Interplay between normal forms and center manifold reduction for homoclinic predictors near Bogdanov-Takens bifurcation

TL;DR

This paper develops accurate third-order homoclinic predictors near Bogdanov-Takens bifurcations in n-dimensional ODEs, integrating advanced normal form and center manifold techniques, and implements them in the MatCont software.

Contribution

It introduces the first correct third-order homoclinic predictors near Bogdanov-Takens points, combining higher-order time approximations and precise transformations.

Findings

Validated predictors with examples

Implemented in open-source MatCont package

Enhanced accuracy of homoclinic bifurcation analysis

Abstract

This paper provides for the first time correct third-order homoclinic predictors in $n$-dimensional ODEs near a generic Bogdanov-Takens bifurcation point. To achieve this, higher-order time approximations to the nonlinear time transformation in the Lindstedt-Poincar\'e 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 needed. A detailed comparison is done between applying different normal forms (smooth and orbital), different phase conditions, and different perturbation methods (regular and Lindstedt-Poincar\'e) 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.