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.