Pattern Formation in a Slowly Flattening Spherical Cap: Delayed Bifurcation

TL;DR

This paper analyzes how patterns form in a reaction-diffusion system on a spherical cap with slowly changing curvature, using centre manifold reduction to understand stability changes and bifurcations.

Contribution

It introduces a method to reduce a nonautonomous reaction-diffusion system on a curved domain to a normal form, capturing stability transitions due to domain evolution.

Findings

Derived evolving domain functions and quasi-patternless solutions

Computed normal form coefficients for stability analysis

Validated reduction solutions against numerical simulations

Abstract

This article describes a reduction of a nonautonomous Brusselator reaction-diffusion system of partial differential equations on a spherical cap with time dependent curvature using the method of centre manifold reduction. Parameter values are chosen such that the change in curvature would cross critical values which would change the stability of the patternless solution in the constant domain case. The evolving domain functions and quasi-patternless solutions are derived as well as a method to obtain this nonautonomous normal form. The coefficients of such a normal form are computed and the reduction solutions are compared to numerical solutions.