一类具时滞同源Nicholson微分方程的Hopf分支的数值逼近

Hopf Bifurcation in Numerical Approximation for a Nicholson Differential Equation with Delay

研究一类具时滞同源的Nicholson微分方程模型的Hopf分支的数值逼近问题.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;利用欧拉方法将得到的差分方程表示为映射,给出差分方程Hopf分支存在的条件及连续模型与其数值逼近间的关系.证明该模型在τ=τ0产生Hopf分支时,其数值逼近也在相应的参数τh处具有Hopf分支,并且τh=τ0+o(h)。

The numerical approximation of a Nicholson differential equation with 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 at τ=τ0, the numerical approximation also has Hopf bifurcation at τh=t0-0（h） is proved.