Seminars and Colloquia by Series

Convergent series and domains of analyticity for response solutions in quasi-periodically forced strongly dissipative systems

Series
CDSNS Colloquium
Time
Monday, March 25, 2013 - 16:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Livia Corsi – University of Naples ``Federico II'' –
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).

On the breakdown mechanisms of Fiberwise Hyperbolic Invariant Tori in skew product systems. Numerical and theoretical results.

Series
CDSNS Colloquium
Time
Tuesday, March 12, 2013 - 16:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jordi-Lluis Figueras Romero – University of Uppsala –
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.

Natural and perturbed dynamics about Trojan bodies

Series
CDSNS Colloquium
Time
Tuesday, March 12, 2013 - 15:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Marta Ceccaroni – University of Rome (Tor Vergata) –
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.

Piecewise linear Fermi-Ulam pingpongs.

Series
CDSNS Colloquium
Time
Wednesday, February 27, 2013 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles Bldg Rm.005
Speaker
Dmitry Dolgopyat – Univ. of Maryland
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

Limiting behaviour for the theorem of Shannon-McMillan-Breiman

Series
CDSNS Colloquium
Time
Monday, February 4, 2013 - 16:05 for 1 hour (actually 50 minutes)
Location
Skiles 005
Speaker
Nicolai Haydn – USC
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.

A parametrization method for invariant manifolds of periodic orbits, with applications to the restricted three body problem.

Series
CDSNS Colloquium
Time
Monday, January 28, 2013 - 16:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Maciej Capinski – Georgia Tech and AGH Univ. Krakow –
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.

Parameterization Methods for Computing Normally Hyperbolic Invariant Tori: some numerical examples

Series
CDSNS Colloquium
Time
Tuesday, January 22, 2013 - 16:00 for 1 hour (actually 50 minutes)
Location
skills 06
Speaker
Marta Canadell – Universitat de Barcelona and Georgia Tech
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.

Quasi-Periodic solutions for conformally symplectic dynamical systems

Series
CDSNS Colloquium
Time
Monday, January 14, 2013 - 16:05 for 1 hour (actually 50 minutes)
Location
Skiles 06
Speaker
Renato Calleja – Georgia Tech and ITAM
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.

Non equilibrium steady state for a simple model of electric conduction

Series
CDSNS Colloquium
Time
Monday, December 3, 2012 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Federico Bonretto – Georgia Tech
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.

A STOCHASTIC EXPANSION-BASED APPROACH FOR DESIGN UNDER UNCERTAINTY

Series
CDSNS Colloquium
Time
Monday, November 5, 2012 - 16:00 for 1 hour (actually 50 minutes)
Location
Skiles 06
Speaker
Miguel Walter – Georgia Tech (Aerospace Eng.)
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.

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