The Numerical Computation of Connecting Orbits in Dynamical Systems
The Numerical Computation of Connecting Orbits in Dynamical Systems
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DOI：
10.1093/imanum/10.3.379
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发表时间：
1990-07
影响因子：
2.1
通讯作者：
W. Beyn
中科院分区：
文献类型：
--
作者：
W. Beyn
动力系统的结构变化通常与连接两个驻点（异宿或同宿）的轨道的出现或消失有关。同宿轨道通常出现在单参数问题中，当在周期解的分支上周期趋于无穷大时（例如Guckenheimer & Holmes，1983）。我们开发了一个直接的数值方法计算的连接轨道及其相关的参数值。我们采用了一般的相位条件，并通过在两端使用渐近边界条件的技巧，将边值问题截断到有限区间;例如，见de Hoog &韦斯（1980），伦蒂尼& Keller（1980）。这种截断所造成的近似误差呈指数衰减。基于此分析和额外的数值研究，我们建立了一个自适应的策略选择截断区间。
Structural changes in dynamical systems are often related to the appearance or disappearance of orbits connecting two stationary points (either heteroclinic or homoclinic). Homoclinic orbits typically arise in one-parameter problems when on a branch of periodic solutions the periods tend to infinity (e.g. Guckenheimer & Holmes, 1983). We develop a direct numerical method for the computation of connecting orbits and their associated parameter values. We employ a general phase condition and truncate the boundary-value problem to a finite interval by using on both ends the technique of asymptotic boundary conditions; see, for example, de Hoog & Weiss (1980), Lentini & Keller (1980). The approximation error caused by this truncation is shown to decay exponentially. Based on this analysis and additional numerical investigations we set up an adaptive strategy for choosing the truncation interval.