Emerging Topics

Arnold diffusion and Mather theory

Ke Zhang
University of Toronto
April 11, 2018
Abstract: Arnold diffusion studies the problem of topological instability in nearly integrable Hamiltonian systems. An important contribution was made my John Mather, who announced a result in two and a half degrees of freedom and developed deep theory for its proof. We describe a recent effort to better conceptualize the proof for Arnold diffusion.

Diffusion along chains of normally hyperbolic cylinders

Marian Gidea
Yeshiva University
April 11, 2018
Abstract: We consider a geometric framework that can be applied to prove the existence of drifting orbits in the Arnold diffusion problem. The main geometric objects that we consider are 3-dimensional normally hyperbolic invariant cylinders with boundary, which admit well-defined stable and unstable manifolds. These enable us to define chains of cylinders i.e., finite, ordered families of cylinders in which each cylinder admits homoclinic connections, and any two consecutive cylinders admit heteroclinic connections.

Some geometric mechanisms for Arnold diffusion

Rafael de la Llave
Georgia Tech
April 10, 2018
Abstract: We consider the problem whether small perturbations of integrable mechanical systems can have very large effects.

It is known that in many cases, the effects of the perturbations average out, but there are exceptional cases (resonances) where the perturbations do accumulate. It is a complicated problem whether this can keep on happening because once the instability accumulates, the system moves out of resonance.

A General Shadowing result for normally hyperbolic invariant manifolds and its application to Arnold diffusion

Tere Seara
April 10, 2018
Abstract: In this talk we present a general shadowing result for normally hyperbolic invariant manifolds. The result does not use the existence of invariant objects like tori inside the manifold and works in very general settings.

We apply this result to establish the existence of diffusing orbits in a large class of nearly integrable Hamiltonian systems. Our approach relies on successive applications of the so called `scattering map' along homoclinic orbits to a normally hyperbolic invariant manifold.