θ方法解滞时微分方程的动力学性质

本文研究求解滞时微分方程的θ-方法数值解的渐近性和方程真实解的关系。首先，我们把数值方法看成以步长为参数的动力系统，考察非线性滞时微分方程θ-方法的数值稳定性。并且证明了A－稳定的θ-方法是NP－稳定的。其次我们证明了θ-方法没有伪不动点，还研究了伪周期2解的存在性。最后我们给出一个例子说明了滞时微分方程θ-方法产生的伪周期2解是不稳定的。

Dynamics of θ Methods for Delay Differential Equations

In this paper we study the relationship between the asymptotic behavior of a numerical simulation by 6 method for delay differential equation and that of the true solution itself for fixed step sizes. The numerical method is viewed as a dynamical system in which the step size acts as a parameter. Numerical stability of 0 method for nonlinear delay differential equation is investigated and we prove that A-stable 9 methods are NP-stable. It is shown that a consistent 0 method does not admit spurious fixed points. The existence of spurious period 2 solution in the time-step is also studied. Finally we give a simple example to illustrate instability of the spurious period two solutions.