Geometric analysis of fast-slow PDEs with fold singularities via Galerkin discretisation

TL;DR

This paper investigates the geometric structure of fast-slow PDE systems with fold singularities using Galerkin discretisation, combining spectral methods with geometric singular perturbation theory to analyze invariant manifolds and their passage through singularities.

Contribution

It extends classical fold analysis to PDEs by integrating Galerkin discretisation with geometric singular perturbation techniques, revealing invariant slow manifolds and their behaviour near singularities.

Findings

Existence of invariant slow manifolds in PDE phase space.

Description of manifold passage through fold singularities via blow-up.

Relation between Galerkin discretised manifolds and original slow manifolds.

Abstract

We study a singularly perturbed fast-slow system of two partial differential equations (PDEs) of reaction-diffusion type on a bounded domain via Galerkin discretisation. We assume that the reaction kinetics in the fast variable realise a generic fold singularity, whereas the slow variable takes the role of a dynamic bifurcation parameter, thus extending the classical analysis of the singularly perturbed fold. Our approach combines a spectral Galerkin discretisation with techniques from geometric singular perturbation theory which are applied to the resulting high-dimensional systems of ordinary differential equations. In particular, we show the existence of invariant slow manifolds in the phase space of the original system of PDEs away from the fold singularity, while the passage past the singularity of the Galerkin manifolds obtained after discretisation is described by geometric desingularisation, or blow-up. Finally, we discuss the relation between these Galerkin manifolds and the underlying slow manifolds.