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

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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
Publication statusPublished - 13 Jul 2022


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


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