TITLES AND ABSTRACTS
Giancarlo Benettin
Nekhoroshev theorem and Fourier spectra of observables
Nekhoroshev theorem is generally referred to as a stability result of
the actions, for exponentially large times. Such a formulation is in a
sense poor, and far from optimal. With some additional work one can work
out more detailed informations, concerning the Fourier spectrum of
generic observables. For resonant initial data (namely when the theorem is
interesting) the spectrum is proven to have a rather peculiar shape,
which is easily observed in numerical experiments. The result is still
far from optimal but, so to speak, less far. The practical result
is that an observation of the spectrum for a relatively short time scale
can provide informations on the behavior of the system for the much
longer Nekhoroshev times.
Sergey Bolotin
A variational construction of connecting orbits for hyperbolic invariant
tori
We construct orbits shadowing chains of heteroclinics connecting
hyperbolic invariant tori. The method is based on a combination
of the variational method of Mather and Shilnikov's lemma.
One of the advantages of this approach is that we do not need
heteroclinics to be minimal. As an application, we estimate
diffusion time in Mather's problem.
Alain Chenciner
Action minimizing periodic orbits in the Newtonian n-body problem
Minimization of the lagrangian action in well chosen spaces of closed
paths has been recently used to prove the existence of interesting
solutions of the n-body problem. This is particularly the case in the
equal mass case where symmetry conditions may be imposed.
Vittorio Coti
Zelati
Multibump solutions for rapidly oscillating second order Hamiltonian
systems
Using a variational approach, we construct multi-bump homoclinic
solutions for a class of rapidly oscillating second order Hamiltonian
systems
L. Hakan Eliasson
Almost reducibility of linear quasi-periodic systems
We prove a result showing that linear quasi-periodic systems which are
analytic and close to constant coefficients
and have Diophantine quasi-periodic frequencies are always almost
reducible in the sense that they can be analytically conjugate to a
system arbitrarily close to constant coefficients. We discuss when this
approach converges and gives full reducibility, and when it does
not.
Michael Herman
Nekhoroshev theorem for Gevrey classes and examples of instabilities
showing the results obtained are almost optimal
Généraliser le théorème de Nekhoroshev en classe
Gevrey et par des examples d'instabilités (très simples) de
montrer en classe de Gevrey le caractère "presque" optimal des
examples: tout au moins montrer qu'un exposant tend vers 0 comme c/n,
lorsque la dimension n tend vers l'infini ou c>0 dépend de la classe
de Gevrey. L'autre exposant (celui du confinement) valant dans le cas
quasi-convexe 1/2. Les examples peuvent être rendus R-analytiques
mais il n'est pas clair d'estimer les examples dans le complexe.
Jacques Laskar
Frequency Map Analysis
I will review the recent development in frequency map analysis, and its
applications.
John Mather
A pseudometric on the union of action minimizing orbits
In this talk, I will define a pseudometric on the union
of action minimizing orbits, and show how it permits a unified treatment
of theorems about connecting orbits.
David
W. McLaughlin
Spectral bifurcations in dispersive wave turbulence
Dispersive wave turbulence is studied numerically for a class of
one-dimensional nonlinear wave equations. Both deterministic and random
(white noise in time) forcings are studied. Four distinct stable spectra
are observed -- the direct and inverse cascades of weak turbulence
(WT) theory, thermal equilibrium, and a fourth spectrum (MMT; Majda,
McLaughlin, Tabak). Each spectrum can describe long-time behavior, and
each can be only metastable (with quite diverse lifetimes) -- depending on
details of nonlinearity, forcing and dissipation. Cases of a long-lived
MMT transient state decaying to a state with WT spectra, and vice-versa,
are displayed. In the case of freely decaying turbulence, without
forcing, both cascades of weak turbulence are observed. These WT states
constitute the clearest and most striking numerical observations of WT
spectra to date -- over four decades of energy, and three decades of
spatial, scales. Energy growth in time will be used to monitor
numerically the selection of MMT or WT spectra.
This is joint work, with David Cai, Andrew Majda and Esteban Tabak.
Mark Pollicott
Statistical properties of geodesic flows
This talk will discuss recent progress on dynamical and
geometric questions for geodesic flows on negatively curved
manifolds. This will include recent results on rates of mixing for the
flow, asymptotics for the number of closed orbits (or equivalently
for the number of closed geodesics on the manifold) and the connection
with the homology of the underlying manifold.
Paul Rabinowitz
A variational shadowing method
A variational method for constructing heteroclinic solutions (of a family of
Hamiltonian systems) which shadow a heteroclinic chain of solutions will
be discussed
Eric Seré
Homoclinic orbits on energy surfaces of Hamiltonian Systems
Carles Simó
Codimension one manifolds as quasiconfiners
Geometrical objects bounding regions with quite different regimes of
motion in the phase space are relevant to understand the global
dynamics. Typically these objects are invariant manifolds of normally
hyperbolic center manifolds of codimension two. If these manifolds have a
small splitting, then they act as quasiconfiners of the motion.
Several examples will be presented, mainly in Celestial Mechanics, in
problems with two and three degrees of freedom. They are relevant to study
the regions where the motion is essentially confined, e.g., in a large
vicinity of the Lagrangian triangular points, and also as a mechanism
leading to the destruction of invariant tori.
Domokos Szasz
Algebraic multidimensional dispersing billiards
In our recent work on correlation decay for multidimensional dispersing
billiards it turned out that --- contrary to the general belief of
experts,
which was based on two-dimensional experience ---
the singularity submanifolds are not smooth everywhere. Indeed, those
arising
from the tangential collisions (forming a one-codimensional manifold in
the
phase space) are even not differentiable on a two-codimensional
submanifold
of the singularity manifold. The picture is somewhat analogous to
Whitney's
umbrella. This phenomenon makes it necessary to rethink the foundations of
the theory of
multidimensional semi-dispersing billiards. According to our present
understanding the fundamental theorem of Chernov and Sinai, the basic tool
to
imply local (and global) ergodicity of the system is valid if one assumes
that the boundaries of the scatterers are algebraic. Fortunately this is
the case, for instance, in the much important model of hard ball systems.
(It is an interesting question to decide whether the algebraicity of
the scatterer boundaries can be relaxed to requiring their analiticity,
only.)
Finally, correlation decay for multidimensional dispersing billiards is
also
discussed.
This research is joint with P. B\'alint, N. Chernov and I. P. T\'oth.
Jeff Xia
Connecting Orbits and Diffusion in Hamiltonian Dynamics
In this talk, we discuss minimum measures and connecting orbits in high
dimensional Lagrangian systems. We extend a result of Mather to give a
sufficient condition for diffusion in positive definite systems, including
"a priori" stable ones. It is conjectured that this condition is generic.
Jean-Christophe Yoccoz
Non uniformly hyperbolic horseshoes
I will describe some joint work with J. Palis on the
dynamics which occur after homoclinic bifurcations. Starting with a
standard
horseshoe of dimension slightly larger than one, we get after the
bifurcation a more complicated invariant set which can be viewed as a
horseshoe with tangencies.
Eduard Zehnder
Pseudoholomorphic curves and dynamics in three dimensions
Surfaces of sections are a classical tool in the study of 3-dimensional
dynamical systems. Their use goes back to the work of Poincaré and
Birkhoff. We shall describe a natural generalization of this concept
by constructing a system of transversal sections in the complement of
finitely many disguished periodic solutions. Such a system is established
for non-degenerate Reeb flows on the tight 3-sphere by means of
pseudoholomorphic curves. The applications cover the non-degenerate
geodesic flows on T1S2:= RP3 via
its double covering S3, and also
non-degenerate Hamiltonian systems in R4 restricted to
sphere-like energy
surfaces.