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

Maikel Bosschaert; Yu.A. Kuznetsov

doi:10.1137/22M151354X

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

Maikel Bosschaert^*, Yu.A. Kuznetsov

^*Corresponding author for this work

Research output: Contribution to journal › Article › Academic › peer-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 language	English
Pages (from-to)	410-439
Number of pages	30
Journal	SIAM Journal on Applied Dynamical Systems
Volume	23
Issue number	1
DOIs	https://doi.org/10.1137/22M151354X
Publication status	Published - 25 Jan 2024

Bibliographical note

Publisher Copyright:
© 2024 Society for Industrial and Applied Mathematics.

Funding

Acknowledgments. The authors would like to thank Prof. Peter De Maesschalck (Hasselt University, Belgium) for multiple useful discussions during this research project, Prof. Wolf-J\" urgen Beyn (Bielefeld University, Germany) for his positive comments on the preprint, and Dr. Hil Meijer (University of Twente, The Netherlands) for multiple suggestions leading to a significant improvement of the paper. Additionally, we would like to express our gratitude to the two anonymous referees for very useful remarks and suggestions.

Funders	Funder number
University of Twente
Universität Bielefeld
Universiteit Hasselt

Keywords

Bogdanov-Takens bifurcation
center manifold reduction
homoclinic asymptotics

Access to Document

10.1137/22M151354X

22m151354xFinal published version, 1.17 MBLicence: Taverne

Cite this

@article{6897945052904c74b3714a9957b639fa,

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

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.",

keywords = "Bogdanov-Takens bifurcation, center manifold reduction, homoclinic asymptotics",

author = "Maikel Bosschaert and Yu.A. Kuznetsov",

note = "Publisher Copyright: {\textcopyright} 2024 Society for Industrial and Applied Mathematics.",

year = "2024",

month = jan,

day = "25",

doi = "10.1137/22M151354X",

language = "English",

volume = "23",

pages = "410--439",

journal = "SIAM Journal on Applied Dynamical Systems",

issn = "1536-0040",

publisher = "Society for Industrial and Applied Mathematics Publications",

number = "1",

}

TY - JOUR

T1 - Interplay between Normal Forms and Center Manifold Reduction for Homoclinic Predictors near Bogdanov-Takens Bifurcation

AU - Bosschaert, Maikel

AU - Kuznetsov, Yu.A.

PY - 2024/1/25

Y1 - 2024/1/25

N2 - 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.

AB - 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.

KW - Bogdanov-Takens bifurcation

KW - center manifold reduction

KW - homoclinic asymptotics

UR - http://www.scopus.com/inward/record.url?scp=85184064927&partnerID=8YFLogxK

U2 - 10.1137/22M151354X

DO - 10.1137/22M151354X

M3 - Article

SN - 1536-0040

VL - 23

SP - 410

EP - 439

JO - SIAM Journal on Applied Dynamical Systems

JF - SIAM Journal on Applied Dynamical Systems

IS - 1

ER -

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

Abstract

Bibliographical note

Funding

Keywords

Access to Document

Other files and links

Fingerprint

Cite this