Open AccessGrowing two-dimensional manifold of nonlinear maps based on generalized Foliation condition
Author(s) -
Huimin Li,
Yangyu Fan,
Sun Heng-Yi,
Jing Zhang,
Jian Meng
Publication year - 2012
Publication title -
wuli xuebao
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.199
H-Index - 47
ISSN - 1000-3290
DOI - 10.7498/aps.61.029501
Subject(s) - foliation (geology) , manifold (fluid mechanics) , computation , stable manifold , nonlinear system , point (geometry) , fixed point , chaotic , mathematics , topology (electrical circuits) , mathematical analysis , geometry , computer science , geology , algorithm , physics , combinatorics , paleontology , artificial intelligence , mechanical engineering , quantum mechanics , engineering , metamorphic rock
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.