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

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