Abstract: The computation of manifold plays an important role in understanding the dynamics of a nonlinear system. This paper presented a revised algorithm for computing two dimensional manifolds of vector fields. The trajectory was grown by solving appropriate initial value problem, then the manifold was grown step by step with equal arc-length along the trajectory and points on the trajectory were selected according to the local curvature. For the first time, both curvature-control technique and distance constraints were employed when interpolation between mesh points was needed. And in this way, the accuracy of interpolated points was guaranteed and it was the main advantage over the existing methods. The simulation results show that the proposed algorithm is effective.

Key words: dynamical system, invariant manifold, hyperbolic, Lorenz manifold

摘要： 提出了一种改进的向量场二维流形计算方法。新算法通过解初值问题来计算轨道,然后等轨道弧长向外扩展来增长流形,保证了计算的快速性;应用曲率控制技术实现了轨道上的离散网格点的优化分布;在进行网格点插值时综合运用了距离控制和曲率控制,用重新计算轨道的方法来确定插值点的位置,一定程度上克服了原二维流形计算方法的不能保证插值点精度的弱点。仿真结果也表明,新算法能够很好地应用于二维稳定流形的计算。

关键词: 动力系统, 不变流形, 双曲, 洛伦兹流形