Linear stability analysis for a system of singular amplitude equations arising in biomorphology

TL;DR

This paper conducts a detailed linear stability analysis of exponential periodic solutions in a complex system of singular amplitude equations relevant to biomorphology, establishing conditions for diffusive and nonlinear stability.

Contribution

It provides a two-parameter matrix perturbation analysis that proves necessary conditions for stability are also sufficient, linking stability to slow manifold dynamics and finite-time approximation.

Findings

Necessary conditions for stability are also sufficient.

Established diffusive and nonlinear stability results.

Connected stability analysis with slow manifold and finite-time approximation theories.

Abstract

We study linear stability of exponential periodic solutions of a system of singular amplitude equations associated with convective Turing bifurcation in the presence of conservation laws, as arises in modern biomorphology models, binary fluids, and elsewhere. Consisting of a complex Ginzburg-Landau equation coupled with a singular convection-diffusion equation in "mean modes" associated with conservation laws, these were shown previously by the authors to admit a constant-coefficient linearized stability analysis as in the classical Ginzburg-Landau case -- albeit now singular in wave amplitude epsilon -- yielding useful necessary conditions for stability, both of the exponential functions as solutions of the amplitude equations, and of the associated periodic pattern solving the underlying PDE. Here, we show by a delicate two-parameter matrix perturbation analysis that (strict) satisfaction of these necessary conditions is also sufficient for diffusive stability in the sense of Schneider, yielding a corresponding result, and nonlinear stability, for the underlying PDE. Moreover, we show that they may be interpreted as stability along a non-normally hyperbolic slow manifold approximated by Darcy-type reduction, together with attraction along transverse mean modes, connecting with finite-time approximation theorems of Hacker-Schneider-Zimmerman.