Weakly coupled two slow- two fast systems, folded node and mixed mode oscillationsM

TL;DR

This paper investigates mixed mode oscillations in weakly coupled slow/fast systems, identifying a key bifurcation mechanism (FSN II) that enables MMOs, with applications to FitzHugh-Nagumo models.

Contribution

It establishes the existence of a transcritical bifurcation (FSN II) in coupled slow/fast systems, providing conditions for MMOs in such models.

Findings

Existence of FSN II bifurcation in coupled systems.

Conditions on coupling for MMOs to occur.

Application to FitzHugh-Nagumo type systems.

Abstract

We study Mixed Mode Oscillations (MMOs) in systems of two weakly coupled slow/fast oscillators. We focus on the existence and properties of a folded singularity called FSN II that allows the emergence of MMOs in the presence of a suitable global return mechanism. As FSN II corresponds to a transcritical bifurcation for a desingularized reduced system, we prove that, under certain non-degeneracy conditions, such a transcritical bifurcation exists. We then apply this result to the case of two coupled systems of FitzHugh- Nagumo type. This leads to a non trivial condition on the coupling that enables the existence of MMOs.