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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.

Series: CDSNS Colloquium

The quasi-ergodic hypothesis, proposed by Ehrenfest and Birkhoff, says
that a typical Hamiltonian system of n degrees of freedom on a typical
energy surface has a dense orbit.
This question is wide open. In this talk I will explain a recent result
by V. Kaloshin and myself which can be seen as a weak form of the
quasi-ergodic hypothesis. We prove that a dense set of perturbations of
integrable Hamiltonian systems of two and a half degrees of freedom
possess orbits which accumulate in sets of positive measure. In
particular, they accumulate in prescribed sets of KAM tori.

Series: CDSNS Colloquium

We provide several explicit examples of 3D quasiperiodic linear skew-products with simple Lyapunov spectrum, that is with 3 different Lyapunov multipliers, for which the corresponding Oseledets bundles are measurable but not continuous, colliding in a measure zero dense set.

Series: CDSNS Colloquium

Let (X, T) be a flow, that is a continuous left action of the group T on the compact Hausdorff space X. The proximal P and regionally proximal RP relations are dened, respectively (assuming X is a metric space) by P = {(x; y) | if \epsilon > 0 there is a t \in T such that d(tx, ty) < \epsilon} and RP = {(x; y) | if \epsilon > 0 there are x', y' \in X and t \in T such that d(x; x') < \epsilon, d(y; y') < \epsilon and t \in T such that d(tx'; ty') < \epsilon}. We will discuss properties of P and RP, their similarities and differences, and their connections with the distal and equicontinuous structure relations. We will also consider a relation V defined by Veech, which is a subset of RP and in many cases coincides with RP for minimal flows.

Series: CDSNS Colloquium

Theoretical aspects: If a smooth dynamical system on a compact invariant
set is structurally stable, then it has the shadowing property, that is,
any pseudo (or approximate) orbit has a true orbit nearby. In fact, the
system has the
Lipschitz shadowing property, that is, the distance between the pseudo and
true orbit is at most a constant multiple of the local error in the pseudo
orbit. S. Pilyugin and S. Tikhomirov showed the converse of this statement
for discrete dynamical systems, that is, if a discrete dynamical system has
the Lipschitz shadowing property, then it is structurally stable. In this
talk this result will be reviewed and the analogous result for flows,
obtained jointly with S. Pilyugin and S. Tikhomirov, will be described.
Numerical aspects: This is joint work with Brian Coomes and Huseyin Kocak.
A rigorous numerical method for establishing the existence of an orbit
connecting two hyperbolic equilibria of a parametrized autonomous system of
ordinary differential equations is presented. Given a suitable approximate
connecting orbit and assuming
that a certain associated linear operator is invertible, the existence of a
true connecting orbit near the approximate orbit and for a nearby parameter
value is proved provided the approximate orbit is sufficiently ``good''. It
turns out that inversion of the operator is equivalent to the solution of a
boundary value problem for a nonautonomous inhomogeneous linear difference
equation. A numerical procedure is given to verify the invertibility of the
operator and obtain a rigorous upper bound for the norm of its inverse (the
latter determines how ``good'' the approximating orbit must be).

Series: CDSNS Colloquium

Many complex models from science and engineering can be studied
in the framework of coupled systems of differential equations on networks.
A network is given by a directed graph. A local system is defined on
each vertex, and directed edges represent couplings among vertex
systems. Questions such as stability in the large, synchronization,
and complexity in terms of dynamic clusters are of interest. A more
recent approach is to investigate the connections between network
topology and dynamical behaviours. I will present some recent results
on the construction of global Lyapunov functions for coupled systems
on networks using a graph theoretic approach, and show how such
a construction can help us to establish global behaviours of compelx
models.

Series: CDSNS Colloquium

In this talk we will discuss recent work, obtained in collaboration with
Jean Bourgain, on new global well-posedness results along Gibbs measure
evolutions for the radial nonlinear wave and Schr\"odinger equations posed
on the unit ball in two and three dimensional Euclidean space, with
Dirichlet boundary conditions.
We consider initial data chosen according to a Gaussian random process
associated to the Gibbs measures which arise from the Hamiltonian structure
of the equations, and results are obtained
almost surely with respect to these probability measures. In particular,
this renders the initial value problem supercritical in the sense that
there is no suitable local well-posedness theory for
the corresponding deterministic problem, and our results therefore rely
essentially on the probabilistic structure of the problem.
Our analysis is based on the study of convergence properties of solutions.
Essential ingredients include probabilistic a priori bounds, delicate
estimates on fine frequency interactions, as well as the use of invariance
properties of the Gibbs measure to extend the relevant bounds to
arbitrarily long time intervals.