Canards, folded nodes and mixed-mode oscillations in piecewise-linear slow-fast systems

TL;DR

This paper investigates how piecewise-linear slow-fast systems can replicate complex canard phenomena and mixed-mode oscillations typically observed in smooth systems, especially near folded-node singularities, and demonstrates their robustness.

Contribution

It extends the analysis of canard-induced MMOs to three-dimensional PWL systems, providing a bridge between smooth and piecewise-linear models with new theoretical insights.

Findings

PWL systems can reproduce folded singularities and canard phenomena.

The phase portraits of PWL and smooth systems are compatible.

A method to construct robust PWL MMOs with global return is demonstrated.

Abstract

Canard-induced phenomena have been extensively studied in the last three decades, both from the mathematical and from the application viewpoints. Canards in slow-fast systems with (at least) two slow variables, especially near folded-node singularities, give an essential generating mechanism for Mixed-Mode oscillations (MMOs) in the framework of smooth multiple timescale systems. There is a wealth of literature on such slow-fast dynamical systems and many models displaying canard-induced MMOs, in particular in neuroscience. In parallel, since the late 1990s several papers have shown that the canard phenomenon can be faithfully reproduced with piecewise-linear (PWL) systems in two dimensions although very few results are available in the three-dimensional case. The present paper aims to bridge this gap by analysing canonical PWL systems that display folded singularities, primary and secondary canards, with a similar control of the maximal winding number as in the smooth case. We also show that the singular phase portraits are compatible in both frameworks. Finally, we show on an example how to construct a (linear) global return and obtain robust PWL MMOs.