Hopf Bifurcation in Numerical Approximation for the Generalized Lienard Equation with Finite Delay

The numerical approximation of the generalized Lienard equation is considered using delay as parameter. First, the delay difference equation obtained by using Euler method is written as a map.According to the theories of bifurcation for discrete dynamical systems,the conditions to guarantee the existence of Hopf bifurcation for numerical approximation are given. The relations of Hopf bifurcation between the continuous and the discrete are discussed. Then when the generalized Lienard equation has Hopf bifurcations at 0 r r  , the numerical approximation also has Hopf bifurcations at 0 ( ) h r r o h   is proved. At last, the text listed an example of numerical simulation, the result shows that system (8) discretized by Euler keeps the dynamic characteristic of former system (1), and the theory is proved.