A Generalized Method for A Posteriori and A Priori Error Estimates for Homoclinic Orbits in Reversible Systems

Abstract

Coherent structures in pattern-forming systems, such as pulses and spikes, are often mathematically represented as homoclinic orbits. We here present a generalized method for finding such homoclinic solutions to 2nd-order ODEs using a posteriori data derived from approximate solutions to boundary value problems on truncated intervals. We then show in the opposite direction that the a priori existence of a non-degenerate homoclinic implies the existence of a family of solutions to a Dirichlet boundary-value problem, with an explicit lower bound on domain size necessary to obtain a solution. In each direction, we also provide explicit error estimates as a function of truncation error. We lastly apply our method to find a family of homoclinics in the Lengyel-Epstein system and compute a minimum domain size for the existence of finite-domain solutions using the a priori proof.