A parametrization method for invariant manifolds of periodic orbits, with applications to the restricted three body problem.Monday, January 28, 2013 - 16:05 , Location: Skiles 006 , Maciej Capinski , Georgia Tech and AGH Univ. Krakow , firstname.lastname@example.org , Organizer:
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.
Tuesday, January 22, 2013 - 16:00 , Location: skills 06 , Marta Canadell , Universitat de Barcelona and Georgia Tech , Organizer:
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.
Monday, January 14, 2013 - 16:05 , Location: Skiles 06 , Renato Calleja , Georgia Tech and ITAM , Organizer:
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.
Monday, December 3, 2012 - 16:00 , Location: Skiles 006 , Federico Bonretto , Georgia Tech , Organizer: Rafael de la Llave
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.
Monday, November 5, 2012 - 16:00 , Location: Skiles 06 , Miguel Walter , Georgia Tech (Aerospace Eng.) , Organizer: Rafael de la Llave
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.
Monday, October 22, 2012 - 16:00 , Location: Skiles Bldg, Room 006 , Nandor Simanyi , U. Alabama Birmingham , email@example.com , Organizer:
Putting in place the last piece of the big mosaic of the proof of the Boltzmann-Sinai Ergodic Hypothesis,we consider the billiard flow of elastically colliding hard balls on the flat $d$-torus ($d>1$), and prove that no singularity manifold can even locally coincide with a manifold describing future non-hyperbolicity of the trajectories. As a corollary, we obtain the ergodicity (actually the Bernoulli mixing property) of all such systems, i.e. the verification of the Boltzmann-sinai Ergodic Hypothesis. The manuscript of the paper can be found at http://people.cas.uab.edu/~simanyi/transversality-new.pdf
Monday, October 8, 2012 - 11:05 , Location: Skiles 006 , Yi Wang , University of Sciences and Technology of China , Organizer: Chongchun Zeng
For a general time-dependent linear competitive-cooperative tridiagonal system of differential equations, we obtain canonical Floquet invariant bundles which are exponentially separated in the framework of skew-product flows. The obtained Floquet theory is applied to study the dynamics on the hyperbolic omega-limit sets for the nonlinear competitive-cooperative tridiagonal systems in time-recurrent structures including almost periodicity and almost automorphy.
Monday, October 1, 2012 - 16:00 , Location: Skiles 06 , Adam Fox , Univ. of Colorado , Organizer: Rafael de la Llave
Invariant tori play a prominent role in the dynamics of symplectic maps. These tori are especially important in two dimensional systems where they form a boundary to transport. Volume preserving maps also admit families of invariant rotational tori, which will restrict transport in a d dimensional map with one action and d-1 angles. These maps most commonly arise in the study of incompressible fluid flows, however can also be used to model magnetic field-line flows, granular mixing, and the perturbed motion of comets in near-parabolic orbits. Although a wealth of theory has been developed describing tori in symplectic maps, little of this theory extends to the volume preserving case. In this talk we will explore the invariant tori of a 3 dimensional quadratic, volume preserving map with one action and two angles. A method will be presented for determining when an invariant torus with a given frequency is destroyed under perturbation, based on the stability of approximating periodic orbits.
Monday, September 24, 2012 - 11:05 , Location: Skiles 006 , Tomas Gedeon , Montana State University , firstname.lastname@example.org , Organizer: Shui-Nee Chow
Joint with Applied and Computational Mathematics Seminar
Bio-polymerization processes like transcription and translation are central to a proper function of a cell. The speed at which the bio-polymer grows is affected both by number of pauses of elongation machinery, as well their numbers due to crowding effects. In order to quantify these effects in fast transcribing ribosome genes, we rigorously show that a classical traffic flow model is a limit of mean occupancy ODE model. We compare the simulation of this model to a stochastic model and evaluate the combined effect of the polymerase density and the existence of pauses on transcription rate of ribosomal genes.
Friday, September 7, 2012 - 11:05 , Location: Skiles 006 , Jason Mireles-James , Rutgers University , Organizer: Rafael de la Llave
I'll discuss some work on rigorous computation of invariant manifolds and computer assisted proof of the existence of transverse connecting orbits for differential equations. I'm also interested in how these computations can be used to obtain global topological data, such as the chain groups and boundary maps of Morse Theory.