The singularity analysis for nearly integrable systems: homoclinic intersections and local multivaluedness

In this study, a new perturbative scheme for nonintegrable ordinary differential equations is proposed. These perturbative expansions are based on the singularity analysis of the unperturbed system and is performed in the neighborhood of its singularities. Under suitable conditions on the homoclinic structure of the unperturbed system, the Melnikov vector can be computed based on the knowledge of the Laurent expansions of the solutions. The existence of transverse homoclinic intersections is therefore explicitly related to the existence of critical points for the solutions in the complex plane of the independent variable.