We present an algorithm to compute the one-dimensional stable manifold of a saddle point for a planar map. In contrast to current standard techniques, it is not necessary to know the inverse or approximate it by using Newton's method. Rather than using the inverse, the manifold is grown starting from the linear eigenspace near the saddle point by adding a point that maps back onto an earlier segment of the stable manifold. The performance of the algorithm is compared to other methods using an example where the inverse map is known explicitly. The strengths of our method are illustrated with examples of a stable manifold that is not simply connected and a piecewise-smooth model of a real life mechanical system. The algorithm has been implemented for use in the DsTool environment and is available to download with this paper.
Additional information: Later published by Society for Industrial and Applied Mathematics (2003), SIAM Journal on Applied Dynamical Systems, 3(2), pp. 161-190, ISSN 1536-0040
Sponsorship: We thank Robert Szalai and Gabor Stepan from the
Budapest University of Technology and Economics for their helpful discussions and assistance with the highly interrupted cutting map.
JE was supported by grant GR/R94572/01 from the Engineering and Physical Sciences Research Council (EPSRC).
- piecewise smooth
- planar map
- discrete-time system
- stable manifold