二维稳定流形的自适应推进算法

稳定和不稳定流形是研究动力系统全局特性的重要工具. 一般系统的稳定和不稳定流形的曲率在全局范围内会有明显变化,应根据流形曲率的变化采用不同尺寸的网格单元计算全局流形. 然而在现有二维流形算法中,流形网格单元的尺寸在全局范围内是统一的. 为持续有效地计算全局稳定流形,提高计算网格对流形曲率变化的适应性. 本文在偏微分方程算法的基础上提出一种二维稳定流形的自适应推进算法. 该算法的基本思想是根据稳定流形曲率的变化自适应地调整网格单元的尺寸. 该算法首先在系统的稳定特征子空间中确定稳定流形的一个初始估计,该初始估计的网格单元尺寸设置为初始大小. 然后根据稳定流形网格前沿的曲率特点自适应地产生新的备选网格单元,继而根据相切性条件更新备选点的坐标,并将距离平衡点最近的备选点接受为已知点,最后更新稳定流形网格的前沿并自适应地产生新的备选网格单元,通过这个迭代过程使流形网格自适应地向前推进. 本文算法通过引入流形单元尺寸自适应,成功实现了洛伦兹流形和类球面流形的计算,并与偏微分方程算法进行了对比,结果表明自适应推进算法的流形计算单元的尺寸可在全局范围内根据流形曲率自适应地调整. 利用自适应推进算法计算二维稳定流形,可实现稳定流形的自适应推进. 

ADAPTIVE FRONT ADVANCING ALGORITHM FOR COMPUTING TWO-DIMENSIONAL STABLE MANIFOLDS

The stable and unstable manifolds are important tools for studying the global characteristics of dynamical systems. The curvature of the stable and unstable manifolds of a general system varies significantly over the global range. Different sizes of simplexes should be used to compute global manifolds. However the size of the computational simplex is globally unified in the existing algorithms. To compute global stable manifolds continuously and efficiently and to improve the adaptability of computational grid simplex to the curvature variation of the manifolds. In this paper, an adaptive front advancing algorithm for computing two-dimensional stable manifolds is proposed based on the PDE method. The basic idea of this algorithm is to adaptively adjust the size of the grid simplex according to the change of the curvature of the stable manifold. First, an initial estimate of the stable manifolds is determined in the stable subspace of the system, and the grid simplex of the initial estimate is set to be the initial size. Then, the considered mesh simplexes are generated adaptively according to the geometric characteristics of the stable manifolds. The coordinates of the considered mesh points are updated according to the tangency conditions and the nearest considered mesh points to the equilibrium point is accepted. Finally, update the front of the stable manifold and adaptively generate new considered mesh simplex. Through this iterative process, the manifold grid is adaptively moved forward. In this paper, the Lorentz manifold and the sphere-like manifold are calculated by introducing the simplex size adaptiveness. Compared with the PDE method, the size of the manifold simplex of the adaptive front advancing algorithm can be adjusted adaptively according to the manifold curvature in the global range. Computing the two-dimensional stable manifold with the adaptive front advancing algorithm can realize adaptive advancement of the stable manifold.