# Unfolding symmetric Bogdanov-Takens bifurcations for front dynamics in a reaction-diffusion system

## TL;DR

This paper analyzes symmetric Bogdanov-Takens bifurcations in a reaction-diffusion system, revealing complex front dynamics and stable periodic motions through advanced spectral analysis and normal form computations.

## Contribution

It proves the triple zero eigenvalue forms a Jordan chain, simplifies the center manifold reduction, and demonstrates unfolding of symmetric Bogdanov-Takens bifurcation in the model.

## Key findings

- Confirmed triple zero eigenvalue forms a Jordan chain of length three.
- Simplified the center manifold reduction using the Evans function.
- Proved unfolding of symmetric Bogdanov-Takens bifurcation leading to stable periodic fronts.

## Abstract

This manuscript 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 stable periodic front motion, including stable traveling breathers, and these results are illustrated by numerical computations.