Friday, September 22, 2017 - 15:00 , Location: Skiles 154 , Jiaqi Yang , Georgia Tech , Organizer: Jiaqi Yang
We will continue from last week's talk. There are many advances toward proof of Arnold diffusion in Mather's setting. In particular, we will study an approach based on recent work of Marian-Gidea and Jean-Pierre Marco.
Friday, September 15, 2017 - 15:00 , Location: Skiles 154 , Jiaqi Yang , Georgia Tech , Organizer: Jiaqi Yang
We will introduce Arnold diffusion in Mather's setting. There are many advances toward proof of this. In particular, we will study an approach based on recent work of Marian-Gidea and Jean-Pierre Marco.
Friday, April 21, 2017 - 15:00 , Location: Skiles 254 , Adrian P. Bustamante , Georgia Tech , Organizer:
A classical theorem of Arnold, Moser shows that in analytic families of maps close to a rotation we can find maps which are smoothly conjugate to rotations. This is one of the first examples of the KAM theory. We aim to present an efficient numerical algorithm, and its implementation, which approximate the conjugations given by the Theorem
Friday, April 7, 2017 - 15:05 , Location: Skiles 254 , Prof. Rafael de la Llave , School of Math, Georgia Tech , Organizer: Jiaqi Yang
It is well known that periodic orbits give all the information about dynamical systems, at least for expanding maps, for which the periodic orbits are dense. This turns out to be true in dimensions 1 and 2, and false in dimension 4 or higher.We will present a proof that two $C^\infty$ expanding maps of the circle, which are topologically equivalent are $C^\infty$ conjugate if and only if the derivatives or the return map at periodic orbits are the same.
Friday, March 31, 2017 - 15:05 , Location: Skiles 254 , Lei Zhang , School of Mathematics, GT , Organizer: Jiaqi Yang
In this talk, we will give an introduction to the variational approach to dynamical systems. Specifically, we will discuss twist maps and prove the classical results that area-preserving twist map has Birkhoff periodic orbits for each rational rotation number.
Friday, March 10, 2017 - 15:00 , Location: Skiles 254 , Rafael de la Llave , GT Math , Organizer: Rafael de la Llave
A classical theorem of Arnold, Moser shows that in analytic families of maps close to a rotation we can find maps which are smoothly conjugate to rotations. This is one of the first examples of the KAM theory. We aim to present a self-contained version of Moser's proof and also to present some efficient numerical algorithms.
Friday, March 3, 2017 - 15:05 , Location: Skiles 254 , Lu Xu , School of Mathematics, Jilin University , Organizer: Jiaqi Yang
My talk is about the quasi-periodic motions in multi-scaled Hamiltonian systems. It consists of four part. At first, I will introduce the results in integrable Hamiltonian systems since what we focus on is nearly-integrable Hamiltonian system. The second part is the definition of nearly-integrable Hamiltonian system and the classical KAM theorem. After then, I will introduce that what is Poincar\'e problem and some interesting results corresponding to this problem. The last part, which is also the main part, I will talk about the definition and the background of nearly-integrable Hamiltonian system, then the persistence of lower dimensional tori on resonant surface, which is our recent result. I will also simply introduce the Technical ingredients of our work.
Friday, February 24, 2017 - 15:05 , Location: Skiles 254 , Simon Berman , School of Physics , Organizer: Jiaqi Yang
In a high harmonic generation (HHG) experiment, an intense laser pulse is sent through an atomic gas, and some of that light is converted to very high harmonics through the interaction with the gas. The spectrum of the emitted light has a particular, nearly universal shape. In this seminar, I will describe my efforts to derive a classical reduced Hamiltonian model to capture this phenomenon. Beginning with a parent Hamiltonian that yields the equations of motion for a large collection of atoms interacting self-consistently with the full electromagnetic field (Lorentz force law + Maxwell's equations), I will follow a sequence of reductions that lead to a reduced Hamiltonian which is computationally tractable yet should still retain the essential physics. I will conclude by pointing out some of the still-unresolved issues with the model, and if there's time I will discuss the results of some preliminary numerical simulations.
Friday, January 20, 2017 - 03:05 , Location: Skiles 254 , Álex Haro , Univ. of Barcelona , Organizer:
We will design a method to compute invariant tori in Hamiltonian systems through the computation of invariant tori for time- T maps. We will also consider isoenergetic cases (i..e. fixing energy).