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

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.