On the Existence of and Relationship between Canards and Torus Canards in Forced Slow/Fast Systems

TL;DR

This paper extends the understanding of canard solutions in forced slow/fast systems, establishing their existence and relationship across different types and broad classes, including the forced FitzHugh-Nagumo model.

Contribution

It generalizes the analytic relationship between folded singularity canards and torus canards to a wider class of systems, including forced FitzHugh-Nagumo.

Findings

Identifies parameter regions for canard solutions

Shows continuation of primary canards into torus canards with increased forcing frequency

Demonstrates results in the forced FitzHugh-Nagumo system

Abstract

Canards are special solutions of slow/fast systems which are ubiquitous in neuroscience and electrical engineering. Two distinct classes of canard solutions have been identified and carefully studied: folded singularity canards and torus canards. Recently, an explicit and analytic relationship between these seemingly unrelated families of solutions was established in the classical forced van der Pol equation (Burke et al., J. Nonlinear Sci. 26:405--451, 2015). In this article, we generalize the results of Burke et al. (2015) to the broader class of time-periodically forced planar slow/fast systems, which includes the forced van der Pol and the forced FitzHugh-Nagumo equations. We analytically determine the parameter values in this class of systems for which the two types of canard solutions exist, and show that the branches of primary canards of folded singularities continue into those of the torus canards as the forcing frequency is increased. We illustrate our results in the paradigm problem of the forced FitzHugh-Nagumo system.