Seminars and Colloquia by Series

Supperdiusion constants for certain nonuniformly hyperbolic systems

Series
CDSNS Colloquium
Time
Monday, October 24, 2016 - 11:06 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Hongkun Zhang – U. Mass Amherst
We investigate deterministic superdiusion in nonuniformly hyperbolic system models in terms of the convergence of rescaled distributions to the normal distribution following the abnormal central limit theorem, which differs from the usual requirement that the mean square displacement grow asymptotically linearly in time. We obtain an explicit formula for the superdiffusion constant in terms of the ne structure that originates in the phase transitions as well as the geometry of the configuration domains of the systems. Models that satisfy our main assumptions include chaotic Lorentz gas, Bunimovich stadia, billiards with cusps, and can be apply to other nonuniformly hyperbolic systems with slow correlation decay rates of order O(1/n)

Pairings between periodic orbits in hyperbolic coupled map lattices.

Series
CDSNS Colloquium
Time
Monday, October 17, 2016 - 11:00 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Boris Gutkin – Georgia Tech (School of Physics)
Upon quantization, hyperbolic Hamiltonian systems generically exhibit universal spectral properties effectively described by Random Matrix Theory. Semiclassically this remarkable phenomenon can be attributed to the existence of pairs of classical periodic orbits with small action differences. So far, however, the scope of this theory has, by and large, been restricted to single-particle systems. I will discuss an extension of this program to hyperbolic coupled map lattices with a large number of sites (i.e., particles). The crucial ingredient is a two-dimensional symbolic dynamics which allows an effective representation of periodic orbits and their pairings. I will illustrate the theory with a specific model of coupled cat maps, where such a symbolic dynamics can be constructed explicitly.

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 Corsi – Georgia 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-James – Florida 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. Groothedde – V.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.

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 Gidea – Yeshiva 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 Zhou – School 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 Montgomery – Univ. 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 Jorba – Univ. 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.

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