Interplay between Normal Forms and Center Manifold Reduction for Homoclinic Predictors near Bogdanov–Takens Bifurcation

Abstract

SIAM Journal on Applied Dynamical Systems, Volume 23, Issue 1, Page 410-439, March 2024.
Abstract.This paper provides for the first time correct third-order homoclinic predictors in [math]-dimensional ODEs near a generic Bogdanov–Takens bifurcation point, which can be used to start the numerical continuation of the appearing homoclinic orbits. To achieve this, higher-order time approximations to the nonlinear time transformation in the Lindstedt–Poincaré method are essential. Moreover, a correct transform between approximations to solutions in the normal form and approximations to solutions on the parameter-dependent center manifold is derived rigorously. A detailed comparison is done between applying different normal forms (smooth and orbital), different phase conditions, and different perturbation methods (regular and Lindstedt–Poincaré) to approximate the homoclinic solution near Bogdanov–Takens points. Examples demonstrating the correctness of the predictors are given. The new homoclinic predictors are implemented in the open-source MATLAB/GNU Octave continuation package MatCont.