Numerical analysis and accurate computation of heteroclinic orbits in the case of center manifolds

Abstract

In earlier papers (see the preceding paper and the references there), Doedel and the author have developed a numerical method for the computation of branches of heteroclinic orbits for a system of autonomous ordinary differential equations in ℝ n in the case that the solution approaches the fixed points exponentially. The idea of the method is to reduce a boundary value problem on the real line to a boundary value problem on a finite interval by using linear approximation of the unstable and stable manifolds. Using the fact that the linearized operator of the problem is Fredholm in Banach spaces with exponential weights, the authors employed the general theory of approximation of nonlinear problems to show that the errors in the approximate solution decay exponentially with the length of the approximating interval. In this paper we extend the analysis in the preceding paper to the case of center manifolds which requires the refinement of the analysis in the preceding paper. The algorithm is applied to a model problem: the DC Josephson Junction. Computations are done using the software package AUTO.