Hopf bifurcation and Stability of Periodic Solutions in the Delayed LiÉnard equation

In this paper, the classical Lienard equation with a discrete delay is considered. Under the assumption that the classical Lienard equation without delay has a unique stable trivial equilibrium, we consider the effect of the delay on the stability of zero equilibrium. It is found that the increase of delay not only can change the stability of zero equilibrium but can also lead to the occurrence of periodic solutions near the zero equilibrium. Furthermore, the stability of bifurcated periodic solutions is investigated by applying the normal form theory and center manifold reduction for functional differential equations. Finally, in order to verify these theoretical conclusions, some numerical simulations are given.