Georgia Institute of Technology College of Sciences

We shall take a look at computer-aided techniques that can be used to prove the existence of stationary solutions of radially symmetric PDEs. These techniques combine existing numerical methods with functional analytic estimates to provide a computer-assisted proof by means of the so-named 'radii-polynomial' approach.

The Parameterization Method is a functional analytic framework for studying invariant manifolds such as stable/unstable manifolds of periodic orbits and invariant tori. This talk will focus on numerical applications such as computing manifolds associated with long periodic orbits, and computing periodic invariant circles (manifolds consisting of several disjoint circles mapping one to another, each of which has an iterate conjugate to an irrational rotation). I will also illustrate how to combine Automatic Differentiation with the Parameterization Method to simplify problems with non-polynomial nonlinearities.