Title: Growing two-dimensional manifold of nonlinear maps based on generalized Foliation condition
Abstract:In this paper we present an algorithm of computing two-dimensional (2D) stable and unstable manifolds of hyperbolic fixed points of nonlinear maps. The 2D manifold is computed by covering it with orbi...In this paper we present an algorithm of computing two-dimensional (2D) stable and unstable manifolds of hyperbolic fixed points of nonlinear maps. The 2D manifold is computed by covering it with orbits of one-dimensional (1D) sub-manifolds. A generalized Foliation condition is proposed to measure the growth of 1D sub-manifolds and eventually control the growth of the 2D manifold along the orbits of 1D sub-manifolds in different directions. At the same time, a procedure for inserting 1D sub-manifolds between adjacent sub-manifolds is presented. The recursive procedure resolves the insertion of new mesh point, the searching for the image (or pre-image), and the computation of the 1D sub-manifolds following the new mesh point tactfully, which does not require the 1D sub-manifolds to be computed from the initial circle and avoids the over assembling of mesh points. The performance of the algorithm is demonstrated with hyper chaotic three-dimensional (3D) Hnon map and Lorenz system.Read More