This note points out that in a wide class of forced relaxation oscillators the hyperbolic behavior, which has up to now been known to exist only on small set, dominates in the Lebesgue sense in a wide class of relaxation oscillators. It is surprising that this phenomenon went unobserved in this class of problems for half a century. This note introduces the simplest physically realistic smooth system where (non-uniform) hyperbolic behavior is expected to dominate in the Lebesgue sense, so that the strange attractor is expected to exist for a set of parameter values of positive measure.
The classical problem of periodically forced realaxation oscillations considered originally by Cartrwright, Littlewood and Levinson stimulated Smale's introduction of the horseshoe map and through that many subsequent developments in dynamical systems.