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Friday, November 10, 2017 - 14:00 ,
Location: Skiles 154 ,
Rafael de la Llave ,
GT Math ,
Organizer: Jiaqi Yang

We consider Hamiltonian systems with normally hyperbolic manifold with a homoclinic connection. The systems are of the form H_0(I, phi, x,y) = h(I) + P(x,y) ,where P is a one dimensional system with a homoclinic intersection. The above Hamiltonian is a standard normal form for near integrable Hamiltonians close to a resonance. We consider perturbations that are time dependent and may be not Hamiltonian. We derive explicit formulas for the first order effects on the stable/unstable manifolds. In particular, we give sufficient conditions for the existence of homoclinic intersections to the normally hyperbolic manifold. Previous treatments in the literature specify the types of the unperturbed orbits considered (periodic or quasiperiodic) and are restricted to periodic or quasi-periodic perturbations. We do not need to distinguish on the perturbed orbits and we allow rather general dependence on the time (periodic, quasiperiodic or random). The effects are expressed by very fast converging improper integrals. This is joint work with M. Gidea. https://arxiv.org/abs/1710.01849

Friday, November 3, 2017 - 15:00 ,
Location: Skiles 154 ,
Hassan Attarchi ,
Georgia Tech ,
Organizer:

This presentation is about the results of a paper by L. Bunimovich in
1974. One considers dynamical systems generated by billiards which are
perturbations of dispersing billiards. It was shown that such dynamical
systems are systems of A. N. Kolmogorov (K-systems), if the perturbation
satisfies certain conditions which have an intuitive geometric
interpretation.

Friday, October 27, 2017 - 15:00 ,
Location: Skiles 154 ,
Hassan Attarchi ,
Georgia Tech ,
Organizer:

This presentation is about the results of a paper by Y. Sinai in
1970. Here, I will talk about dynamical systems which resulting from the
motion of a material point in domains with strictly convex boundary,
that is, such that the operator of the second quadratic form is
negative-definite at each point of the boundary, where the boundary is
taken to be equipped with the field of inward normals. It was proved
that such systems are ergodic and are K-systems. The basic method of
investigation is the construction of transversal foliations for such
systems and the study of their properties.

Friday, October 13, 2017 - 15:00 ,
Location: Skiles 154 ,
Bhanu Kumar ,
GT Math ,
Organizer: Jiaqi Yang

Birkhoff's Theorem is a result useful in characterizing the boundary of certain open sets U ⊂ T^1 x [0, inf) which are invariant under "vertical-tilting" homeomorphisms H. We present the method used by A. Fathi to prove Birkhoff's theorem, which develops a series of lemmas using topological arguments to prove that this boundary is a graph.

Friday, October 6, 2017 - 15:00 ,
Location: Skiles 154 ,
Sergio Mayorga ,
Georgia Tech ,
Organizer: Jiaqi Yang

We will look at a system of hamiltonian equations on the torus, with an
initial condition in momentum and a terminal condition in position, that
arises in mean field game theory. Existence of and uniqueness of
solutions will be shown, and a few remarks will be made in regard to its
connection to the minimization problem of a cost functional. This is the second part of lasrt week's talk.

Friday, October 6, 2017 - 15:00 ,
Location: Skiles 154 ,
Prof. Rafael de la Llave ,
School of Mathematics, Georgia Tech ,
Organizer: Jiaqi Yang

We will present an introduction to the results of S. Aubry and J. Mather who used variational methods to prove the existence of quasi-periodic orbits in twist mappings and in some models appearing in solid state Physics.

Friday, September 29, 2017 - 15:00 ,
Location: Skiles 154 ,
Sergio Mayorga ,
Georgia Tech ,
Organizer: Jiaqi Yang

We will look at a system of hamiltonian equations on the torus, with an initial condition in momentum and a terminal condition in position, that arises in mean field game theory. Existence of and uniqueness of solutions will be shown, and a few remarks will be made in regard to its connection to the minimization problem of a cost functional.

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