Improved homoclinic predictor for Bogdanov-Takens bifurcation

Abstract

An improved homoclinic predictor at a generic codim 2 Bogdanov-Takens (BT) bifucation is derived. We use the classical ‘blow-up’ technique to reduce the canonical smooth normal form near a generic BT bifurcation to a perturbed Hamiltonian system. With a simple perturbation method, we derive explicit first-and second-order corrections of the unperturbed homoclinic orbit and parameter value. To obtain the normal form on the center manifold, we apply the standard parameter-dependent center manifold reduction combined with the normalization, that is based on the Fredholm solvability of the homological equation. By systematically solving all linear systems appearing from the homological equation, we remove an ambiguity in the parameter transformation existing in the literature. The actual implementation of the improved predictor in MatCont and numerical examples illustrating its efficiency are discussed.