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Unfolding Symmetric Bogdanov–Takens Bifurcations for Front Dynamics in a Reaction–Diffusion System

  • M. Chirilus-Bruckner
  • P. van Heijster
  • H. Ikeda
  • J. D. M. RademacherEmail author
Article

Abstract

This paper extends the analysis of a much studied singularly perturbed three-component reaction–diffusion system for front dynamics in the regime where the essential spectrum is close to the origin. We confirm a conjecture from a preceding paper by proving that the triple multiplicity of the zero eigenvalue gives a Jordan chain of length three. Moreover, we simplify the center manifold reduction and computation of the normal form coefficients by using the Evans function for the eigenvalues. Finally, we prove the unfolding of a Bogdanov–Takens bifurcation with symmetry in the model. This leads to the appearance of stable periodic front motion, including stable traveling breathers, and these results are illustrated by numerical computations.

Keywords

Three-component reaction–diffusion system Front solution Singular perturbation theory Evans function Center manifold reduction Normal forms 

Mathematics Subject Classification

35C07 37L10 35K57 34D15 

Notes

Acknowledgements

PvH thanks Leiden University for its hospitality. JR notes this paper is a contribution to project M2 of the Collaborative Research Centre TRR 181 ‘Energy Transfer in Atmosphere and Ocean’ funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)—Project Number 274762653. The authors also acknowledge that a crucial part of this paper was established during the first and second joint Australia–Japan workshop on dynamical systems with applications in life sciences.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Mathematisch InstituutLeiden UniversityLeidenThe Netherlands
  2. 2.Mathematical Sciences SchoolQueensland University of TechnologyBrisbaneAustralia
  3. 3.Department of MathematicsUniversity of ToyamaToyamaJapan
  4. 4.Fachbereich MathematikUniversität BremenBremenGermany

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