Computing the Maslov index from singularities of a matrix Riccati equation

TL;DR

This paper links the Maslov index to stability analysis of reaction-diffusion equations by analyzing singularities in a matrix Riccati equation, providing a new method to detect eigenvalue changes.

Contribution

It introduces a novel approach to compute the Maslov index via singularities in a Riccati equation, connecting geometric and spectral stability analysis.

Findings

Singularities in the Riccati solution correspond to changes in the Maslov index.

The change in Maslov index equals the net count of eigenvalues diverging to infinity.

The method offers a new tool for stability analysis of reaction-diffusion systems.

Abstract

We study the Maslov index as a tool to analyze stability of steady state solutions to a reaction-diffusion equation in one spatial dimension. We show that the path of unstable subspaces associated to this equation is governed by a matrix Riccati equation whose solution $S$ develops singularities when changes in the Maslov index occur. Our main result proves that at these singularities the change in Maslov index equals the number of eigenvalues of $S$ that increase to $+\infty$ minus the number of eigenvalues that decrease to $-\infty$.