Chaotic attractors of relaxation oscillators

We develop a general technique for proving the existence of chaotic attractors for three dimensional vector fields with two time scales. Our results connect two important areas of dynamical systems: the theory of chaotic attractors for discrete two-dimensional Henon-like maps and geometric singular perturbation theory. Two dimensional Henon-like maps are diffeomorphisms that limit on non-invertible one dimensional maps. Wang and Young formulated hypotheses that suffice to prove the existence of chaotic attractors in these families. Three dimensional singularly perturbed vector fields have return maps that also are two dimensional diffeomorphisms limiting on one dimensional maps. We describe a generic mechanism that produces folds in these return maps and demonstrate that the Wang–Young hypotheses are satisfied. Our analysis requires a careful study of the convergence of the return maps to their singular limits in the Ck topology for k ≥ 3. The theoretical results are illustrated with a numerical study of a variant of the forced van der Pol oscillator. AMS classification scheme numbers: 34C26,34E15,37D45,37E10 Submitted to: Nonlinearity