Geometric analysis of fast-slow PDEs with fold singularities

Maximilian Engel, Felix Hummel, Christian Kuehn, Nikola Popović, Mariya Ptashnyk, Thomas Zacharis

Research output: Working paperPreprint

74 Downloads (Pure)

Abstract

We study a singularly perturbed fast-slow system of two partial differential equations (PDEs) of reaction-diffusion type on a bounded domain. We assume that the reaction terms in the fast variable contain a fold singularity, whereas the slow variable assumes the role of a dynamic bifurcation parameter, thus extending the classical analysis of a fast-slow dynamic fold bifurcation to an infinite-dimensional setting. Our approach combines a spectral Galerkin discretisation with techniques from Geometric Singular Perturbation Theory (GSPT) which are applied to the resulting high-dimensional systems of ordinary differential equations (ODEs). In particular, we show the existence of invariant manifolds away from the fold singularity, while the dynamics in a neighbourhood of the singularity is described by geometric desingularisation, via the blow-up technique. Finally, we relate the Galerkin manifolds that are obtained after the discretisation to the invariant manifolds which exist in the phase space of the original system of PDEs.
Original languageEnglish
PublisherarXiv
Publication statusPublished - 13 Jul 2022

Keywords

  • math.AP
  • math.DS
  • 35B25 (Primary), 34D15, 34E13

Fingerprint

Dive into the research topics of 'Geometric analysis of fast-slow PDEs with fold singularities'. Together they form a unique fingerprint.

Cite this