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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).

Friday, November 11, 2016 - 15:05 ,
Location: Skiles 170 ,
Adrián P. Bustamante ,
Georgia Tech ,
Organizer:

In this talk I am going to present a way to study
numerically the complex domains of invariant Tori for the dissipative standar map. The numerical approach is based on a Nash Moser method. This is work in
progress jointly with R. Calleja.

Friday, November 4, 2016 - 15:05 ,
Location: Skiles 170 ,
Adrián P. Bustamante ,
Georgia Tech ,
Organizer:

In the first part of the talks we are going to present a way to study
numerically the complex domains of invariant Tori for the standar map.
The numerical method is based on Padé approximants. For this part we are
going to follow the work of C. Falcolini and R. de la LLave.In the
second part we are going to present how the numerical method, developed
earlier, can be used to study the complex domains of analyticity of
invariant KAM Tori for the dissipative standar map. This part is work in
progress jointly with R. Calleja (continuation of last talk).

Friday, October 28, 2016 - 15:05 ,
Location: Skiles 170 ,
Adrián P. Bustamante ,
Georgia Tech ,
Organizer:

In the first part of the talk(s) we are going to present a way to study
numerically the complex domains of invariant Tori for the standar map.
The numerical method is based on Padé approximants. For this part we are
going to follow the work of C. Falcolini and R. de la LLave.In the
second part we are going to present how the numerical method, developed
earlier, can be used to study the complex domains of analyticity of
invariant KAM Tori for the dissipative standar map. This part is work in
progress jointly with R. Calleja (continuation of last talk).

Friday, October 14, 2016 - 15:05 ,
Location: Skiles 170 ,
Adrián P. Bustamante ,
Georgia Tech ,
Organizer:

In the first part of the talk(s) we are going to present a way to study numerically the complex domains of invariant Tori for the standar map. The numerical method is based on Padé approximants. For this part we are going to follow the work of C. Falcolini and R. de la LLave.In the second part we are going to present how the numerical method, developed earlier, can be used to study the complex domains of analyticity of invariant KAM Tori for the dissipative standar map. This part is work in progress jointly with R. Calleja.