# Geometric analysis of fast-slow PDEs with fold singularities via Galerkin discretisation Source: [Geometric analysis of fast-slow PDEs with fold singularities via Galerkin discretisation]() Collection index: [llms.txt]() Data updated: 2025-10-25T17:37:24.753Z Authors: Maximilian Engel, Felix Hummel, Christian Kuehn, Nikola Popovi\\'c, Mariya Ptashnyk, Thomas Zacharis Published: 2022-07-13T11:40:09.000Z ## 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. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. ## Original sources and implementation [Original paper]() [PDF]()