Slow-fast systems with an equilibrium near the folded slow manifold

TL;DR

This paper analyzes a slow-fast dynamical system with a fold in the slow manifold and an equilibrium nearby, deriving a normal form, studying bifurcations, and applying results to a generalized FitzHugh-Nagumo model.

Contribution

It introduces a normal form for systems with a fold and nearby equilibrium, and provides asymptotic analysis of bifurcations in such systems.

Findings

Derived a normal form near the fold-equilibrium pair.

Obtained an asymptotic formula for the Poincaré map.

Calculated parameter values for the first period-doubling bifurcation.

Abstract

We study a slow-fast system with two slow and one fast variables. We assume that the slow manifold of the system possesses a fold and there is an equilibrium of the system in a small neighbourhood of the fold. We derive a normal form for the system in a neighbourhood of the pair "equilibrium-fold" and study the dynamics of the normal form. In particular, as the ratio of two time scales tends to zero we obtain an asymptotic formula for the Poincar\'e map and calculate the parameter values for the first period-doubling bifurcation. The theory is applied to a generalization of the FitzHugh-Nagumo system.