Hopf and resonant codimension two bifurcation in van der Pol equation with two time delays

The van der Pol equation with two discrete delays is considered. The local stability of the zero solution of this equation is investigated by analyzing the corresponding transcendental characteristic equation of the linearized equation and employing Nyquist criteria. Moreover, some general stability criteria involving the delays and the system parameters are also derived. By choosing one of the delays as a bifurcation parameter, the model is found to undergo a sequence of Hopf bifurcation. The direction and the stability criteria of the bifurcating periodic solutions are determined by using the normal form theory and the center manifold theorem. We have also found that resonant codimension two bifurcation occur in this model. A complete description of the location of points in parameter space where the transcendental characteristic equation possesses two pairs of pure imaginary roots, ±iω1, ±iω2 with ω1:ω2=m:n, m,n∈Z+ (Z+ is the set of positive integers), is given. Some numerical simulation examples for justifying the theoretical results are also illustrated.