Periodic solutions for van der Pol equation with time delay

For van der Pol equation with distributed delay, when the distributed delay kernel is the general Gamma distributed delay kernel, the existence and the stability of periodic solution are obtained by using the linear chain trick and geometric singular perturbation theory. On the other hand, for van der Pol equation with discrete delay, the existence and the stability are proved by employing a technique which is based on inertial manifold. Finally, numerical simulations are also given to illustrate the results.