Discrete geometric singular perturbation theory

Abstract

<p style='text-indent:20px;'>We propose a mathematical formalism for discrete multi-scale dynamical systems induced by maps which parallels the established <i>geometric singular perturbation theory</i> for continuous-time fast-slow systems. We identify limiting maps corresponding to both 'fast' and 'slow' iteration under the map. A notion of normal hyperbolicity is defined by a spectral gap requirement for the multipliers of the fast limiting map along a critical fixed-point manifold <inline-formula><tex-math id="M1">\begin{document}$ S $\end{document}</tex-math></inline-formula>. We provide a set of Fenichel-like perturbation theorems by reformulating pre-existing results so that they apply near compact, normally hyperbolic submanifolds of <inline-formula><tex-math id="M2">\begin{document}$ S $\end{document}</tex-math></inline-formula>. The persistence of the critical manifold <inline-formula><tex-math id="M3">\begin{document}$ S $\end{document}</tex-math></inline-formula>, local stable/unstable manifolds <inline-formula><tex-math id="M4">\begin{document}$ W^{s/u}_{loc}(S) $\end{document}</tex-math></inline-formula> and foliations of <inline-formula><tex-math id="M5">\begin{document}$ W^{s/u}_{loc}(S) $\end{document}</tex-math></inline-formula> by stable/unstable fibers is described in detail. The practical utility of the resulting <i>discrete geometric singular perturbation theory (DGSPT)</i> is demonstrated in applications. First, we use DGSPT to identify singular geometry corresponding to excitability, relaxation, chaotic and non-chaotic bursting in a map-based neural model. Second, we derive results which relate the geometry and dynamics of fast-slow ODEs with non-trivial time-scale separation and their Euler-discretized counterpart. Finally, we show that fast-slow ODE systems with fast rotation give rise to fast-slow Poincaré maps, the geometry and dynamics of which can be described in detail using DGSPT.</p>

Pitchfork bifurcation along a slow parameter ramp: Coherent structures in the critical scaling

“…It would also be of interest how pattern-forming phenomena arising from slow spatial ramps compare with those explored in temporally dynamic bifurcations (see, e.g., Refs. [22][23][24].…”