Computation and Continuation of Homoclinic and Heteroclinic Orbits with Arclength Parameterization

In this paper, we study a numerical method for the computation and continuation of homoclinic and heteroclinic orbits based upon the arclength parameterization of the orbits. Unlike most other methods, this method utilizes the geometric structure of the homoclinic and heteroclinic orbits and does not require solving a boundary value problem on an innnite interval. However, the boundary value problem 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 the convergence properties of our collocation method and the implementation of the method which uses the software AUTO. For several examples we show that the arclength parameterization compares very favorably with the other numerical methods, although there are some limitations in the Sil'nikov case.