Hopf Bifurcation and Stability of Periodic Solutions for van der Pol Equation with Distributed Delay

The van der Pol equation with a distributed time delay is analyzed. Itslinear stability is investigated by employing the Routh–Hurwitzcriteria. Moreover, the local asymptotic stability conditions are alsoderived. By using the mean time delay as a bifurcation parameter, themodel is found to undergo a sequence of Hopf bifurcations. The directionand the stability criteria of the bifurcating periodic solutions areobtained by the normal form theory and the center manifold theorem. Somenumerical simulation examples for justifying the theoretical analysisare also given.