Chaotic motion and singularity structures of front solutions in multi-component FitzHugh-Nagumo-type systems

TL;DR

This paper analyzes the complex dynamics of front solutions in multi-component reaction-diffusion systems, revealing how coupling and system dimension influence singularity structures and chaotic behaviors, supported by rigorous analysis and numerical simulations.

Contribution

It demonstrates how to control and unfold singularity structures and chaos in multi-component reaction-diffusion systems using geometric and analytical methods.

Findings

Control of front motion via coupling functions and system dimension.

Unfolding of scalar singularities and chaotic behavior for N≥3.

Numerical validation guided by analytical results.

Abstract

We study the dynamics of front solutions in a certain class of multi-component reaction-diffusion systems, where one fast component governed by an Allen-Cahn equation is weakly coupled to a system of $N$ linear slow reaction-diffusion equations. By using geometric singular perturbation theory, Evans function analysis and center manifold reduction, we demonstrate that and how the complexity of the front motion can be controlled by the choice of coupling function and the dimension $N$ of the slow part of the multi-component reaction-diffusion system. On the one hand, we show how to imprint and unfold a given scalar singularity structure. On the other hand, for $N\geq 3$ we show how chaotic behaviour of the front speed arises from the unfolding of a nilpotent singularity via the breaking of a Shil'nikov homoclinic orbit. The rigorous analysis is complemented by a numerical study that is heavily guided by our analytic findings.