The hyperbolic umbilic singularity in fast-slow systems

TL;DR

This paper provides a detailed local analysis of hyperbolic umbilic singularities in fast-slow systems with three slow variables, revealing how slow manifolds behave near these singularities using blow-up techniques.

Contribution

It introduces a novel application of blow-up methods to analyze hyperbolic umbilic singularities in fast-slow systems with gradient structure, connecting to catastrophe theory.

Findings

Attracting slow manifolds jump onto the fast regime near the singularity.

Slow manifolds fan out as they cross the hyperbolic umbilic point.

The analysis relates classical catastrophe theory to fast-slow dynamics.

Abstract

Fast-slow systems with three slow variables and gradient structure in the fast variables have, generically, hyperbolic umbilic, elliptic umbilic or swallowtail singularities. In this article we provide a detailed local analysis of a fast-slow system near a hyperbolic umbilic singularity. In particular, we show that under some appropriate non-degeneracy conditions on the slow flow, the attracting slow manifolds jump onto the fast regime and fan out as they cross the hyperbolic umbilic singularity. The analysis is based on the blow-up technique, in which the hyperbolic umbilic point is blown up to a 5-dimensional sphere. Moreover, the reduced slow flow is also blown up and embedded into the blown-up fast formulation. Further, we describe how our analysis is related to classical theories such as catastrophe theory and constrained differential equations.