The dynamics of flows in 3-dimensions

Abstract:

The dynamics of flows in 2 dimensions is a well-understood field of study. The celebrated Poincaré–Bendixson theorem describes the long-term behaviour of orbits of continuous dynamical systems on the plane, cylinder, or two-sphere. However, for flows in 3 dimensions, much less is known about the structure of the long-term behavior of orbits, and the stability of the behavior of the dynamical system for nearby flows, unless strong dynamical hypotheses are imposed on the system, such that it is Hamiltonian, or hyperbolic in the sense of Smale's Axiom A condition. There is a special class of systems, the aperiodic flows, which have no periodic orbits, so their long-term behavior is expected to be quite complicated.  All known examples of aperiodic systems are constructed using a celebrated method of Krystyna Kuperberg, discovered in 1993. The speaker, in collaboration with Ana Rechtman, investigated the long-term behavior of the orbits of these systems, and also the problem of stability of these systems under smooth perturbations.  In this talk, I will present two of our discoveries, and how they are related, which suggests the richness of the dynamics of flows in 3-dimensions which remain to be discovered.