Canards and horseshoes in the forced van der Pol equation

Cartwright and Littlewood discovered "chaotic" solutions in the periodically forced van der Pol equation in the 1940's. Subsequent work by Levinson, Levi, and others has made this singularly perturbed system one of the archetypical dissipative systems with chaotic dynamics. Despite the extensive history of this system, many questions concerning its bifurcations and chaotic dynamics remain. We use a combination of analysis of the singular limit and numerical simulation to describe a horseshoe map that arises in the three-dimensional phase space. The canards that form at a "folded saddle" play a crucial role in this analysis.