COMPUTATION AND CONTINUATION OF HOMOCLINIC AND HETEROCLINIC ORBITS WITH ARCLENGTH PARAMETERIZATION

In this paper, we study a numerical method for the computation and con tinuation of homoclinic and heteroclinic orbits based upon the arcleng th parameterization of the orbits. Unlike most other methods, this met hod utilizes the geometric structure of the homoclinic and heteroclini c orbits and does not require solving a boundary value problem (BVP) o n an infinite interval. However, the BVP formulated by this method can have a singularity at the end of the domain, and thus we introduce a special collocation method to handle such a singularity. We discuss th e convergence properties of our collocation method and the implementat ion of the method which uses the software AUTO. For several examples w e show that the arclength parameterization compares very favorably wit h the other numerical methods, although there are some limitations in the Sil'nikov case.