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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.

Series: CDSNS Colloquium

(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.

Series: CDSNS Colloquium

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.

Series: CDSNS Colloquium

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.

Series: CDSNS Colloquium

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.