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

Maikel M. Bosschaert, Yuri A. Kuznetsov

Research output: Contribution to journalArticleAcademicpeer-review

Abstract

This paper provides for the first time correct third-order homoclinic predictors in n-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-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 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-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.

Original languageEnglish
Pages (from-to)410-439
Number of pages30
JournalSIAM journal on applied dynamical systems
Volume23
Issue number1
DOIs
Publication statusPublished - 2024

Keywords

  • n/a OA procedure
  • Center manifold reduction
  • Homoclinic asymptotics
  • Bogdanov-Takens bifurcation

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