Bifurcation to instability through the lens of the Maslov index

TL;DR

This paper applies a recently developed Maslov index framework to analyze the stability of standing wave solutions in a three-component activator-inhibitor model, providing new insights into instability mechanisms.

Contribution

It demonstrates how the Maslov index framework can be used to understand bifurcations to instability in singularly perturbed systems.

Findings

Maslov index correlates with stability changes in the model

New insight into the mechanism of instability in activator-inhibitor systems

Potential to identify previously unknown instabilities

Abstract

The Maslov index is a powerful tool for assessing the stability of solitary waves. Although it is difficult to calculate in general, a framework for doing so was recently established for singularly perturbed systems. In this paper, we apply this framework to standing wave solutions of a three-component activator-inhibitor model. These standing waves are known to become unstable as parameters vary. Our goal is to see how this established stability criterion manifests itself in the Maslov index calculation. In so doing, we obtain new insight into the mechanism for instability. We further suggest how this mechanism might be used to reveal new instabilities in singularly perturbed models.