ELSEVIER Physica D 85 (1995) 93-125 PHYSICA The singularity analysis for nearly integrable systems" homoclinic intersections and local multivaluedness Alain Goriely a'b'l, Michael Tabor a "University of Arizona, Program in Applied Mathematics, Building #89, Tucson, AZ 85721, USA bUniversitO Libre de Bruxelles, Service de Physique Statistique, Campus Plaine CP231, 1050 Brussels, Belgium Received 10 February 1994; revised 5 May 1994; accepted 7 May 1994 Communicated by H. Flaschka Abstract 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. 1. Introduction More than a century ago Henri Poincar6 and Paul Painlev6 introduced new concepts in order to study nonlinear systems of differential equations. In his study of celestial mechanics, Poincar6 developed the geometric picture of phase space. His idea was to study the asymptotic solutions as a geometric set which define the global qualitative behavior of solutions in the long time limit. Poincar6 also introduced the concept of homoclinic and heteroclinic orbits which connect fixed points to themselves. He showed that perturbations of these orbits was the source of complex behaviors in dynamical systems [1]. In the meantime, Painlev6 and co-workers found that nonlinear differential equations can be classified according to the types of singularities exhibited by the solutions in the complex plane. Painlev6 was specifically interested in the simplest case where the solution has no movable critical points (i.e. branch points whose location depend upon the initial conditions and around which the solution is multi-valued), the so-called Painlev~ property [2]. As a consequence he defined new transcendents related to the solution of certain second-order differential equations which are character- ized by the Painlev6 property. 1E-mail address: agoriel@ulb.ac.be 0167-2789/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0167-2789(94)00137-F