Computing stable and unstable manifolds of typical chaotic maps

Homoclinic intersections are source of chaos for a map. It is convenient to determine whether a given map is chaotic or not by computing stable and unstable manifolds of its hyperbolic fixed point and observing if there are homoclinic intersections. A new algorithm is presented to compute one-dimensional stable and unstable manifolds of a map. Inspired by a unique property that derivative is transported along the orbit of one-dimensional manifold, position of new point is located quickly with a two-step "prediction and correction" scheme. Tangent component of the manifold is used as reference line to check if the new point is acceptable. Performance of the algorithm is demonstrated with several typical chaotic maps. It shows that the algorithm is capable of computing both one-dimensional stable and unstable manifolds of maps.