Seminars and Colloquia by Series

Friday, February 7, 2014 - 15:00 , Location: Skiles 06 , Alex Haro , Univ. of Barcelona , Organizer: Rafael de la Llave
This talk is devoted to quasi-periodic Schrödinger operators beyond theAlmost Mathieu, with more general potentials and interactions. The  linksbetween the spectral properties of these operators and the dynamicalproperties of the associated quasi-periodic linear skew-products rule thegame. In particular, we present a Thouless formula  and some consequencesof Aubry duality. This is a joint work with Joaquim Puig~                                                                   
Monday, February 3, 2014 - 11:00 , Location: Skiles05 , Marta Canadell , Univ. of Barcelona , Organizer: Rafael de la Llave
We present a KAM-like theorem for the existence of quasi-periodic tori with a prescribed Diophantine rotation for a discrete family of dynamical system. The theorem is stated in an a posteriori format, so it can be used to validate numerical computations. The method of proof provides an efficient algorithm for computing quasi-periodic tori. We also present implementations of the algorithm, illustrating them throught several examples and observing different mechanisms of breakdown of qp invariant tori. This is a joint work with Alex Haro.
Wednesday, January 22, 2014 - 11:00 , Location: Skiles 006 , Renato Calleja , IIMAS UNAM , Organizer: Rafael de la Llave
We present a numerical  study of the dynamics of a state-dependent delay equation with two state dependent delays that are linear in the state. In particular, we study some of the the dynamical behavior driven by the existence of two-parameter families of invariant tori. A formal normal form analysis predicts the existence of torus bifurcations and the appearance of a two parameter family of stable invariant tori. We investigate the dynamics on the torus thought a Poincaré section. We find some boundaries of Arnold tongues and indications of loss of normal hyperbolicity for this stable family. This is joint work with A. R. Humphries and B. Krauskopf.
Friday, January 17, 2014 - 11:05 , Location: Skiles 005 , Christian H. Sadel , University of British Columbia, Vancouver. , Organizer:
(Joint work with A. Avila and S. Jitomirskaya). An analytic, complex, one-frequency cocycle is given by a pair $(\alpha,A)$ where $A(x)$ is an analytic and 1-periodic function that maps from the torus $\mathbb(R) / \mathbb(Z)$ to the complex $d\times d$ matrices and $\alpha \in [0,1]$ is a frequency. The pair is interpreted as the map $(\alpha,A)\,:\, (x,v) \mapsto (x+\alpha), A(x) v$. Associated to the iterates of this map are (averaged) Lyapunov exponents $L_k(\alpha,A)$ and an Osceledets filtration. We prove joint-continuity in $(\alpha,A)$ of the Lyapunov exponents at irrational frequencies $\alpha$, give a criterion for domination and prove that for a dense open subset of cocycles, the Osceledets filtration comes from a dominated splitting which is an analogue to the Bochi-Viana Theorem.
Tuesday, January 7, 2014 - 15:05 , Location: Skiles 005 , Xifeng Su , Beijing Normal University , Organizer:
We consider the semi-linear elliptic PDE driven by the fractional Laplacian: \begin{equation*}\left\{%\begin{array}{ll}    (-\Delta)^s u=f(x,u) & \hbox{in $\Omega$,} \\    u=0 &  \hbox{in $\mathbb{R}^n\backslash\Omega$.} \\\end{array}% \right.\end{equation*}An $L^{\infty}$ regularity result is given, using De Giorgi-Stampacchia iteration  method.By the Mountain Pass Theorem and some other nonlinear analysis methods, the existence and multiplicity of non-trivial solutions for the above equation are established. The validity of the Palais-Smale condition without Ambrosetti-Rabinowitz condition for non-local elliptic equations is proved. Two non-trivial solutions are given under some weak hypotheses. Non-local elliptic equations with concave-convex nonlinearities are also studied, and existence of at least six solutions are obtained. Moreover, a global result of Ambrosetti-Brezis-Cerami type is given, which  shows that the effect of  the parameter $\lambda$ in the nonlinear term changes considerably the nonexistence, existence and multiplicity of solutions.
Monday, January 6, 2014 - 11:00 , Location: Skiles 005 , James Meiss* , Department of Applied Mathematics, University of Colorado, Boulder , Organizer:
Synchronization of coupled oscillators, such as grandfather clocks or metronomes, has been much studied using the approximation of strong damping in which case the dynamics of each reduces to a phase on a limit cycle. This gives rise to the famous Kuramoto model.  In contrast, when the oscillators are Hamiltonian both the amplitude and phase of each oscillator are dynamically important. A model in which all-to-all coupling is assumed, called the Hamiltonian Mean Field (HMF) model, was introduced by Ruffo and his colleagues. As for the Kuramoto model, there is a coupling strength threshold above which an incoherent state loses stability and the oscillators synchronize. We study the case when the moments of inertia and coupling strengths of the oscillators are heterogeneous. We show that finite size fluctuations can greatly modify the synchronization threshold by inducing correlations between the momentum and parameters of the rotors. For unimodal parameter distributions, we find an analytical expression for the modified critical coupling strength in terms of statistical properties of the parameter distributions and confirm our results with numerical simulations. We find numerically that these effects disappear for strongly bimodal parameter distributions. *This work is in collaboration with Juan G. Restrepo.
Monday, November 25, 2013 - 16:00 , Location: Skiles 05 , Rodrigo Trevino , Cornell Univ./Tel Aviv Univ. , Organizer: Rafael de la Llave
The three objects in the title come together in the study of ergodic properties of geodesic flows on flat surfaces. I will go over how these three things are intimately related, state some classical results about the unique ergodicity of translation flows and present new results which generalize much of the classical theory and also apply to non-compact (infinite genus) surfaces.
Monday, October 28, 2013 - 15:05 , Location: Skiles 202 , Dmitry Todorov , Chebyshev laboratory, Saint-Petersburg , Organizer: Rafael de la Llave
There is known a lot of information about classical or standard shadowing. Itis also often called a pseudo-orbit tracing property (POTP).  Let M be a closedRiemannian manifold. Diffeomorphism f : M → M is said to have POTPif for a given accuracy any pseudotrajectory with errors small enough can beapproximated (shadowed) by an exact trajectory. Therefore, if one wants to dosome numerical investiagion of the system one would definitely prefer it to haveshadowing property.However, now it is widely accepted that good (qualitatively strong) shad-owing is present only in hyperbolic situations. However it seems that manynonhyperbolic systems still could be well analysed numerically.As a step to resolve this contradiction I introduce some sort of weaker shad-owing. The idea is to restrict a set of pseudotrajectories to be shadowed. Onecan consider only pseudotrajectories that resemble sequences of points generatedby a computer with floating-point arithmetic.I will tell what happens in the (simplified) case of “linear” two-dimensionalsaddle connection. In this case even stochastic versions of classical shadowing(when one tries to ask only for most pseudotrajectories to be shadowed) do notwork. Nevertheless, for “floating-point” pseudotrajectories one can prove somepositive results.There is a dichotomy: either every pseudotrajectory stays close to the un-perturbed trajectory forever if one carefully chooses the dependence betweenthe size of errors and requested accuracy of shadowing, or there is always apseudotrajectory that can not be shadowed.
Monday, October 21, 2013 - 16:05 , Location: Skiles 005 , Bastien Fernandez , CPT Luminy , Organizer:
To identify and to explain coupling-induced phase transitions in Coupled Map Lattices (CML) has been a lingering enigma for about two decades. In numerical simulations, this phenomenon has always been observed preceded by a lowering of the Lyapunov dimension, suggesting that the transition might require changes of linear stability. Yet, recent proofs of co-existence of several phases in specially designed models work in the expanding regime where all Lyapunov exponents remain positive. In this talk, I will consider a family of CML composed by piecewise expanding individual map, global interaction and finite number N of sites, in the weak coupling regime where the CML is uniformly expanding. I will show, mathematically for N=3 and numerically for N>3, that a transition in the asymptotic dynamics occurs as the coupling strength increases. The transition breaks the (Milnor) attractor into several chaotic pieces of positive Lebesgue measure, with distinct empiric averages. It goes along with various symmetry breaking, quantified by means of magnetization-type characteristics. Despite that it only addresses finite-dimensional systems, to some extend, this result reconciles the previous ones as it shows that loss of ergodicity/symmetry breaking can occur in basic CML, independently of any decay in the Lyapunov dimension.
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. 

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