Convergence estimates for the numerical approximation of homoclinic solutions

Abstract

This article is concerned with the numerical computation of homoclinic solutions converging to a hyperbolic or semi-hyperbolic equilibrium of a system u = f(u, μ). The approximation is done by replacing the original problem with a boundary value problem on a finite interval and introducing an additional phase condition to make the solution unique. Numerical experiments have indicated that the parameter μ is much better approximated than the homoclinic solution. This was proved in Schecter (1995 IMA J. Numer Anal. 15, 23–60) for phase conditions satisfying an additional ‘niceness’ assumption, which is unfortunately not satisfied for the phase condition most commonly used in numerical experiments and which actually suggested the super-convergence result. Here, this result is proved for arbitrary phase conditions. Moreover, it is shown that it suffices to approximate the original boundary value problem to first order when considering semi-hyperbolic equilibria, extending a result of Schecter (1993 SIAM J. Numer Anal. 30, 1155–78).