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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.

Series: CDSNS Colloquium

An analysis of the dynamics of a mass-less spacecraft (or point mass) around an in-homogeneousTrojan body in a system composed of three primaries lying at the
vertexes of an equilateral triangle, with their mutual positions fixed over
the course of the motion is here presented. To this end two suitable models
are identified to represent the system, depending on the distance from the
primary. The first model, adopted for use close to the asteroid, where the
dynamics is dominated by this sole body, is the Restricted Two Body Problem.
In this model the in-homogeneities of the asteroid are taken into account as
they have a dominant effect on the dynamics of the spacecraft. The second
model is the Lagrangian Circular Restricted Four Body Problem (CR4BP), which
is adopted far from the asteroid, where the gravitational perturbations of the
Sun and Jupiter are dominant while the in-homogeneities of the asteroid are
negligible.
Low-thrust propulsion perturbations are incorporated into this model. The
possibility to determine the range of validity of each model using an
application of a Weak Stability Boundary (WSB) theory is investigated and
applied.
Applications are shown for the main example of Lagrangian configuration in the
Solar system, the Sun-Jupiter-Trojan-spacecraft system.

Series: CDSNS Colloquium

Piecewise linear Fermi-Ulam pingpongs.
We consider a particle moving freely between two periodically moving
infinitely heavy walls. We assume that one wall is fixed and the
second one
moves with piecewise linear velocities. We study the question about existence
and abundance of accelerating orbits for that model. This is a joint work with
Jacopo de Simoi

Series: CDSNS Colloquium

The theorem of Shannon-McMillan-Breiman states that for every
generating partition on an ergodic system,
the exponential decay rate of the measure of cylinder sets
equals the metric entropy almost everywhere (provided the entropy is finite).
We show that the measure of cylinder sets are lognormally
distributed for strongly mixing systems and infinite partitions and show that the rate of convergence
is polynomial provided the fourth moment of the information function is finite.
We also show that it satisfies the almost sure invariance principle.
Unlike previous results by Ibragimov and others which only apply to finite partitions,
here we do not require any regularity of the conditional entropy function.

Series: CDSNS Colloquium

We present a method for the detection
of stable and unstable fibers of invariant manifolds of periodic
orbits.
We show how to propagate the fibers to prove transversal
intersections of invariant manifolds. The method can be applied using
interval arithmetic
to produce rigorous, computer assisted estimates
for the manifolds. We apply the method to prove transversal
intersections of stable and unstable manifolds of Lyapunov orbits in
the restricted three body problem.

Series: CDSNS Colloquium

We explain numerical algorithms for the computation of normally hyperbolic invariant manifolds and their invariant bundles, using the parameterization method. The framework leads to solving invariance
equations, for which one uses a Newton method adapted to the dynamics and the geometry of the invariant manifolds. We illustrate the algorithms with several examples. The algorithms are inspired in current work with A. Haro
and R. de la Llave. This is joint work with Alex Haro.

Series: CDSNS Colloquium

Conformally symplectic systems send a symplectic form into a
multiple of itself. They appear in mecanical systems with friction proportional to the velocity and as Euler-Lagrange equations of the time discounted actions common in economics. The conformaly symplectic structure
provides identities that we use to prove "a-posteriori" theorems that
show that if we have an approximate solution which satisfies some non-degeneracy conditions, we can obtain a true solution close to the approximate one. The identities used to prove the theorem, also lead to very efficient algorithms with small
storage and operation counts. We will also present implementations of the algorithms.