From Canards of Folded Singularities to Torus Canards in a Forced van der Pol Equation

TL;DR

This paper investigates canard solutions in the forced van der Pol equation across different forcing frequencies, deriving explicit formulas for their existence and revealing a unifying formula for canard folds, with implications for slow/fast systems.

Contribution

It provides the first explicit formulas for primary maximal canards and torus canards in all frequency regimes, unifying their descriptions and analyzing their geometric properties.

Findings

Explicit formulas for canard fold points across frequency regimes

Identification of torus canards as long segments near limit cycles

Unification of canard formulas across regimes

Abstract

We study canard solutions of the forced van der Pol (fvdP) equation in the relaxation limit for low-, intermediate-, and high-frequency periodic forcing. A central numerical observation is that there are two branches of canards in parameter space which extend across all positive forcing frequencies. For low-frequency forcing, we demonstrate the existence of primary maximal canards induced by folded saddle-nodes of type I, and establish explicit formulas for the parameter values at which the primary maximal canards and their folds exist. We then turn to the intermediate- and high-frequency forcing regimes, and show that the fvdP equation possesses torus canards instead. These torus canards consist of long segments near families of attracting and repelling limit cycles of the fast system, in alternation. We also derive explicit formulas for the parameter values at which the maximal torus canards and their folds exist. Primary maximal canards and maximal torus canards correspond geometrically to the situation in which the persistent manifolds near the family of attracting limit cycles coincide to all orders with the persistent manifolds that lie near the family of repelling limit cycles. The formulas derived for the folds of maximal canards in all three frequency regimes turn out to be representations of a single formula in the appropriate parameter regimes, and this unification confirms our numerical observation. In addition, we study the secondary canards induced by the folded singularities in the low-frequency regime and find that their fold curves turn around in the intermediate-frequency regime. We identify the mechanism responsible for this turning. Finally, we show that the fvdP equation is a normal form for a class of single-frequency periodically-driven slow/fast systems with two fast variables and one slow variable which possess a nondegenerate fold of limit cycles.