of two van der Pol equations with a finite delay are investigated. We show that there exist the stability switches and a sequence of Hopf bifurcations occur at the zero equilibrium when the delay varies. Using the theory of normal form and the center manifold theorem, the explicit expression for determining the direction of the Hopf bifurcations and the stability of the bifurcating periodic solutions are derived.