Entry-exit functions in fast-slow systems with intersecting eigenvalues

TL;DR

This paper investigates delayed loss of stability in certain fast-slow dynamical systems with intersecting eigenvalues, proposing new formulas for entry-exit functions where traditional methods fail.

Contribution

It introduces novel formulae for entry-exit functions in systems with intersecting eigenvalues, addressing limitations of existing approaches.

Findings

Identifies limitations of known entry-exit relations in specific systems.

Proposes new formulas for accurately predicting trajectory exit points.

Illustrates various qualitative behaviors in the studied class of systems.

Abstract

We study delayed loss of stability in a class of fast-slow systems with two fast variables and one slow one, where the linearisation of the fast vector field along a one-dimensional critical manifold has two real eigenvalues which intersect before the accumulated contraction and expansion are balanced along any individual eigendirection. That interplay between eigenvalues and eigendirections renders the use of known entry-exit relations unsuitable for calculating the point at which trajectories exit neighbourhoods of the given manifold. We illustrate the various qualitative scenarios that are possible in the class of systems considered here, and we propose novel formulae for the entry-exit functions that underlie the phenomenon of delayed loss of stability therein.