Tomas Johnson: Computer-aided proof of a tangency bifurcation involving a slow manifold

Slow-fast dynamical systems have two time scales and an explicit parameter representing the ratio of these time scales. Locally invariant slow manifolds along which motion occurs on the slow time scale are a prominent feature of slow-fast systems. These manifolds are defined asymptotically in the time scale ratio, and their existence is difficult to verify in actual systems with a fixed time scale ratio. I will introduce the concept of a computable slow manifold, which is an attempt to define an object with similar properties for a fixed ratio of the time scales. The existence of these computable slow manifolds can be verified rigorously. As an application, I will use it to prove the existence of tangencies of invariant manifolds in the problem of singular Hopf bifurcation and to give bounds on the location of one such tangency. Such tangencies are important in multiple-time scale systems, as they are involved in global bifurcations that mark the onset of special periodic solutions with motion on both time-scales -- so called mixed-mode oscillations. The talk is based on joint work with John Guckenheimer and Philipp Meerkamp.