we give a geometric analysis of relaxation oscillations and canard cycles i
n singularly perturbed planar vector fields. The transition from small Hopf
-type cycles to large relaxation cycles, which occurs in an exponentially t
hin parameter interval, is described as a perturbation of a family of singu
lar cycles. The results are obtained by means of two blow-up transformation
s combined with standard tools of dynamical systems theory. The efficient u
se of various charts is emphasized. The results are applied to the van der
Pol equation. (C) 2001 Academic Press.