This talk discusses the discovery of a Shilnikov homoclinic bifurcation in the Koper model, a canonical family of vector fields having slow-fast structure. Our result answers an early question in one of the burgeoning programs in applied dynamics: the classification of routes to complex and chaotic mixed-mode oscillations (MMOs). This elusive orbit becomes the organizing center of a countable family of MMOs and return maps exhibiting Smale horseshoes.
The difficulty in locating the Shilnikov orbit is related to the extreme sensitivity of the dynamics near the slow manifolds of the system, a situation characteristic of investigations of slow-fast systems. This talk explores how our methods, which use careful integration and continuation procedures to study intersections of invariant manifolds, can accommodate these challenges.