Convergence estimates for the numerical approximation of homoclinic solutions

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; ). The approximation is done by replacing the original problem by a boundary value problem on a nite interval and introducing an additional phase condition to make the solution unique. Numerical experiments have indicated that the parameter is much better approximated than the homoclinic solution. This was proved in Schecter (1995) for phase conditions ful lling an additional 'niceness' assumption, which is unfortunately not satis ed 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 is su cient to approximate the original boundary value problem to rst order when considering semi-hyperbolic equilibria extending a result of Schecter (1993).