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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.

Series: CDSNS Colloquium

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.

Series: CDSNS Colloquium

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.

Series: CDSNS Colloquium

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.

Series: CDSNS Colloquium

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.

Series: CDSNS Colloquium

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.

Series: CDSNS Colloquium

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.

Series: CDSNS Colloquium

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.