  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.
