Preprint des Projektes: SFB 343: Diskrete Strukturen in der Mathematik

Theorie und Numerik von Aufgaben der linearen Algebra und diskreter dynamischer Systeme

95-012 Wolf-Jürgen Beyn, Jan-Martin Kleinkauf.
The Numerical Computation of Homoclinic Orbits for Maps

Transversal homoclinic orbits of maps are known to generate shift dynamics on a set with Cantor like structure. In this paper a numerical method is developed for computation of the corresponding homoclinic orbits. They are approximated by finite orbit segments subject to asymptotic boundary conditions. We provide a detailed error analysis including a shadowing type result by which one can infer the existence of a transversal homoclinic orbit from a finite segment. This approach is applied to several examples. In some of them parameters appear and closed loops of homoclinic orbits are found by a path-following algorithm.