Hopf bifurcation of a Lienard differential equation with delay-dependent coefficients

A Lienard differential equation with delay dependent coefficients is investigated. The analysis on the stability of the trivial solution is completed. As the time delay increases, the trivial may change from being stable to unstable. The sufficient criterion for the appearance of the oscillation periodic solution is also given out. By means of the center manifold theory, the direction of the Hopf bifurcation is computed. Finally, some examples enrich the contents and numerical simulations show the predicted properties of periodic oscillations.