## Seminars and Colloquia by Series

Monday, April 8, 2013 - 16:30 , Location: Skiles 005 , Bob W. Rink , Vrije Universiteit Amsterdam , Organizer: Rafael de la Llave
A classical result of Aubry and Mather states that Hamiltonian twist maps have orbits of all rotation numbers. Analogously, one can show that certain ferromagnetic crystal models admit ground states of every possible mean lattice spacing. In this talk, I will show that these ground states generically form Cantor sets, if their mean lattice spacing is an irrational number that is easy to approximate by rational numbers. This is joint work with Blaz Mramor.
Tuesday, March 26, 2013 - 15:05 , Location: Skiles 006 , , Universiteit Utrecht , , Organizer: Shui-Nee Chow
A new approach based on Wasserstein distances, which are numerical costs ofan optimal transportation problem, allows to analyze nonlinear phenomena ina robust manner. The long-term behavior is reconstructed from time series, resulting in aprobability distribution over phase space. Each pair of probabilitydistributions is then assigned a numerical distance that quantifies thedifferences in their dynamical properties. From the totality of all these distances a low-dimensional representation ina Euclidean spaceis derived. This representation shows the functional relationships betweenthe dynamical systems under study. It allows to assess synchronizationproperties and also offers a new way of numerical bifurcation analysis.
Monday, March 25, 2013 - 16:05 , Location: Skiles 006 , , University of Naples Federico II'' , , Organizer:
We study the ordinary differential equation \varepsilon \ddot x + \dot x + \varepsilon g(x) = \e f(\omega t), with f and g analytic and f quasi-periodic in t with frequency vector \omega\in\mathds{R}^{d}. We show that if there exists c_{0}\in\mathds{R} such that g(c_{0}) equals the average of f and the first non-zero derivative of g at c_{0} is of odd order \mathfrak{n}, then, for \varepsilon small enough and under very mild Diophantine conditions on \omega, there exists a quasi-periodic solution "response solution" close to c_{0},  with the same frequency vector as f. In particular if f is a trigonometric polynomial the Diophantine condition on \omega can be completely removed. Moreover we show that for \mathfrak{n}=1 such a solution depends analytically on \e in a domain of the complex plane tangent more than quadratically to the imaginary axis at the origin. These results have been obtained in collaboration with Roberto Feola (Universit\a di Roma La Sapienza'') and Guido Gentile (Universit\a di Roma Tre).
Tuesday, March 12, 2013 - 16:05 , Location: Skiles 006 , Jordi-Lluis Figueras Romero , University of Uppsala , , Organizer:
In this talk we will first present several breakdown mechanisms of Uniformly Hyperbolic Invariant Tori (FHIT) in area-preserving skew product systems by means of numerical examples. Among these breakdowns we will see that there are three types: Hyperbolic to elliptic (smooth bifurcation), the Non-smooth breakdown and the Folding breakdown. Later, we will give a theoretical explanation of the folding breakdown. Joint work with Alex Haro.
Tuesday, March 12, 2013 - 15:05 , Location: Skiles 006 , , University of Rome (Tor Vergata) , , Organizer:
An analysis of the dynamics of a mass-less spacecraft (or point mass) around an in-homogeneousTrojan body in a system composed of three primaries lying at the vertexes of an equilateral triangle, with their mutual positions fixed over the course of the motion is here presented. To this end two suitable models are identified to represent the system, depending on the distance from the primary. The first model, adopted for use close to the asteroid, where the dynamics is dominated by this sole body, is the Restricted Two Body Problem. In this model the in-homogeneities of the asteroid are taken into account as they have a dominant effect on the dynamics of the spacecraft. The second model is the Lagrangian Circular Restricted Four Body Problem (CR4BP), which is adopted far from the asteroid, where the gravitational perturbations of the Sun and Jupiter are dominant while the in-homogeneities of the asteroid are negligible. Low-thrust propulsion perturbations are incorporated into this model. The possibility to determine the range of validity of each model using an application of a Weak Stability Boundary (WSB) theory is investigated and applied. Applications are shown for the main example of Lagrangian configuration in the Solar system, the Sun-Jupiter-Trojan-spacecraft system.
Wednesday, February 27, 2013 - 16:00 , Location: Skiles Bldg Rm.005 , Dmitry Dolgopyat , Univ. of Maryland , Organizer:
Piecewise linear Fermi-Ulam pingpongs. We consider a particle moving freely between two periodically moving infinitely heavy walls. We assume that one wall is fixed and the second one moves with piecewise linear velocities. We study the question about existence and abundance of accelerating orbits for that model. This is a joint work with Jacopo de Simoi
Monday, February 4, 2013 - 16:05 , Location: Skiles 005 , Nicolai Haydn , USC , Organizer:
The theorem of Shannon-McMillan-Breiman states that for every generating partition on an ergodic system, the exponential decay rate of the measure of cylinder sets equals the metric entropy almost everywhere (provided the entropy is finite). We show that the measure of cylinder sets are lognormally distributed for strongly mixing systems and infinite partitions and show that the rate of convergence is polynomial provided the fourth moment of the information function is finite. We also show that it satisfies the almost sure invariance principle. Unlike previous results by Ibragimov and others which only apply to finite partitions, here we do not require any regularity of the conditional entropy function.
Monday, January 28, 2013 - 16:05 , Location: Skiles 006 , Maciej Capinski , Georgia Tech and AGH Univ. Krakow , , Organizer:
We present a method for the detection of stable and unstable fibers of invariant manifolds of periodic orbits. We show how to propagate the fibers to prove transversal intersections of invariant manifolds. The method can be applied using interval arithmetic to produce rigorous, computer assisted estimates for the manifolds. We apply the method to prove transversal intersections of stable and unstable manifolds of Lyapunov orbits in the restricted three body problem.
Tuesday, January 22, 2013 - 16:00 , Location: skills 06 , Marta Canadell , Universitat de Barcelona and Georgia Tech , Organizer:
We explain numerical algorithms for the computation of normally hyperbolic invariant manifolds and their invariant bundles, using the parameterization method. The framework leads to solving invariance equations, for which one uses a Newton method adapted to the dynamics and the geometry of the invariant manifolds. We illustrate the algorithms with several examples. The algorithms are inspired in current work with A. Haro and R. de la Llave. This is joint work with Alex Haro.
Monday, January 14, 2013 - 16:05 , Location: Skiles 06 , Renato Calleja , Georgia Tech and ITAM , Organizer:
Conformally symplectic systems send a symplectic form into a multiple of itself. They appear in mecanical systems with friction proportional to the velocity and as Euler-Lagrange equations of the time discounted actions common in economics. The  conformaly symplectic structure provides identities that we use to prove "a-posteriori" theorems that show that if we have an approximate solution which satisfies some non-degeneracy conditions, we can obtain a true solution close to the approximate one.  The  identities  used to prove the theorem, also lead   to very efficient algorithms with small storage and operation counts. We will also present implementations of the algorithms.