Homoclinic Orbits of Planar Maps: Asymptotics and Mel'nikov Functions

Abstract

Homoclinic orbits to saddle fixed points of planar diffeomorphisms generically imply complicated dynamics due to Smale Horseshoes. Such orbits can be computed only numerically, which is time-consuming. The aim of this project is to explore an alternative method to compute homoclinic orbits near degenerate fixed points of codimension 2 with a double multiplier 1. The method will be based on approximating the map near the bifurcation by the time-1-shift along orbits of a planar ODE and evaluating the Mel'nikov function along its homoclinic loop. Zeroes of this Mel'nikov function are intersections of the stable and unstable manifold and will approximate the homoclinic orbit. The main part of this project is devoted to the derivation of a prediction for both the homoclinic orbit and the homoclinic bifurcation curve in the case of the normal form of 1:1 resonance.