尺度自适应准则计算二维流形

Adaptive Scale Criterion for Computation of Two-dimensional Manifolds

针对流形计算过程中各方向流间距变化不均衡而导致的插值难题,文中提出一种基于多阈值尺度自适应插值准则.该准则通过学习前一时刻的流形增长信息指导当前时刻的插值选取,每对相邻轨迹拥有自己的插值阈值.阈值的大小由前一时刻的流形环上的最小阈值和流形尺度增长情况决定.控制因子为流形尺度增长与最小阈值间的比例系数.控制因子的引入可以使阈值的变化更好的适应流形尺度变化,从而在不同几何尺度下构建流形.以洛伦兹系统中二维稳定流形的计算为例对插值准则进行了验证.实验表明,尺度自适应准则能够有效地学习流分离信息,并准确地为下一时刻准备插值网格点.

In order to solve the interpolation problem caused by the disequilibrium of the distances between adjacent flows in different directions,a multi-threshold criterion based adaptive scale criterion is proposed for the computation of two-dimensional manifolds. By studying previous distance changesbetween adjacent flows the criterion can guide the current interpolation and then the data for the next loop can be prepared. Each pair of adjacent flows has its own threshold and thresholds are determined by the scale of manifold and previous minimum threshold. The ratio of the increase of manifold scale to the minimum threshold is recorded as the control factor. The introduction of the control factor can make the changes of thresholds better adapt to the changes of manifold, and so the manifold can be constructed in different geometric scales. Computation of two-dimensional stable manifold in Lorenz system is taken as an example to demonstrate that adaptive scale criterion can effectively study the separation information of flows and precisely prepare interpolation points for next loop.