有限时滞Lienard方程Hopf分支的数值逼近

Hopf Bifurcation in Numerical Approximation for a Lienard Equation with Finite Delay

研究了一类有限时滞的Lienard方程Hopf分支的数值逼近问题.首先,以滞量为参数,应用Cooke和J Hale的方法,得到Hopf分支存在的条件;然后,利用欧拉方法将得到的差分方程表示为映射,利用离散动力系统的分支理论,给出了差分方程Hopf分支存在的条件和连续系统与其数值逼近间的关系.证明了当该系统在r=r0产生Hopf分支时,其数值逼近也在相应的参数rh处具有Hopf分支,并且rh=r0+o(h).

The numerical approximation of a Lienard equotion with finite delay is discussed. Firstly, regarding the delay as a parameter and employing the method of Cooke and J. Hale ,the conditions to the existence of Hopf bifurcation at some valus of the delay are given. Then ,the dalay deference equation obtained by using Euler method is written as a map. According to the theories of bifurcation for discrete dynamical systems,the conditions to the existence of Hopf bifurcation for numerical approximation are given. The relations of Hopf bifurcation between the continuous and the discrete are discussed. That when this model has Hopf bifurcation ,the numerical approximation also has Hopf bifurcation is proved.