Extending Melnikov theory to invariant manifolds on non-compact domains

Abstract

Melnikov theory is usually examined under the hypothesis of hyperbolicity of critical points. We are extending this theory for regular perturbation problems of autonomous systems in arbitrary dimensions to the case of non-hyperbolic critical points that are located at infinity. The heteroclinic orbit of the unperturbed problem connecting these non-hyperbolic equilibria is of at most algebraic growth. By using a weaker dichotomy property than in the classical approach we obtain by a Lyapunov-Schmidt reduction a bifurcation equation that describes the existence of solutions of at most algebraic growth under small perturbations. We apply this extended Melnikov theory to problems arising in singularly perturbed systems to prove the existence of 'canard solutions'.