Jump-induced mixed-mode oscillations through piecewise-affine maps

TL;DR

This paper introduces a new class of slow-fast systems exhibiting mixed-mode oscillations driven by jump mechanisms, linking them to piecewise affine maps and providing a classification of oscillation signatures.

Contribution

It presents a novel family of three-dimensional systems with jump-induced MMOs and establishes their connection to one-dimensional piecewise affine maps, expanding understanding of MMO mechanisms.

Findings

Established correspondence between slow-fast systems and piecewise affine maps

Classified admissible mixed-mode oscillation signatures

Validated findings with numerical examples

Abstract

Mixed-mode oscillations (MMOs) are complex oscillatory patterns in which large-amplitude relaxation oscillations (LAOs) alternate with small-amplitude oscillations (SAOs). MMOs are found in singularly perturbed systems of ordinary differential equations of slow-fast type, and are typically related to the presence of so-called folded singularities and the corresponding canard trajectories in such systems. Here, we introduce a canonical family of three-dimensional slow-fast systems that exhibit MMOs which are induced by relaxation-type dynamics, and which are hence based on a "jump mechanism", rather than on a more standard canard mechanism. In particular, we establish a correspondence between that family and a class of associated one-dimensional piecewise affine maps (PAMs) which exhibit MMOs with the same signature. Finally, we give a preliminary classification of admissible mixed-mode signatures, and we illustrate our findings with numerical examples.