## Seminars and Colloquia by Series

Monday, August 26, 2013 - 16:00 , Location: Skiles 006 , Adam M. Fox , Department of Mathematics, Georgia Institute of Technology , Organizer:
Volume preserving maps naturally arise in the study of many natural phenomena including incompressible fluid-flows, magnetic field-line flows, granular mixing, and celestial mechanics. Codimension one invariant tori play a fundamental role in the dynamics of these maps as they form boundaries to transport; orbits that begin on one side cannot cross to the other. In this talk I will present a Fourier-based, quasi-Newton scheme to compute the invariant tori of three-dimensional volume-preserving maps. I will further show how this method can be used to predict the perturbation threshold for their destruction and study the mechanics of their breakup.
Thursday, August 22, 2013 - 14:00 , Location: Skiles 006 , Jordi-Lluis Figueras , Uppsala Univ. , Organizer: Rafael de la Llave
Abstract: We develop techniques for the verication of the Chebyshev property of Abelian integrals. These techniques are a combination of theoretical results, analysis of asymptotic behavior of Wronskians, and rigorous computations based on interval arithmetic. We apply this approach to tackle a conjecture formulated by Dumortier and Roussarie in [Birth of canard cycles, Discrete Contin. Dyn. Syst. 2 (2009), 723781], which we are able to prove for q <= 2.
Wednesday, August 14, 2013 - 15:30 , Location: Skiles 269 (Tentative) , Timothy Blass , Carnegie Mellon , Organizer: Rafael de la Llave
I will present a KAM theorem on the existence of codimension-one invariant tori with Diophantine rotation vector for volume-preserving maps. This is an a posteriori result, stating that if there exists an approximately invariant torus that satisfies some non-degeneracy conditions, then there is a true invariant torus near the approximate one. Thus, the theorem can be applied to systems that are not close to integrable. The method of proof provides an efficient algorithm for numerically computing the invariant tori which has been implemented by A. Fox and J. Meiss. This is joint work with Rafael de la Llave.
Wednesday, May 29, 2013 - 11:00 , Location: Skiles 05 , Alex Haro , Univ. of Barcelona , Organizer: Rafael de la Llave
In recent times there have appeared  a variety of efficient algorithms to compute quasi-periodic solutions and their invariant manifolds.  We will present a review of the main ideas and some of the implementations.
Wednesday, May 15, 2013 - 16:30 , Location: Skiles 05 , Daniel Blazevski , ETH Zurich , Organizer: Rafael de la Llave
Building on recent work on hyperbolic barriers (generalized stable and unstable manifolds) and elliptic barriers (generalized KAM tori) for two-dimensional unsteady flows, we present Lagrangian descriptions of shearless barriers (generalized nontwist KAM tori) and barriers in higher dimensional flows.  Shearless barriers (generalized nontwist KAM tori) capture the core of Rossby waves appearing in atmospheric and oceanic flows, and their robustness is appealing in the theory of magnetic confinement of plasma.  For three-dimensional flows, we give a description of hyperbolic barriers as Lagrangian Coherent Structures (LCSs) that maximally repel in the normal direction, while shear barriers are LCSs that generate shear along the LCS and act as boundaries of Lagrangian vortices in unsteady fluid flows.  The theory is illustrated on several models.
Tuesday, April 16, 2013 - 16:00 , Location: Skiles 005 , Mark Pollicott , Univ. of Warwick , Organizer: Rafael de la Llave
In joint work with P. Guilietti and C. Liverani, we show that the Ruelle zeta function for C^\infty Anosov flows has a meromorphic extension to the entire complex plane.  This generalises results of Selberg (for geodesic flows in constant curvature) and Ruelle. I
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.