The Hopf Bifurcation Formulas Of Generalized Liénard Equation With Delay

In this paper, we discuss the Hopf bifurcation for the generalized Lienard equation x(t) + f(x(t))x(t) 4- g(x(t — r)) = 0. First, we perform systematic analysis in the current results of Hopf bifurcation for the generalized Lienard equation. Then we discuss the possible cases of Hopf bifurcation under the cases of different parameters. Because the Hopf bifurcation formula with delay r in [4] is so complex, we use the method of Hassard " normal form " to consider the Hopf bifurcations with delay k for the generalized Lienard equation, and we obtain simple and applicable Hopf bifurcation formula for the generalized Lienard equation, and the further simplification of the formula we gave is also made. Making use of the formula, we can decide the stability of one periodic solution and give the approximate expression of one periodic solution. Finally, we summarize the Hopf bifurcation formulas under different parameters for the generalized Lienard equation, and give corresponding examples to illustrate.