具有限时滞Van der pol方程Hopf分支的数值逼近

Hopf Bifurcation in Numerical Approximation for Van der pol Equation with Finite Delay

研究了欧拉方法对以滞量为参数的具有Hopf分支的Van der pol方程的数值逼近问题。首先,利用欧拉方法将得到的时滞差分方程表示为映射,然后以滞量为分支参数,利用离散动力系统的分支理论,在Van der pol方程具有Hopf分支的条件下,给出了差分方程Hopf分支存在的条件及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 Van der pol equation is considerd using delay as parameter.the Delay deference equation obtained by using Euler method is written as a map.Accoding 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.That when the Van der pol equation has Hopf bifurcations at r=r0,the numerical approximation also has Hopf bifurcations at rh=r0＋o（h） is proved.