- You are here:
- GT Home
- Home
- News & Events

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.

Series: CDSNS Colloquium

A very simple model for electric conduction consists of N particles movingin a periodic array of scatterers under the influence of an electric field and of aGaussian thermostat that keeps their energy fixed. I will present analytic result for the behaviourof the steady state of the system at small electric field, where the velocity distribution becomesindependent of the geometry of the scatterers, and at large N, where the system can bedescribed by a linear Boltzmann type equation.

Series: CDSNS Colloquium

A common practice in aerospace engineering has been to carry out deterministicanalysis in the design process. However, due to variations in design condition suchas material properties, physical dimensions and operating conditions; uncertainty isubiquitous to any real engineering system. Even though the use of deterministicapproaches greatly simplifies the design process since any uncertain parameter is setto a nominal value, the final design can have degraded performance if the actualparameter values are slightly different from the nominal ones.Uncertainty is important because designers are concerned about performance risk.One of the major challenges in design under uncertainty is computational efficiency,especially for expensive numerical simulations. Design under uncertainty is composedof two major parts. The first one is the propagation of uncertainties, and the otherone is the optimization method. An efficient approach for design under uncertaintyshould consider improvement in both parts.An approach for robust design based on stochastic expansions is investigated. Theresearch consists of two parts : 1) stochastic expansions for uncertainty propagationand 2) adaptive sampling for Pareto front approximation. For the first part, a strategybased on the generalized polynomial chaos (gPC) expansion method is developed. Acommon limitation in previous gPC-based approaches for robust design is the growthof the computational cost with number of uncertain parameters. In this research,the high computational cost is addressed by using sparse grids as a mean to alleviatethe curse of dimensionality. Second, in order to alleviate the computational cost ofapproximating the Pareto front, two strategies based on adaptive sampling for multi-objective problems are presented. The first one is based on the two aforementionedmethods, whereas the second one considers, in addition, two levels of fidelity of theuncertainty propagation method.The proposed approaches were tested successfully in a low Reynolds number airfoilrobust optimization with uncertain operating conditions, and the robust design of atransonic wing. The gPC based method is able to find the actual Pareto front asa Monte Carlo-based strategy, and the bi-level strategy shows further computationalefficiency.