Numerical Computation and Continuation of Invariant Manifolds Connecting Fixed Points

Abstract

A numerical method for the computation of an invariant manifold that connects two fixed points of a vector field in $\mathbb{R}^n $ is given, extending the results of an earlier paper [Comput. Appl. Math., 26 (1989), pp. 159’170] by the authors. Basically, a boundary value problem on the real line is truncated to a finite interval. The method applies, in particular, to the computation of heteroclinic orbits. The emphasis is on the systematic computation of such orbits by continuation. Using the fact that the linearized operator of our problem is Fredholm in appropriate Banach spaces, the general theory of approximation of nonlinear problems is employed to show that the errors in the approximate solution decay exponentially with the length of the approximating interval. Several applications are considered, including the computation of traveling wave solutions to reaction diffusion problems. Computations were done using the software package AUTO.