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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.

Series: CDSNS Colloquium

In joint work with P. Guilietti and C. Liverani, we show that the Ruelle
zeta function for C^\infty Anosov flows has a meromorphic extension to
the entire complex plane. This generalises results of Selberg (for
geodesic flows in constant curvature) and Ruelle.
I

Series: CDSNS Colloquium

A classical result of Aubry and Mather states that Hamiltonian
twist maps have orbits of all rotation numbers. Analogously, one can
show that certain ferromagnetic crystal models admit ground states of
every possible mean lattice spacing. In this talk, I will show that
these ground states generically form Cantor sets, if their mean lattice
spacing is an irrational number that is easy to approximate by rational
numbers. This is joint work with Blaz Mramor.

Series: CDSNS Colloquium

A new approach based on Wasserstein distances, which are numerical costs ofan optimal transportation problem, allows to analyze nonlinear phenomena ina robust manner. The long-term behavior is reconstructed from time series, resulting in aprobability distribution over phase space. Each pair of probabilitydistributions is then assigned a numerical distance that quantifies thedifferences in their dynamical properties. From the totality of all these distances a low-dimensional representation ina Euclidean spaceis derived. This representation shows the functional relationships betweenthe dynamical systems under study. It allows to assess synchronizationproperties and also offers a new way of numerical bifurcation analysis.

Series: CDSNS Colloquium

We study the ordinary differential equation
\varepsilon \ddot x + \dot x + \varepsilon g(x) = \e f(\omega t),
with f and g analytic and f quasi-periodic in t
with frequency vector \omega\in\mathds{R}^{d}.
We show that if there exists c_{0}\in\mathds{R} such that
g(c_{0}) equals the average of f and the first non-zero
derivative of g at c_{0} is of odd order \mathfrak{n},
then, for \varepsilon small enough and under very mild Diophantine
conditions on \omega, there exists a quasi-periodic solution
"response solution" close to c_{0}, with the same
frequency vector as f. In particular if f is a trigonometric
polynomial the Diophantine condition on \omega can be completely
removed. Moreover we show that for \mathfrak{n}=1 such a solution
depends analytically on \e in a domain of the complex plane tangent
more than quadratically to the imaginary axis at the origin.
These results have been obtained in collaboration with Roberto
Feola (Universit\`a di Roma ``La Sapienza'') and Guido Gentile
(Universit\`a di Roma Tre).

Series: CDSNS Colloquium

In this talk we will first present several breakdown mechanisms of Uniformly Hyperbolic Invariant Tori (FHIT) in
area-preserving skew product systems by means of numerical examples. Among these breakdowns we will
see that there are three types: Hyperbolic to elliptic (smooth bifurcation), the Non-smooth breakdown
and the Folding breakdown. Later, we will give a theoretical explanation of the folding breakdown. Joint work with Alex Haro.