Relaxation oscillation and canard explosion

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.