Seminars and Colloquia by Series

Locally integrable non-Liouville analytic geodesic flows on T^2

Series
CDSNS Colloquium
Time
Monday, August 29, 2016 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Livia CorsiGeorgia Tech - School of Math
A metric on the 2-torus T^2 is said to be "Liouville" if in some coordinate system it has the form ds^2 = (F(q_1) + G(q_2)) (dq_1^2 + dq_2^2). Let S^*T^2 be the unit cotangent bundle.A "folklore conjecture" states that if a metric is integrable (i.e. the union of invariant 2-dimensional tori form an open and dens set in S^*T^2) then it is Liouville: l will present a counterexample to this conjecture.Precisely I will show that there exists an analytic, non-separable, mechanical Hamiltonian H(p,q) which is integrable on an open subset U of the energy surface {H=1/2}. Moreover I will show that in {H=1/2}\U it is possible to find hyperbolic behavior, which in turn means that there is no analytic first integral on the whole energy surface.This is a work in progress with V. Kaloshin.

Parameterization of periodic invariant objects for maps

Series
CDSNS Colloquium
Time
Monday, May 2, 2016 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
J. Mireles-JamesFlorida Atlantic Univ.
The Parameterization Method is a functional analytic framework for studying invariant manifolds such as stable/unstable manifolds of periodic orbits and invariant tori. This talk will focus on numerical applications such as computing manifolds associated with long periodic orbits, and computing periodic invariant circles (manifolds consisting of several disjoint circles mapping one to another, each of which has an iterate conjugate to an irrational rotation). I will also illustrate how to combine Automatic Differentiation with the Parameterization Method to simplify problems with non-polynomial nonlinearities.

Rigorous validation of Radially Symmetric Stationary Solutions of PDEs.

Series
CDSNS Colloquium
Time
Monday, May 2, 2016 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
C.M. GrootheddeV.U. Amsterdam
We shall take a look at computer-aided techniques that can be used to prove the existence of stationary solutions of radially symmetric PDEs. These techniques combine existing numerical methods with functional analytic estimates to provide a computer-assisted proof by means of the so-named 'radii-polynomial' approach.

Oscillatory motions in a restricted three body problem

Series
CDSNS Colloquium
Time
Monday, April 25, 2016 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Tere M. SearaUniv Polit. Catalunya

Hamiltonian Instability in a Four-Body Problem

Series
CDSNS Colloquium
Time
Monday, April 25, 2016 - 10:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Marian GideaYeshiva Univ.
We consider a restricted four-body problem, modeling the dynamics of a light body (e.g., a moon) near a Jupiter trojan asteroid. We study two mechanisms of instability. For the first mechanism, we assume that the orbit of Jupiter is circular, and we investigate the hyperbolic invariant manifolds associated to periodic orbits around the equilibrium points. The conclusion is that the light body can undergo chaotic motions inside the Hill sphere of the trojan, or well outside that region. For the second mechanism, we consider the perturbative effect due to the eccentricity of the orbit of Jupiter. The conclusion is that the size of the orbit of the light body around the trojan can keep increasing, or keep decreasing, or undergo oscillations. This phenomenon is related to the Arnold Diffusion problem in Hamiltonian dynamics

New Dynamical System Models for Games Inspired by the Fokker-Planck Equations on Graphs

Series
CDSNS Colloquium
Time
Monday, April 18, 2016 - 11:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Haomin ZhouSchool of Math, Georgia Tech
In this talk, I will present new models to describe the evolution of games. Our dynamical system models are inspired by the Fokker-Planck equations on graphs. We will present properties of the models, their connections to optimal transport on graphs, and computational examples for generalized Nash equilibria. This presentation is based on a recent joint work with Professor Shui-Nee Chow and Dr. Wuchen Li.

The shape sphere: a new vista on the three body problem (David Alcaraz conference: Video conference)

Series
CDSNS Colloquium
Time
Tuesday, April 12, 2016 - 13:30 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Richard MontgomeryUniv. California Santa Cruz
Video Conference David Alcaraz confernce. Newton's famous three-body problem defines dynamics on the space of congruence classes of triangles in the plane. This space is a three-dimensional non-Euclidean rotationally symmetric metric space centered'' on the shape sphere. The shape sphere is a two-dimensional sphere whose points represent oriented similarity classes of planar triangles. We describe how the sphere arises from the three-body problem and encodes its dynamics. We will see how the classical solutions of Euler and Lagrange, and the relatively recent figure 8 solution are encoded as points or curves on the sphere. Time permitting, we will show how the sphere pushes us to formulate natural topological-geometric questions about three-body solutions and helps supply the answer to some of these questions. We may take a brief foray into the planar N-body problem and its associated shape sphere'' : complex projective N-2 space.

Dynamical systems tools for Solar sails

Series
CDSNS Colloquium
Time
Monday, April 11, 2016 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Angel JorbaUniv. of Barcelona
Dynamical systems have proven to be a useful tool for the design of space missions. For instance, the use of invariant manifolds is now common to design transfer strategies. Solar Sailing is a proposed form of spacecraft propulsion, where large membrane mirrors take advantage of the solar radiation pressure to push the spacecraft. Although the acceleration produced by the radiation pressure is smaller than the one achieved by a traditional spacecraft it is continuous and unlimited. This makes some long term missions more accessible, and opens a wide new range of possible applications that cannot be achieved by a traditional spacecraft. In this presentation we will focus on the dynamics of a Solar sail in a couple of situations. We will introduce this problem focusing on a Solar sail in the Earth-Sun system. In this case, the model used will be the Restricted Three Body Problem (RTBP) plus Solar radiation pressure. The effect of the solar radiation pressure on the RTBP produces a 2D family of "artificial'' equilibria, that can be parametrised by the orientation of the sail. We will describe the dynamics around some of these "artificial'' equilibrium points. We note that, due to the solar radiation pressure, the system is Hamiltonian only for two cases: when the sail is perpendicular to the Sun - Sail line; and when the sail is aligned with the Sun - sail line (i.e., no sail effect). The main tool used to understand the dynamics is the computation of centre manifolds. The second example is the dynamics of a Solar sail close to an asteroid. Note that, in this case, the effect of the sail becomes very relevant due to the low mass of the asteroid. We will use, as a model, a Hill problem plus the effect of the Solar radiation pressure, and we will describe some aspects of the natural dynamics of the sail.

Several analytical properties of Camassa-Holm type equations.

Series
CDSNS Colloquium
Time
Monday, March 7, 2016 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Qingtian ZhangPenn State University
Abstract: In this talk, I will present the uniqueness of conservative solutions to Camassa-Holm and two-component Camassa-Holm equations. Generic regularity and singular behavior of those solutions are also studied in detail. If time permitting, I will also mention the recent result on wellposedness of cubic Camassa-Holm equations.

Index theory for symplectic matrix paths and periodic solutions of Hamiltonian systems with prescribed energy

Series
CDSNS Colloquium
Time
Wednesday, March 2, 2016 - 13:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Yiming LongNankai University
One of the major tools in the study of periodic solutions of Hamiltonian systems is the Maslov-type index theory for symplectic matrix paths. In this lecture, I shall give first a brief introduction on the Maslov-type index theory for symplectic matrix paths as well as the iteration theory of this index. As an application of these theories I shall give a brief survey about the existence, multiplicity and stability problems on periodic solution orbits of Hamiltonian systems with prescribed energy, especially those obtained in recent years. I shall also briefly explain some ideas in these studies, and propose some open problems.