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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.

Series: CDSNS Colloquium

In 1994, Dumortier,
Roussarie and Rousseau launched a program aiming at proving the
ﬁniteness part of Hilbert’s 16th problem for the quadratic
system. For the program, 121 graphics need to be proved to have ﬁnite
cyclicity. In this presentation, I will show that 4 families of
HH-graphics with a triple nilpotent singularity of saddle or elliptic
type have finite cyclicity. Finishing the proof of the cyclicity of
these 4 families of HH-graphics represents one important step towards
the proof of the finiteness part of Hilbert’s 16th problem for
quadratic systems. This is a joint work with Professor Christiane
Rousseau and Professor Huaiping Zhu.

Series: CDSNS Colloquium

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~

Series: CDSNS Colloquium

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.

Series: CDSNS Colloquium

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.