## Seminars and Colloquia by Series

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.
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.