Hunting for Homoclinic orbits (with a geometric PDE)

A Hamiltonian system is a special kind of ODE, classically used to model a physical system in which energy is conserved. One of the big questions is to understand what special orbits can occur and how many of them there are - examples of special orbits include periodic orbits and homoclinic orbits.

Until the mid-80's, most of the advances in the field came by studying a variational problem. Then, Floer turned the field on its head by studying a first-order elliptic PDE and using this to relate periodic orbits of the Hamiltonian system to certain geometric properties.

I will present a quick overview of the successes achieved by this means and will then show how the existing theory can be extended to prove the existence of a homoclinic orbit in certain special circumstances.

The talk will not assume any significant geometry background. Instead, the primary emphasis will be on the PDE methods and problems that arise.