基于Foliation条件的离散动力系统二维流形计算

研究离散动力系统双曲不动点的二维流形计算,利用不变流形轨道上Jacobian矩阵能够传递导数这一特殊性质,提出一种新的一维流形计算方法,通过预测-校正两个步骤迅速确定流形上新网格点,避免重复计算,并简化精度控制条件.在此基础上,将基于流形面Foliation条件进行推广,推广后的Foliation条件能够控制二维流形上的一维子流形的增长速度,从而实现二维流形在各个方向上的均匀增长.此外,算法可以同时用于二维稳定和不稳定流形的计算.以超混沌三维Hénon映射和具有蝶形吸引子的Lorenz系统为例验证了算法的有效性.

Growing 2D Manifold of Discrete Dynamical System Based on Foliation Condition

An algorithm for computing 2D stable and unstable manifolds of hyperbolic fixed points of discrete dynamical systems is shown. With the fact that Jacobian transports derivative along orbit of an invariant manifold, an algorithm for computing 1D manifold is proposed. The mesh point is located with a Prediction-Correction scheme which reduces searching time and at the same time gives rise to a simplified accuracy condition. 2D manifold is computed by covering it with orbits of 1D sub-manifold. A generalized Foliation condition is used to guarantee that 2D manifold is growing equally along orbits of 1D sub-manifold in different directions. Performance of the algorithm is demonstrated with hyper chaotic 3D Hénon map and Lorenz system.