Nathan Duignan, Department of Applied Mathematics, University of Colorado Boulder
Non-Existence of Invariant Surfaces Transverse to Foliations
Of fundamental importance to the qualitative understanding of dynamical systems are invariant manifolds. In this presentation we will explore a recent paper of MacKay on a condition which guarantees the non-existence of invariant manifolds that are transverse to a chosen foliation—a family of curves. In particular, the foliation can be chosen to prove the non-existence of invariant manifolds of a particular homology, just like a set of radial curves can detect invariant circles not enclosing the origin.
A numerical implementation of this non-existence condition will then be constructed. We will investigate two key systems through this method; a model of a charged particle in two waves and a family of Beltrami flows that can have quasicrystal symmetry. Ultimately, the method detects regions of chaos and island structures within these systems.
This is joint work with Jim Meiss.