|March 4, 2005: Samuel Lisi, CIMS
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.