8:30 AM-9:00 AM - Workshop Reception (Skiles, Room 236)
 |
9:00 AM - 10:00 AM Rafael de la Llave (University of Texas at Austin, USA) Geometric Mechanisms of Diffusion Without Whiskered Tori Abstract: We will discuss several mechanisms of diffusion in nearly integrable (a-priori unstable) systems which do not involve whiskered tori. The main tool is perturbation theory of normally hyperbolic manifolds and laminations. These mechanisms allow quantitative estimates on diffusion times and on the Hausdorff dimension of diffusing orbits. |
|
10:00 AM - 10:15 AM Break (Skiles, Room 236) |
|
10:15 AM - 11:15 AM M. Alejandra Gonzalez (University of Texas at Austin, USA) Invariant Tori for Symplectic Maps Abstract |
|
11:15 AM - 11:30 AM Break (Skiles, Room 236) |
|
11:30 AM - 12:30 PM Tere Seara (Universitat Politecnica de Catalunya, Spain) A Rigorous Proof of the Existence of Diffusion in a Problem with Large Gaps Abstract: We consider a two and a half degrees of freedom Hamiltonian, which consists on a small coupling, periodic in time, of a pendulum and a rotator. In this classical model the existence of diffusion is not trivial, because the transition chains between the usual whiskered tori do not cover any finite interval in the action (of the rotator). The reason is that when the interval contains some resonance, a gap without such tori appears. We prove that, under some non degeneracy conditions, one can construct transition chains along any interval using, besides the usual whiskered tori, what we call secondary objects, that can be secondary tori or (un)stable manifolds of periodic orbits which appear in the gaps which are devoid of whiskered tori. |
|
12:30 PM - 2:30 PM Lunch |
|
2:30 PM - 3:30 PM Turgay Uzer (Georgia Institute of Technology, USA) Phase-Space Transition States Abstract: Dynamical systems theory is used to construct a general phase space version of transition state theory. Special multidimensional separatrices are found which act as impenetrable barriers in phase space between reacting and nonreacting trajectories. The elusive momentum-dependent transition state between reactants and products is thereby characterized. A practical algorithm is presented and applied to a strongly coupled Hamiltonian. |
|
3:30 PM - 3:45 PM Break (Skiles, Room 236) |
|
3:45 PM - 4:45 PM Federico Bonetto (Georgia Institute of Technology, USA) Some Results on a Simple Dissipative System Arising in Celestial Mechanics Abstract: We consider a class of one and a half degrees of freedom Hamiltonian which could describe the spin orbit interaction of a sattelite moving around a planet. The introduction of a small friction stabilizes few periodic orbit providing a possible explanation for the resonance-locking between the revolution and rotation period. Although the model for friction we use is probably too simple a comparison with real data gives interesting results. |
|
4:45 PM - 5:00 PM Break (Skiles, Room 236) |
|
5:00 PM - 6:00 PM Vieri Mastropietro (Universita degli Studi Roma, Italy) Arnold Diffusion and the D'Alembert Precession Problem Abstract: A planet can be described by an homogeneous rigid ellipsoid with flatness h, moving on a Keplerian orbit around a star and subject only to Newtonian forces. It was proposed in 1994 by Chierchia and Gallavotti that, for suitable initial data, the precession cone can change O(1) in a finite time, no matter how small h is, as a consequence of Arnold diffusion mechanism. One can start introducing some simplifications in the original model, neglecting a term in its Hamiltonian so that the problem is reduced to a priori unstable three time scale system; for such systems a general theory of Arnold diffusion can indeed be developed. I will review in this talk the main results obtained by Gallavotti, Gentile and myself about Arnold diffusion in three time scale a priori unstable systems and I discuss its relevance for a complete understanding of the precession problem. |
7:00 PM-9:00 PM - Workshop Banquet
Dinho Chinese Restaurant,
no charge for invited speakers,
$25/person for non-speakers
|
9:00 AM - 10:00 AM C. Eugene Wayne (Boston University, USA) Reduced Equations for Models of Laminated Materials in Thin Domains Abstract: Many problems involving the motion of elastic structures occur in domains where one or more of the dimensions of the body is significantly small than the others, e.g. the vibrations of a long metal beam or rod. In such situations the true equations of elasticity have often been replaced by simpler model equations such as the Bernoulli or Timoshenko models of beams. In this lecture I will describe how one can use ideas from Hamiltonian mechanics to justify such approximations in the context of a model for laminated materials introduced by Schwab and Babuska. This is joint work with R. L. DeVille. |
|
10:00 AM - 10:15 AM Break (Skiles, Room 236) |
|
10:15 AM - 11:15 AM Yingfei Yi (Georgia Institute of Technology, USA) Destruction of Resonant Tori in Hamiltonian Systems Abstract: It is well known that resonant tori in an integrable Hamiltonian system tend to be destroyed under arbitrary generic perturbations and give rise to a resonance zone containing both stochastic under arbitrary generic perturbations and give rise to a resonance zone containing both stochastic trajectories and regular orbits. This lecture will present a Poincaré-Treshchev mechanism for the destruction of resonant tori and the onset of regular orbits in nearly integrable Hamiltonian systems. The persistence of the majority of Poincaré non-degenerate n-tori on any resonant surface of dimension n will be shown under various non-degenerate conditions. |
|
11:15 AM - 11:30 AM Break (Skiles, Room 236) |
|
11:30 AM - 12:30 PM Jiangong You (Nanjing University, China) On Eliasson's Full Measure Reducibility Problem Abstract |
|
12:30 PM - 2:30 PM Lunch |
|
2:30 PM - 3:30 PM Mark Levi (Pennsylvania State University, USA) Stability Webs in Schrodinger's Equations Abstract: We describe an interesting web-like structure of the stability diagram of the 1D Schroedinger equation with periodic potential. This structure does not arise in the well-known Mathieu equation, whose potential contains only one harmonic. An observation of this web-like structure led to the discovery of a certain isochronous property of trigonometic potentials. This is a joint work with Carles Simo and Henk Broer. |
|
3:30 PM - 3:45 PM Break (Skiles, Room 236) |
|
3:45 PM - 4:45 PM Martijn van Noort (Georgia Institute of Technology, USA) Two-Quasiperiodicity in the Planar Pendulum with Periodic Forcing Abstract: The parametrically forced pendulum is a Hamiltonian one-and-a-half degree of freedom system, with parameters controlling the amplitude and frequency of the forcing. For any value of these parameters the system has invariant 2-tori of "rotational" type. The measure of the set of such tori converges to full measure exponentially fast as the velocity of the pendulum goes to (plus or minus) infinity. As a consequence, the phase space can be divided into three parts at each parameter point: two parts where the dynamics is mostly quasi-periodic, and one bounded "region of interest" in between. Numerical analysis provides a bound on this region, depending on the parameters. A method to prove a bound by theoretical means will be discussed. |
|
4:45 PM - 5:00 PM Break (Skiles, Room 236) |
|
5:00 PM - 6:00 PM Sergey Bolotin (University of Wisconsin-Madison, USA) Shadowing Collision Orbits of the Elliptic 3-Body Problem Abstract: Using Levi-Civita regularization and variational methods we construct solutions of the elliptic 3 body problem shadowing chains of collision orbits. |