Extending Discrete Geometric Singular Perturbation Theory to Non-Hyperbolic Points

TL;DR

This paper extends discrete geometric singular perturbation theory to non-hyperbolic points using Takens embedding, enabling approximation of complex fast-slow map dynamics by vector fields with singularities.

Contribution

It introduces a method to approximate non-hyperbolic fast-slow map dynamics with vector fields, broadening the scope of geometric singular perturbation theory.

Findings

Approximation of reduced maps by time-one maps of vector fields.

Description of local dynamics near non-hyperbolic singularities.

Application to high-dimensional center manifolds with singularities.

Abstract

We extend the recently developed discrete geometric singular perturbation theory to the non-normally hyperbolic regime. Our primary tool is the Takens embedding theorem, which provides a means of approximating the dynamics of particular maps with the time-1 map of a formal vector field. First, we show that the so-called reduced map, which governs the slow dynamics near slow manifolds in the normally hyperbolic regime, can be locally approximated by the time-one map of the reduced vector field which appears in continuous-time geometric singular perturbation theory. In the non-normally hyperbolic regime, we show that the dynamics of fast-slow maps with a unipotent linear part can be locally approximated by the time-1 map induced by a fast-slow vector field in the same dimension, which has a nilpotent singularity of the corresponding type. The latter result is used to describe (i) the local dynamics of two-dimensional fast-slow maps with non-normally singularities of regular fold, transcritical and pitchfork type, and (ii) dynamics on a (potentially high dimensional) local center manifold in $n$-dimensional fast-slow maps with regular contact or fold submanifolds of the critical manifold. In general, our results show that the dynamics near a large and important class of singularities in fast-slow maps can be described via the use of formal embedding theorems which allow for their approximation by the time-1 map of a fast-slow vector field featuring a loss of normal hyperbolicity.