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Series: CDSNS Colloquium

We present a methodology to rigorously validate
a given approximation of a quasi-periodic Lagrangian torus of a
symplectic map. The approach consists in verifying the hypotheses of
a-posteriori KAM theory based of the parameterization method (following
Rafael de la Llave and collaborators).
A crucial point of our imprementation is an analytic Lemma that allows
us to control the norm of periodic functions using their discrete
Fourier transform. An outstanding consequence of this approach it that
the computational cost of the validation is assymptotically equivalent
of the cost of the numerical computation of invariant tori using the
parametererization method.
We pretend to describe some technical aspects of our
implementation. This is a work in progress
joint with Jordi-Lluis Figueras and Alejandro Luque.

Series: CDSNS Colloquium

The iteration theory for Lagrangian Maslov index is a very useful tool
in studying the multiplicity of brake orbits of Hamiltonian systems.
In this talk, we show how to use this theory to prove that there exist
at least $n$ geometrically distinct brake orbits on every
$C^2$ compact convex symmetric hypersurface in $\R^{2n}$
satisfying the reversible condition. As a consequence, we show that if the Hamiltonian
function is convex and even, then Seifert conjecture of 1948 on the
multiplicity of brake orbits holds for any positive integer $n$.

Series: CDSNS Colloquium

The Standard Map is a discrete time area-preserving dynamical
system and is one of the simplest of such systems to exhibit chaotic
dynamics. Traditional studies of the Standard Map have employed
symmetric forcing functions that do not induce a net flux. Although the
dynamics of these maps is rich there are many systems which cannot be
modeled with these restrictions. In this talk we will explore the
dynamics of the Standard Map when the forcing is asymmetric and induces a
positive flux on the system. We will introduce new numerical methods
to study these dynamics and give an overview of how transport in the
system changes under these new forces.

Series: CDSNS Colloquium

A Penrose tiling is an example of an aperiodic tiling and its
vertex set is an example of an aperiodic point set (sometimes known as a
quasicrystal). There are higher rank dynamical systems associated with any
aperiodic tiling or point set, and in many cases they define a uniquely
ergodic action on a compact metric space. I will talk about the ergodic
theory of these systems. In particular, I will state the results of an
ongoing work with S. Schmieding on the deviations of ergodic averages of
such actions for point sets, where cohomology plays a big role. I'll relate
the results to the diffraction spectrum of the associated quasicrystals.

Series: CDSNS Colloquium

We show that all time changes of the horocycle flow on compact surfaces of
constant negative curvature have purely absolutely continuous spectrum in
the orthocomplement of the constant functions. This provides an answer to a
question of A. Katok and J.-P. Thouvenot on the spectral nature of time
changes of horocycle flows. Our proofs rely on positive commutator methods
for self-adjoint operators and the unique ergodicity of the horocycle flow.
www.mat.uc.cl/~rtiedra/download/Horocycles_Bordeaux_2014.pdf
<http://www.mat.uc.cl/%7Ertiedra/download/Horocycles_Bordeaux_2014.pdf
<http://www.mat.uc.cl/~rtiedra/download/Horocycles_Bordeaux_2014.pdf>>

Series: CDSNS Colloquium

We develop a mathematical model for ultra-short pulse propagation in
nonlinear metamaterials characterized by a weak Kerr-type nonlinearity
in their dielectric response. The fundamental equation in the model is
the short-pulse equation (SPE) which will be derived in frequency band
gaps. We use a multi-scale ansatz to relate the SPE to the nonlinear
Schroedinger equation, thereby characterizing the change of width of the
pulse from the ultra short regime to the classical slow varying
envelope approximation. We will discuss families of solutions of the SPE
in characteristic coordinates, as well as discussing the global
wellposedness of generalizations of the model that describe uni- and
bi-directional nonlinear waves.

Series: CDSNS Colloquium

We consider an atomic model of deposition of materials over a quasi-periodic medium. The atoms of the deposited material interact with the medium (a quasi-periodic interaction) and with their nearest neighbors (a harmonic interaction). This is a quasi-periodic version of the well known Frenkel-Kontorova model. We consider the problem of whether there are quasi-periodic equilibria with a frequency that resonates with the frequencies of the medium. We show that there are always perturbative expansions. We also prove a KAM theorem in a-posteriori form. We show that if there is an approximate solution of the equilibrium equation satisfying non-degeneracy conditions, we can adjust one parameter and obtain a true solution which is close to the approximate solution. The proof is based on an iterative method of the KAM type. The iterative method is not based on transformation theory as the most usual KAM theory, but it is based on a novel technique of supplementing the equilibrium equation with another equation that factors the linearization of the equilibrium equilibrium equation.

Series: CDSNS Colloquium

Given a holomorphic map of C^m to itself that fixes a point, what happens to points near that fixed point under iteration? Are there points attracted to (or repelled from) that fixed point and, if so, how? We are interested in understanding how a neighborhood of a fixed point behaves under iteration. In this talk, we will focus on maps tangent to the identity. In dimension one, the Leau-Fatou Flower Theorem provides a beautiful description of the behavior of points in a full neighborhood of a fixed point. This theorem from the early 1900s continues to serve as inspiration for this study in higher dimensions. In dimension 2 our picture of a full neighborhood of a fixed point is still being constructed, but we will discuss some results on what is known, focusing on the existence of a domain of attraction whose points converge to that fixed point.

Series: CDSNS Colloquium

We consider several possibilities on how to select a Filippov sliding
vector field on a co-dimension 2 singularity manifold, intersection of
two co-dimension 1 manifolds, under the assumption of general
attractivity. Of specific interest is the selection of a smoothly
varying Filippov sliding vector field. As a result of our analysis and
experiments, the best candidates of the many possibilities explored are
based on the so-called barycentric coordinates: in particular, we choose
what we call the moments solution. We then examine the behavior of the
moments vector field at first order exit points, and show that it aligns
smoothly with the exit vector field. Numerical experiments illustrate
our results and contrast the present method with other choices of
Filippov sliding vector field. We further present some minimum variation
properties, related to orbital equivalence, of Filippov solutions for
the co-dimension 2 case in \R^{3}.

Series: CDSNS Colloquium

In this talk, we will discuss a question
posed by Vladimir Arnold some twenty years ago, in a subject he called
"dynamics of intersections." In the simplest setting, the question is
the following: given a (discrete time) holomorphic dynamical system on a
complex manifold X and two holomorphic curves C and D in X which pass
through a fixed point P of the system, how quickly can the local
intersection multiplicies at P of C with the iterates of D grow in time?
Questions like this arise naturally, for instance, when trying to count
the periodic points of a dynamical system. Arnold conjectured that this
sequence of intersection multiplicities can grow at most exponentially
fast, and in fact we can show this conjecture is true if the curves are
chosen to be suitably generic. However, as we will see, for some (even
very simple) dynamical systems one can choose curves so that the
intersection multiplicities grow as fast as desired. We will see how to
construct such counterexamples to Arnold's conjecture, using geometric
ideas going back to work of Yoshikazu Yamagishi.