映射的二维不稳定流形计算

Computation for two-dimension unstable manifold of map

针对二维流形求解较困难的问题,提出一种新的离散映射系统二维不稳定流形的算法。该算法以成熟的数值算法为基础,首先通过求初值曲线计算均匀分布的一维子流形,再用三角形有限元逼近相邻一维子流形之间的流形面。计算一维子流形的关键思想是在流形面上找到与当前点相距合适步长的下一点,从而逐步增长流形,该步长根据当前点附近流形的弯曲程度调整。该算法不但可以快速求得流形的直观图像,而且能够准确地反映流形的变化过程。并用超混沌广义Hénon映射不动点的不稳定流形的计算验证了本算法的有效性,此外,通过计算出的直观流形图验证了稳定流形和不稳定流形的相交。

This paper presented a new algorithm for computing the global two-dimensional unstable manifolds of a fixed point of a map.On the basis of numerical computing,the algorithm firstly found the initial values of those sub manifolds by well distributed one-dimensional sub manifolds,then approximated the manifolds between sub-manifolds through triangle limited elements.The key idea was to find a new point on the manifolds in correct step to increase the manifolds.The value of step was limited by the curve.With this algorithm,we can get a picture of a manifold efficiently with much information of manifolds.The performance of the algorithm was demonstrated with hyperchaotic generalized Hénon map,and the intersection of stable and unstable manifolds was demonstrated.