\documentclass{article} \parindent=0pt \begin{document}

\newcommand{\lig}[4]{\item[#1 to #2] {\it #3} : {#4}}
\newcommand{\cof}[2]{\item[#1 to #2] {\bf Coffee break}}
\newcommand{\lun}[2]{\item[#1 to #2] {\bf Lunch and discussion}}


\begin{center} \fbox{ \parbox{10cm}{\bf \begin{center} International
Workshop\\[2ex] CELESTIAL MECHANICS, SEPARATRIX SPLITTING, DIFFUSION \\[2ex]
Aussois, France June 21-27, 1998 \end{center} } }

\bigskip

\bigskip

{\bf  Scientific Organizing Committee ~:}

A. Chenciner, J. Laskar (Chair), C. Simo, S. Wiggins.

\bigskip

{\bf  Local Organizing Committee ~:}

S. Boatto, J. Laskar (Chair), P. Robutel, D. Sauzin, M. Tellier.

\end{center}

\bigskip\bigskip\bigskip

\begin{center} {\large PROGRAM} \bigskip \bigskip
\end{center}


{\bf Monday : }\\
\begin{description}
\lig{09:00}{09:30}{Jacques LASKAR}{Welcome address}
\lig{09:30}{10:30}{Carles SIM\'O}{Geometry of Hamiltonian Dynamics: Goals and
Methods}
\lig{10:30}{11:00}{Vered ROM-KEDAR}{Big islands appearing in near-ergodic
flows}
\cof{11:00}{11:30}
\lig{11:30}{12:00}{Luigi CHIERCHIA}{Action stability in perturbations of
properly degenerate Hamiltonian systems}
\lig{12:00}{12:30}{yoshihiro HIRATA}{Asymptotic expansion of the 2-dimensional
unstable manifold in a 4-dimensional symplectic mapping}
\lun{12:30}{16:30}
\lig{16:30}{17:15}{Zhihong Jeff XIA}{Arnold Diffusion, Variational Method and
Arnold-Mather Mechanism}
\lig{17:15}{17:45}{Luca SBANO}{The topology of the planar three-body problem with
zero total angular momentum and the existence of periodic orbits}
\cof{17:45}{18:15}
\lig{18:15}{18:45}{Angel JORBA }{On quasiperiodic solutions close to the
equilateral points of the real Earth-Moon system}
\lig{18:45}{19:15}{Robert MACKAY}{Hamiltonian Dynamics of Poincare's "orbites �
chocs"}
\end{description}

\bigskip\bigskip

�
{\bf Tuesday : }\\
\begin{description}
\lig{09:00}{09:30}{Amadeu DELSHAMS}{Perturbed ellipsoidal billiards}
\lig{09:30}{10:00}{Jean-Pierre MARCO}{Dynamics in the vicinity of double resonances}
\lig{10:00}{10:30}{Rafael de la LLAVE}{Some geometric approaches to problems in
diffusion }
\lig{10:30}{11:00}{Giovanni GALLAVOTTI}{Arnold's diffusion in presence of
strong perturbations}
\cof{11:00}{11:30}
\lig{11:30}{12:00}{Guido GENTILE}{Lower bounds for homoclinic splitting. Some
examples.I}
\lig{12:00}{12:30}{Vieri MASTROPIETRO}{Lower bounds for homoclinic
splitting. Some examples. II}
\lun{12:30}{16:30}
\lig{16:30}{17:00}{Stephen WIGGINS}{Exponential small separatrix splittings in
hamiltonian systems with three and more degrees of freedom}
\lig{17:00}{17:30}{Michael RUDNEV}{A geometric method for the splitting of
separatrices }
\cof{17:30}{18:00}
\lig{18:00}{18:30}{Jacky CRESSON}{Instability times along transition chain}
\lig{18:30}{19:00}{Pau MARTIN}{Arnold diffusion in perturbations of analytic
Hamiltonian systems}
\end{description}

�\bigskip\bigskip
\eject

{\bf Wednesday :}\\
\begin{description}
\lig{09:00}{10:00}{Stephen CREAGH}{Tunnelling in KAM systems}
\lig{10:00}{10:30}{Armando BAZZANI}{Diffusion in slowly modulated
Hamiltonian systems} 
\cof{10:30}{11:00}
\lig{11:00}{11:30}{Tetsuro KONISHI}{Itinerant behavior in 1-dimensional
mass-sheet model}
\lig{11:30}{12:00}{Stefano RUFFO}{Time scales for the relaxation to thermodynamical
equilibrium}
\lig{12:00}{12:30}{Pierre MOUSSA}{Real and complex Brjuno functions}
\end{description}

\bigskip\bigskip

�
{\bf Thursday :}\\
\begin{description}
\lig{09:00}{10:00}{Sergey V.BOLOTIN}{Diffusion trajectories with unbounded energy
for time periodic Hamiltonian systems}
\lig{10:00}{10:30}{Heinz HANSSMANN}{Bifurcations of lower-dimensional tori close to
an internal-normal resonance in quasi-periodically forced Hamiltonian systems}
\lig{10:30}{11:00}{Stefano MARMI}{Materialization of resonances in the linear
homological equation and in the semistandard map}
\cof{11:00}{11:30}
\lig{11:30}{12:00}{David SAUZIN}{Averaging and Borel transformation}
\lig{12:00}{12:30}{H.Scott DUMAS}{Averaging method for perturbations of mixing
flows}

\lun{12:30}{16:30}

\lig{16:30}{17:15}{Lev LERMAN}{Hyperbolic and elliptic behavior in Hamiltonian
systems with saddle-foci or saddle-centers }
\lig{17:15}{17:45}{Dario BAMBUSI}{Nekhoroshev theorem for small amplitude
solutions in  some nonlinear PDE's}
\cof{17:45}{18:15}
\lig{18:15}{18:45}{Giancarlo BENETTIN}{Nekhoroshev stability of elliptic
equilibria } 
\lig{18:45}{19:15}{Laurent NIEDERMAN}{Nonlinear stability around an
elliptic equilibrium point in a Hamiltonian system}
�\end{description}

\bigskip\bigskip

{\bf Friday :}\\
\begin{description}
\lig{09:00}{10:00}{Haruo YOSHIDA}{ A new necessary condition for the integrability
of Hamiltonian systems with a two dimensional homogeneous potential}
\lig{10:00}{10:30}{Juan Jose MORALES-RUIZ}{Non-integrability of Hamiltonian
systems by means of the variational equations}
\lig{10:30}{11:00}{Emmanuelle JULLIARD-TOSEL}{A non integrability result for the
inverse square potential}
\cof{11:00}{11:30}
\lig{11:30}{12:00}{Sergei DOVBYSH}{Some non-integrable multi-dimensional systems}
\lig{12:00}{12:30}{Satoru KUROSAKI}{The breakup of KAM tori and diffusion in
four-dimensional symplectic maps}
\lun{12:30}{16:30}
\lig{16:30}{17:15}{Dmitry TRESCHEV}{Multidimensional separatrix maps}
\lig{17:15}{17:45}{Vassili GELFREICH}{Examples of exponentially small transversality
of separatrices}
\cof{17:45}{18:15}
\lig{18:15}{18:45}{Nils BERGLUND}{Invariant manifolds and Nekhoroshev estimates in
adiabatic dynamical systems}
\end{description}

\bigskip\bigskip



\vfill\eject \begin{center} {\large PARTICIPANTS AND ABSTRACTS} \bigskip \bigskip
\end{center}





{\bf \large Simonetta  ABENDA\\} {\it Dipartimento di Matematica \& C.I.R.A.M.,
Universit� di Bologna, Piazza San Donato 5, I 40127 Bologna    Italie \\
abenda@ciram.ing.unibo.it\\[0.2cm]} {\bf  \\[0.2cm]} We  consider so called
hyperelliptically separable systems (h.s.s.) arising in various physical problems
whose generic inviariant manifolds are open subsets of strata of hyperelliptic
Jacobians or their coverings. It turns out that algebraic geometrical structure
of such systems has much in common with that of algebraic completely integrable
systems (a.c.i.s.). Using this property we study formal singular solutions of
a.c.i.s. and h.s.s. and give estimates for the number and leading behaviour of
their principal and lower balances.This can be regarded as a useful extension of
Kowalewski-  Painlev� integrability test. Some model examples are considered,
such as generalizations of the integrable case of the Henon-Heiles system.This
can be regarded as a useful extension of Kowalewski-Painlev� integrability test.
Some model examples are considered, such as generalizations of the integrable
case of the Henon-Heiles system. \\

{\bf \large Taehoon  AHN\\} {\it The Nonlinear Centre, Department of Applied
Mathematics and Theoretical Physics, University of Cambridge Silver Street,
Cambridge, CB3 9EW     UK \\ ath@euclid.postech.ac.kr\\[0.2cm]} {\bf  \\[0.2cm]}
\\

{\bf \large Alain  ALBOUY\\} {\it  Bureau des Longitudes Astronomie et Syst�mes
Dynamiques 77 avenue Denfert-Rochereau 75014 Paris \\  albouy@bdl.fr\\[0.2cm]}
{\bf  \\[0.2cm]} \\



{\bf \large Claude  BAESENS\\} {\it Dept of Applied Maths \& Theoretical Physics,
University of Cambridge, Silver Street, Cambridge CB3 9EW UK  \\
c.baesens@damtp.cam.ac.uk\\[0.2cm]} {\bf Gradient dynamics of tilted
Frenkel-Kontorova models \\[0.2cm]} The archetype of a Frenkel-Kontorova model is
a chain of balls and springs in a periodic potential.  Equilibrium states of such
a system are in one-to- one correspondence with the orbits of an area-preserving
twist map on the cylinder, via a Legendre transform. Here we consider the
gradient dynamics of Frenkel-Kontorova chains in a tilted periodic potential.  If
the tilt is sufficiently strong then there  are no equilibria but there is a
globally attracting periodic orbit, in the sense of a configuration which repeats
itself after a certain time shifted in space by one period.  This result is
proved in the following spaces of configurations: finite chains, spatially
periodic configurations, androtationally ordered configurations of irrational
mean spacing. A fundamental tool in our analysis is monotonicity of the gradient
flow with respect to a partial order on configurations, directly  analogous to
the use of the maximum principle for parabolic PDEs.  A second tool is Birkhoff
recurrence theorem. \\

{\bf \large Dario  BAMBUSI\\} {\it Dipartimento di Matematica Via Saldini 50
20133 Milano Italie \\ bambusi@mat.unimi.it\\[0.2cm]} {\bf Nekhoroshev theorem
for small amplitude solutions in some nonlinear PDE's \\[0.2cm]} Consider a
perturbation of linear Hamiltonian system with the following properties: (i) the
linear dynamics is periodic (ii) there exists a finite order Birkhoff normal form
which is integrable and quasi convex as a function of the action variables. I
will consider small amplitude initial data in which the majority of the energy is
stored in a finite number of oscillators, and show that along the corresponding
solutions all the actions of the linearized system are approximatively constant
up to times growing exponentially with a suitable small parameter. An application
to the nonlinear Shr\"odinger equation on a segment will be given. In particular
it will be proved that given initial data which are intere analytic functions of
finite order then all the Fourier coefficients are approximate constants of
motions for times longer than any power of a suitable small parameter.\\

{\bf \large Teresa  BARRAGAN\\} {\it Departament de Matematica Aplicada  I
Universitat Politecnica de Catalunya. Diagonal 647, E-08028 Barcelona Spain \\
barragan@ma1.upc.es\\[0.2cm]} {\bf  \\[0.2cm]} \\

{\bf \large Armando  BAZZANI\\} {\it Department of Mathematics University of
Bologna Piazza di Porta San Donato 5 40127 Bologna Italy \\
bazzani@bologna.infn.it\\[0.2cm]} {\bf DIFFUSION IN SLOWLY MODULATED HAMILTONIAN
SYSTEMS \\[0.2cm]} We consider the diffusion in the space of the adiabatic
invariants, when a parameter is slowly modulated. By using the neoadiabatic
theory we introduce a time energy map which turns out to be the integrator of a
quasi-hamiltonian syste, in order to study the evolution of the distribution
function of the invariants. We consider the case of 2 degrees of freedom systems
and we study a model which is interesting for the application to accelerator
Physics.\\

{\bf \large Gian Carlo  BENETTIN\\} {\it Universita di Padova Dipartimento di
Matematica Pura e Applicata Via G. Belzoni, 7 35131 Padova Italy \\
benettin@pdmat1.math.unipd.it\\[0.2cm]} {\bf  Nekhoroshev stability of elliptic
equilibria \\[0.2cm]} \\

{\bf \large Nils  BERGLUND\\} {\it Institut de Physique Th�orique EPFL,
PHB-Ecublens CH-1015 Lausanne Switzerland \\ berglund@iptsg.epfl.ch\\[0.2cm]}
{\bf Invariant manifolds and Nekhoroshev estimates in adiabatic dynamical systems
\\[0.2cm]} We consider n-dimensional adiabatic dynamical systems of the form
epsilon dx/dt = f(x,t). We prove existence of linear and nonlinear invariant
manifolds near hyperbolic fixed points. These manifolds admit asymptotic series
in the adiabatic parameter epsilon, which can be truncated in such a way that
they approach these manifolds exponentially closely.\\


{\bf \large Stefanella  BOATTO\\} {\it Astronomie  \& Syst�mes dynamiques CNRS
URA 707 Bureau des Longitudes 77 avenue Denfert-Rochereau 75014 Paris \\
lella@bdl.fr\\[0.2cm]} {\bf  \\[0.2cm]} \\

{\bf \large Sergey V.   BOLOTIN\\} {\it Department of Mathematics and Mechanics,
Moscow State University. Vorob'evy Gory, Moscow 119899 Russia \\
bolotin@mech.math.msu.su\\[0.2cm]} {\bf Diffusion trajectories with unbounded
energy for time periodic Hamiltonian systems. \\[0.2cm]} The result of Mather on
the existence of trajectories with unbounded energy for time periodic Hamiltonian
systems on a torus is generalized to a class of Hamiltonian systems possessing a
hyperbolic invariant torus. Instead of the variational methods of Mather, a
geometrical approach based on KAM theory and the Poincare-Melnikov method is
used. This makes it possible to study a more general Hamiltonian systems, but
requires additional regularity assumptions on the Hamiltonian.\\



{\bf \large Timoteo  CARLETTI\\} {\it Dept of mathematics �U. Dini� University of
Florence Viale Morgagni  67/A 50134 Firenze Italy \\ carletti@bdl.fr
carletti@udini.math.unifl.it\\[0.2cm]} {\bf Scaling properties and functional
properties of the e-critic in the standard map. \\[0.2cm]} We study the scaling
low of the function e-critic when the frequency approaches a rational value. We
give also a numerical proof of the fact that log e-critic is a bmo(p4) function.
These results ae a joint work with prof. J. Laskar \& S. Marmi.\\

{\bf \large Paolo  CARTA\\} {\it Via E. Checchi 61 (C/O Longo), 00157 ROMA \\
carta@ipparco.roma1.infn.it\\[0.2cm]} {\bf  \\[0.2cm]} \\

{\bf \large Enric  CASTELLA\\} {\it Cami Ral, 430 Mataro (Barcelona) Espagne
08302 \\ enric@maia.ub.es\\[0.2cm]} {\bf  \\[0.2cm]} \\

{\bf \large Cristel  CHANDRE\\} {\it Laboratoire de Physique Universit� de
Bourgogne 21011 Dijon France \\ cchandre@jupiter.u-bourgogne.fr\\[0.2cm]} {\bf
Critical strange attractor of an approximate renormalization-group scheme for
quasi- periodically driven rotators \\[0.2cm]} We study an approximate
renormalization-group transformation to analyze the breakup of invariant tori for
three dimensional Hamiltonian systems. Two basic approximations are involved in
this transformation: we truncate the Hamiltonians to quadratic order in the
action variables, and we neglect all the Fourier modes except the three most
relevant ones at each iteration of the transformation. The scheme is implemented
for the spiral mean torus. We find numerically that the critical surface is the
stable manifold of a critical strange attractor. We compute some scaling
exponents associated with this fixed set, and find that they can be expected to
be universal. \\


{\bf \large Yi-Chiuan  CHEN\\} {\it Dept. of Applied Maths. and Theoretical Phys.
University of Cambridge Silver street, Cambridge  CB3 9EW UK   \\
Y-C.Chen@damtp.cam.ac.uk\\[0.2cm]} {\bf  \\[0.2cm]} \\

{\bf \large Alain  CHENCINER\\} {\it  Bureau des Longitudes Astronomie et
Syst�mes Dynamiques 77 avenue Denfert-Rochereau 75014 Paris \\
chenciner@bdl.fr\\[0.2cm]} {\bf  \\[0.2cm]} \\

{\bf \large Luigi  CHIERCHIA\\} {\it Dipartimento di Matematica, Universita' di
Roma III Largo San Leonardo Murialdo 1, 00146 Roma            Italie \\
luigi@matrm3.mat.uniroma3.it\\[0.2cm]} {\bf Action stability in perturbations of
properly degenerate Hamiltonian systems \\[0.2cm]} Various stability and
instability results (including Arnold diffusion) are proved in the context of
perturbations of  properly degenerate Hamiltonian systems. \\

{\bf \large Vittorio  COTI ZELATI\\} {\it Dipartimento di Matematica �R.
Caccioppoli� Universit� di Napoli �Federico II� Via Cintia 80126 Napoli Italy \\
zelati@unina.it\\[0.2cm]} {\bf Homoclinic solutions and chaotic behaviour for a
class of rapidly forced second order Hamiltonian system \\[0.2cm]} We prove that
a class of systems of the form $-\ddot{q} + q = (1 + a(\omega t) + b(t /
\omega))V'(q)$ has a chaotic behaviour provided: $a$ and $b$ are periodic,
$\omega$ is small, $V: R^n \to R$ is superquadratic.\\


{\bf \large Stephen  CREAGH\\} {\it Service de Physique Theorique CEA/Saclay
F91191 Gif-sur-Yvette Cedex \\ \\[0.2cm]} {\bf Tunnelling in KAM systems
\\[0.2cm]} Formal calculations of quantum mechanical tunnelling effects   weakly
nonintegrable systems require the analytic  continuation of invariant tori to
complex angles. Natural boundaries, formed by dense accumulations of
singularities along lines in the complex plane prevent  a satisfactory resolution
of this problem, and the classical aspect remains essentially ununderstood. One
class of  problem for which some analysis might be possible are rapidly forced
systems such are responsible for exponentially small separatrix splittings. It is
hoped that the same techniques as used there might allow determination of the
classical structures demanded by semiclassical calculations, at least in
restricted cases.\\

{\bf \large Jacky  CRESSON\\} {\it Equipe de Mathematiques de Besancon CNRS-UMR
6623 Universite de Franche-Comte 16 route de Gray 25030 Besancon Cedex France \\
cresson@math.jussieu.fr\\[0.2cm]} {\bf  Instability times along transition
chain\\[0.2cm]} We discuss several geometrical methods to compute time of drift
along transition chains. The first one, initiated by J.P. Marco used the Easton's
windowing method. The second one is based on a new geometric tool called the
transfer lemma and the proof of the transitivity property for Treshchev tori. The
third one, is based on symbolic dynamics and periodic orbits for Treshchev tori
with a transversal homoclinic connection. We prove that near a transition chain
of Treshchev tori there exits a "dual" chain of periodic hyperbolic orbits. Then,
we compute the time of instability along this chain. These methods can be used to
prove the Chirikov's conjecture about the optimality of the stability exponent of
the Nekhoroshev theorem. \\

{\bf \large Maria Gabriella  DELLA PENNA\\} {\it Observatoire de la C�te d' Azur,
Dpt. Cassini B.P. 4229 06304 Nice cedex  4 France \\
gabry@purcell.obs-nice.fr\\[0.2cm]} {\bf  \\[0.2cm]} \\

{\bf \large Amadeu  DELSHAMS\\} {\it Departament de Matematica Aplicada I
Universitat Politecnica de Catalunya Diagonal 647, 08028 Barcelona Espagne \\
amadeu@ma1.upc.es\\[0.2cm]} {\bf Perturbed ellipsoidal billiards (with Y.~Fedorov
and R.~Ram{\'\i}rez-Ros) \\[0.2cm]} We consider perturbations of general
ellipsoidal billiards as a paradigm of a nearly-integrable 4D-discrete Lagrangian
system. In the integrable case, we give the explicit biasymptotic solutions and
describe the corresponding asymptotic variety. The persistence of some of these
biasymptotic solutions under perturbations is studied by means of the Melnikov
potential. For non-trivial polynomial peturbations, the Melnikov potential is
non-constant and the variety of biasymptotic solutions breaks up. Some examples
with explicit computations are presented.\\


{\bf \large Florin  DIACU\\} {\it Department of Mathematics and Statistics
University of Victoria P.O.Box 3045 Victoria, B.C. Canada, V8W 3P4 \\
diacu@math.uvic.ca\\[0.2cm]} {\bf  On the Anisotropic Manev Problem \\[0.2cm]}
Abstarct: The anisotropic Kepler problem describes the motion of two  bodies in
an Euclidean space in which the classical gravitational  force acts differently
in each direction. It has been considered in order to find connections between
classical and quantum mechanics. The Manev problem is isotropic, but the
potential is the sum between the inverse and the inverse square of the distance.
Here we combine the above two aspects, that is, in the Manev potential we include
an anisotropy of the space. Using McGehee coordinates, we blow up the collision
singularity, paste a collision manifold to the phase space, study the flow on and
near the collision manifold, and find a positive-measure set of collision orbits.
In the zero-energy case we study the escape solutions by defining an infinity
manifold. Then we describe all possible connections between equilibria and/or
cycles at collision and at infinity. In this way we find the main  qualitative
features of the global flow on the zero-energy level.\\

{\bf \large Sergei  DOVBYSH\\} {\it Institute of Mechanics, Moscow State
University, Michurinskij pr-t, d.1. Moscow 117192 RUSSIE \\
dovbysh@inmech.msu.su\\[0.2cm]} {\bf  Some non-integrable multi-dimensional
systems \\[0.2cm]} Recently, the author established the non-integrability
conditions for multi- dimensional systems which are related to the presence of a
set of quasi-random motions possessing a ``rough structure''. These conditions
concerns either 1) homoclinic structure formed by transversally intersecting
separatrices of hyperbolic periodic solutions, or 2) the phenomenon of branching
of solutions in the complex domain as being considered from the viewpoint of
symbolic dynamics methods. Here, the non-integrability is understood in the
strongest analytic sense, i.e. as the absence of a non-trivial analytic or
meromorphic first integral on a level of a priori first integrals. Moreover, it
implies the absence of a non-trivial analytic one-parameter symmetry group, or,
more generally, meromorphic vector field generating a local phase flow commuting
with the system under consideration. Another new faithful generalization is the
discreteness of the analytic centralizer in the compact-open topology. We discuss
applications of the non-integrability conditions in various dynamical systems
arising in physics and mechanics (such as a three-component homogeneous
Yang--Mills field, a spherical pendulum with the horizontally oscillating point
of suspension, the planar problem of more than three point vortices in an ideal
incompressible liquid, the planar problem of more than two bodies attracting by
the Newton law). The following observation can be very useful in the
consideration of systems of many weakly interacting particles in an external
time-periodic force field. If a given diffeomorphism $S$ possesses a homoclinic
structure satisfying the non-integrability conditions then the same is valid for
the direct product of any finite set of copies of $S$. Using the robust character
of the non-integrability conditions (i.e. the fact that they persist under small
perturbations) one can transfer immediately the non-integrability result for a
single particle to the system of many particles. This approach was utilized in
proving the non-integrability of the many-vortex the and many-body systems.\\



{\bf \large H.  Scott  DUMAS\\} {\it Department of Mathematics University of
Cincinnati Cincinnati, Oh  45221 - 0025 USA \\ Scott.Dumas@uc.edu   or
dumas@dmi.ens.fr\\[0.2cm]} {\bf Averaging method for perturbations of mixing
flows  (joint work with F. Golse) \\[0.2cm]} For a class of multiphase averaging
problems in which  the unperturbed flow of the fast variables is a transitive
Anosov flow or is sufficiently rapidly mixing, we obtain what is probably an
optimal power-law estimate, in the small parameter, of the rate at which the
average maximum difference between solutions of the exact and averaged problems
converges to zero.  This verifies and extends an earlier conjectural remark by
V.I. Arnold.\\



{\bf \large Yves  ELSKENS\\} {\it �quipe Turbulence Plasma du Lab. de Physique
des interactions ioniques et mol�culaires, UMR 6633 CNRS-Universite de Provence
Centre univ. de Saint-J�r�me, ser. B 22, case postale 321, Av. Escadrille
Normandie-Niemen, 13397 Marseille cedex 20 \\
elskens@newsup.univ-mrs.fr\\[0.2cm]} {\bf Mean field hamiltonian models in  one
dimension : rotators, plasma, gravitation and wave-particle self-consistent dynamics
\\[0.2cm]}

A family of one-dimensional
spatially periodic systems of $N$ classical particles
interacting by a long-range pair force is investigated.
This force is the (smooth)
long-range part of the 1-D Coulomb or gravitational interaction ;
the family includes the mean-field hamiltonian rotator model.

For initial conditions close to thermal equilibrium,
the dynamics generates long-wavelength excitations
corresponding to Langmuir modes of the plasma,
with the Bohm-Gross dispersion relation.
The coupling between these modes and near-resonant particles
is described in the same way as for the Coulomb case,
and the resulting wave-particle dynamics is described by
a self-consistent effective hamiltonian.

The self-consistent dynamics of $N$ classical particles
interacting with $M$ modes is therefore investigated.
It is usually described by a kinetic model
in the limit $N \to \infty$, and this limit is proved to
commute with the time evolution. However, finite $N$ effects
are important to understand the actual evolution of
molecular dynamics simulations.
Such simulations, performed for the classical beam-mode instability
(Landau growth), have revealed a destabilization of the first
`trapping oscillations' saturation regime,
pointing to finite-$N$ effects.
Similar effects are observed in the linear and nonlinear regimes
of collisionless Landau damping.
The implementation of numerical schemes for kinetic equations
leads to effects similar to finite $N$.

Initial conditions generating turbulent structures
are also considered.
These structures are density holes in $(x,v)$-space
which produce non-gaussian probability distribution of fluctuations
of the particle distribution function $f(x,v,t)$,
as predicted by the kinetic theory of clumps in plasmas.
Their evolution is shown to be controlled by
the motion of the particles in the
$(x,v)$-space domain swept by the separatrix
associated with the longest-range coupling field components,
which implies that their lifetime is
proportional to the number $N$ of particles.
Relaxation times are found to be
quite insensitive to the presence of turbulent structures
and to spatial scales smaller than the Debye length. \\



{\bf \large Corrado  FALCOLINI\\} {\it Dipartimento di Matematica Universita' di
Roma "Tor Vergata" Via della Ricerca Scientifica 00133 Roma Italy \\
falcolin@mat.utovrm.it\\[0.2cm]} {\bf  \\[0.2cm]} \\


{\bf \large Yuri  FEDOROV\\} {\it Departament de Matematica Aplicada II
Universitat Politecnica de Catalunya Edifici U, c.P. Gargallo 5 08208 Barcelona
Espagne \\ fedorov@grec.upc.es\\[0.2cm]} {\bf  \\[0.2cm]} \\

{\bf \large Jacques   FEJOZ\\} {\it Astronomie et Syst�mes Dynamiques Bureau des
Longitudes 77 avenue Denfert-Rochereau 75014 Paris \\ fejoz@bdl.fr\\[0.2cm]} {\bf
Isotropic Invariant Tori in the Planar Three-Body Problem \\[0.2cm]}

I have studied the secular system of the planar three body problem when the ratio
of the semi major axes is small. It is a completely integrable hamiltonian system
on the \(2\)-sphere. It has the prominent role in the way the disturbing function
pertubs the \(2\)-frequency degenerate keplerian motion.

Indeed, in the (non averaged) three body problem, a positive measure of regular
points of the secular system gives rise to Lagrangian invariant \(4\)-tori. On
the other hand a positive measure of non degenerate singular points of the
secular system gives rise to isotropic invariant \(3\)-tori. So, these tori have
intermediate dimension between Lagrangian tori and degenerate Keplerian tori, and
their normal dynamics is elliptic or hyperbolic. The proof relies on Herman's
versions of isotropic invariant tori theorems. \\

{\bf \large Sebastian  FERRER\\} {\it Departamento de Matem�tica Aplicada,
Universidad de Murcia, Avenida Alfonso XIII, 42 30203 Cartagena, Murcia Espagne
\\ ferrer@plc.um.es\\[0.2cm]} {\bf Reduction of harmonic oscillators in 1-1-1
resonance: Relative equilibria and bifurcations of Quartic perturbations
\\[0.2cm]} This communication deals with the dynamics of a Hamilton system with
three equal modes around an elliptic equilibrium. The Taylor expansion is
truncated at fourth order. The process of the reduction, due to the oscillator
symmetry, brings the system to the orbit space $\mbox{\bf CP}2$. In system with
axial symmetry a singular reduction can be carried out. This permits to make a
global study of one-degree of freedom system which lies on a double reduced
space. As an application we analyze a parametric family modelling an elliptical
galaxy, showing its relative equilibria, bifurcations and stability.\\

{\bf \large Giovanni  GALLAVOTTI\\} {\it Dipartimento di Fisica Universita' La
Sapienza P.le Moro 2 00185 Roma Italie \\
giovanni@ipparco.roma1.infn.it\\[0.2cm]} {\bf Arnold's diffusion in presence of
strong perturbations (title subject to change) \\[0.2cm]} An illustration in a
simple case of the general method used in the work by Chierchia, Gallavotti and
discussion of related problems in a priori stable systems\\

{\bf \large Vassili  GELFREICH\\} {\it Institut fur MathematikFreie Universitat
Berlin Arnimallee 2-6 D-14195 Berlin Germany \\ gelf@maia.ub.es\\[0.2cm]} {\bf
Examples of exponentially small transversality of separatrices \\[0.2cm]} The
condition, which ensure the exponential smallness of the separatrices splitting,
are rather well known. On the other hand establishing the transversality of the
separatrices for such systems is an extremely difficult problem. In the talk some
examples are described. Each system has a transversal homoclinic trajectory and
the corresponding splitting is exponentially small with respect  to the
perturbation parameter. This includes the famous standard map, periodic and
quasiperiodic perturbations of the pendulum and some other examples. The special
attention is given to comparing of the actual value of the splitting with the
Poiancare-Arnold- Melnikov theory. \\

{\bf \large Guido  GENTILE\\} {\it Universit� di Roma Dipartimento di Matematica
Largo San Leonardo Murialdo 1 00146 Roma Italie \\ gentileg@ipparco.roma1.infn.it
gentile@matrm3.mat.uniroma3.it\\[0.2cm]} {\bf Lower bounds for homoclinic splitting.
Some examples. I. \\[0.2cm]} 
 The problem of finding lower bounds for the homoclinic splitting of whiskers is
discussed in some simple asymptotic models, in which the first non trivial order
dominance can be shown to hold. The proof uses the Lindstedt series expansion,
relying on field theory techniques for discussing convergence and exhibiting the
cancellations which, operating to all perturbative orders, allow a detailed analysis
of the asymptotics.\\

{\bf \large Giordano  GIORGI\\} {\it Via Sirtori, 69 00149 Roma Italia \\
giordagi@tin.it\\[0.2cm]} {\bf  \\[0.2cm]} \\

{\bf \large Pere  GUTIERREZ\\} {\it Departament de Matematica Aplicada II
Universitat Politecnica de Catalunya Pau Gargallo 5 E-08028 Barcelona Spain \\
gutierrez@ma2.upc.es\\[0.2cm]} {\bf  On Melnikov theory for the splitting
potential of whiskered tori in regular and singular hyperbolic Hamiltonians (with
Amadeu Delshams) \\[0.2cm]} A perturbation of an integrable Hamiltonian system,
having hyperbolic invariant tori with coincident whiskers, is considered. Our
goal is to detect homoclinic intersections between the perturbed  whiskers, and
to measure the splitting distance. Using a geometric approach (based on a paper
by Eliasson), the splitting distance can be shown to be the gradient of a (scalar
and periodic) splitting potential. For a regular (or strongly hyperbolic) system,
a first order approximation for this potential is given through Melnikov theory.
This approach is designed also in the singular (or weakly hyperbolic) case, which
is more involved and constitutes a model for a nearly-integrable Hamiltonian near
a single resonance. The Melnikov potential in this singular case is studied, and
is expected to be useful in order to give predictions of the splitting under some
weak hypotheses on the perturbation. Explicit computations in concrete examples
are also given.\\



{\bf \large Heinz  HANSSMANN\\} {\it Institut f\" ur Reine und angewandte
Mathematik RWTH Aachen 52056 Aachen Germany \\
Heinz@iram.rwth-aachen.de\\[0.2cm]} {\bf Bifurcations of lower-dimensional tori
close to an internal-normal resonance in quasi-periodically forced Hamiltonian
systems \\[0.2cm]} Let us consider a quasi-periodic time-dependent perturbation
of an autonomous Hamiltonian system. Let us focus on an elliptic equilibrium
point of the unperturbed Hamiltonian. Under suitable hypotheses, this equilibrium
point becomes an invariant torus under the effect of the perturbation, with the
same frequencies as the perturbation. The normal frequencies of this torus are a
perturbation of the normal frequencies around the initial equilibrium point. In
this work we consider the interaction between these frequencies. We add a
parameter to the system such that the internal frequencies are always fixed
(hence, the torus is not destroyed because of an internal resonance) and the
normal ones move with the parameter. In the talk we will present a numerical
study about several bifurcations that take place when the normal and internal
frequencies are in resonance. We will also discuss the theoretical support for
those bifurcations.\\

{\bf \large Alex  HARO\\} {\it Departament de Matematica Aplicada i Analisi
Facultat de Matematiques (Univ. Barcelona) Gran Via de les Corts Catalanes 585
08007 Barcelona (SPAIN) \\ haro@cerber.mat.ub.es\\[0.2cm]} {\bf  \\[0.2cm]} \\

{\bf \large Michael  HERMAN\\} {\it \\ \\[0.2cm]} {\bf  \\[0.2cm]} \\

{\bf \large Yoshihiro  HIRATA\\} {\it R-lab., Dept. of Phys., Nagoya Univ.,
Furo-cho Nagoya, 464-01, Japan  \\ yhirata@allegro.phys.nagoya-u.ac.jp\\[0.2cm]}
{\bf Asymptotic expansion of the 2-dimensional unstable manifold in a
4-dimensional symplectic mapping \\[0.2cm]} It is a very interesting problem to
construct analytical representations of the unstable manifold in symplectic
mappings. Recently an analytical method for constructing the unstable manifold of
2-dimensional symplectic mappings is reported in various systems. This method is
called ``asymptotic expansions beyond all orders'', which is a kind of singular
perturbation methods. The phase space structure of 4- or more-dimensional
symplectic mappings is different from that of 2-dimensional symplectic mappings.
Furthermore it is difficult to understand global structure of phase space of the
high- dimensional systems and it is important to compose analytical approaches.
Therefore we have tried to extend and apply the method mentioned above to
4-dimensional symplectic mappings.  In the previous paper
(http://xxx.lanl.gov/abs/chao-dyn/9802009), we analytically obtained a
1-dimensional sub-manifold of the unstable manifold in a 4-dimensional symplectic
mapping. And we showed that the analytically obtained manifold agrees well with
numerical trajectories, and discussed the high-dimensionality of the
4-dimensional symplectic mapping. In this paper we shall construct the
2-dimensional unstable manifold in a 4-dimensional symplectic mapping. We will
discuss the phase space structure of 4-dimensional symplectic mappings from the
view point of the unstable manifold.\\

{\bf \large Poul G.  HJORTH\\} {\it Department of Mathematics, B-303 Technical
University  of  Denmark DK-2800  Lyngby \\ P.G.Hjorth@mat.dtu.dk\\[0.2cm]} {\bf
Breaking of Adiabatic Invariance in Separatrix Crossing \\[0.2cm]} The change in
the adiabatic invariants associated with orbits that undergo separatrix crossing
is studied for a  pendulum Hamiltonian, where either the gravitational field
direction or the field strength is allowed to be a  slowly- varying function of
time.\\


{\bf \large Maylis  IRIGOYEN\\} {\it Universit� Paris 6, Tour 46-00, 5� �tage,
case 247 4 Place Jussieu 75252 Paris Cedex 05 \\
irigoyen@math.jussieu.fr\\[0.2cm]} {\bf  \\[0.2cm]} \\

{\bf \large Sylvie  JANCART\\} {\it 8 Rempart de la Vierge FundP Math 5000 Namur
Belgique \\ sja@math.fundp.ac.be\\[0.2cm]} {\bf  \\[0.2cm]} \\

{\bf \large Hans-Rudolf  JAUSLIN\\} {\it Laboratoire de Physique Universit� de
Bourgogne BP 400 21011 Dijon France \\ jauslin@u-bourgogne.fr\\[0.2cm]} {\bf
KAM-renormalization Group approach to instabilities in Hamiltonian flows
\\[0.2cm]} We present some recent work on the stability of KAM tori in
Hamiltonian flows. The analysis is based on a combination of  KAM transformations
and renormalization group ideas. The scheme is implemented numerically for a
family of Hamiltonians quadratic in the actions. We obtained results that
indicate that the KAM iteration can converge up to the critical coupling at which
the torus breaks up. Adding a renormalization scheme consisting of  a rescaling
and a shift of the relevant resonances, we obtain a much more efficient method
that allows to determine the critical coupling with high accuracy. We determine a
non-trivial fixed point of the renormalization transformation, and discuss the
associated universality properties. \\

{\bf \large Angel  JORBA\\} {\it Departament de Matematica aplicada i analisi
Universitat de Barcelona Gran Via 585 08007 Barcelona Espagne \\
angel@maia.ub.es\\[0.2cm]} {\bf On quasiperiodic solutions close to the
equilateral points of the real Earth-Moon system \\[0.2cm]} In this talk we will
consider the existence of invariant tori near the equilateral libration points of
the Earth-Moon system. We will use several models for this study, ranging from
the simplest one (the spatial Restricted Three Body Problem, from now on RTBP) to
the one provided by the JPL ephemeris (this is the model we refer to as �real
system�). The RTBP model is an autonomous three degrees of freedom Hamiltonian
system  where the equilateral points are two elliptic equilibrium points. Then,
KAM methods can be used to derive the existence of invariant tori near the
points, and using standard Nekhoroshev estimates one can produce a region of
effective stability around them. However, this estimated region is much smaller
than the region obtained by purely numerical methods. On the other hand,
numerical simulations for the JPL model (where the geometrically defined
equilateral points are no longer equilibrium solutions) show that there are no
effective stability regions around these points. This is because the
perturbations to the RTBP coming from the non circular motion of Earth and Moon
plus the effect of Sun and the other planets is big enough to produce a change of
stability. However, numerical simulations seem to show the existence of maximal
dimension invariant tori at some distance from the points. This implies the
existence of regions with effective stability. During the talk we will discuss
these issues using some intermediate models, trying to elucidate the relevant
dynamics fro them. An important tool in the methodology presented is the
numerical computation of normal forms and invariant tori. We will also present
theoretical results about the existence of quasiperiodic solutions in order to
give support to some of the numerical results presented.\\

{\bf \large Emmanuelle  JULLIARD-TOSEL\\} {\it Astronomie et Systeme Dynamiques.
Bureau des longitudes. Paris. France.\\ julliard@bdl.fr\\[0.2cm]} {\bf A non
integrability result for the inverse square potential \\[0.2cm]} We show the
analytical non-integrability of the $4$-body problem on the line for an inverse
square potential and masses $(1,m,m,1),m\ne 1$. This is the first case left open
by the integrability results of Jacobi ($3$bodies with arbitrary masses) and
Calogero-Moser ($n$bodies with equal masses). \\

{\bf \large Tetsuro  KONISHI\\} {\it R-Lab., Department of Physics, School of
Science, Nagoya University Nagoya, 464-01 Japan \\
tkonishi@allegro.phys.nagoya-u.ac.jp\\[0.2cm]} {\bf Itinerant behavior in
1-dimensional mass-sheet model  \\[0.2cm]} Long time behavior of Hamiltonian
systems with many degrees of freedom are often well described by  classical
statistical mechanics. This implies that we observe simple diffusive behavior for
those systems.  On the other hand, there are many Hamiltonian  systems which are
chaotic and at the same time do not relax to  thermal equilibrium uniformly. They
have several states which are macroscopically distinguishable and their equations
of motion take them from one state to another by purely deterministic way. This
itinerant behavior is significant in the understanding of  Hamiltonian chaos for
many degrees of freedom. Physical examples  are found in microclusters,
gravitational systems, and so on. The itinerant behavior is different from
1/f-type long time tail behavior, because the long time tail is considered to be
caused by motions trapped to local hierarchy of KAM tori and islands, whereas the
itinerancy takes the systems to various parts in the whole phase space. We
introduce several models which show the itinerant behaviors. One is a simplectic
coupled map which have finite range interaction, which shows collision and
reconfiguration of clusters. Another is one-dimensional self-gravity model
(``sheet model''). We numerically analyse the dynamics of these models.
Non-thermal  transition among states are found, and relation to phase space
structure is discussed.  For the sheet model, we found that  mechanism which
trigger the itinerancy  is resonance between particles and mean field potential.

Key words: chaotic itinerancy, relaxation, non-thermal behavior, cluster, sheet
model, self-gravitational system, phase space structure.\\

{\bf \large Wang Sang  KOON\\} {\it Control and Dynamical Systems California
Institute of Technology 116-81 Pasadena, CA 91125 USA \\
koon@cds.caltech.edu\\[0.2cm]} {\bf  \\[0.2cm]} \\

{\bf \large Satoru  KUROSAKI\\} {\it Department of Applied Physics School of
Science and Engineering Waseda University 3-4-1, Okubo Shinjuku-ku Tokyo 169
Japan \\ kurosaki@aizawa.phys.waseda.ac.jp\\[0.2cm]} {\bf The breakup of KAM tori
and diffusion in four-dimensional symplectic maps \\[0.2cm]} The breakup of KAM
tori in four-dimensional symplectic maps is  studied. KAM tori are constructed
explicitly by the conjugation technique. The succession to the breakup depends on
the perturbations which determine the dominant type of small divisors in the
conjugation function. The secular component of orbits is sensitive to the breakup
of   KAM tori close to it. We construct the resonance structure on the frequency
plane based on that observation. The resonance structure is related to the Arnold
web. The region fulfilled by KAM tori prevents orbits diffusing but elliptic
normal modes do not.\\

{\bf \large Leonid  KUZNETSOV\\} {\it 4 Washington Place, Box 52 New York
University New York, NY 10003, USA \\ kuzntsvl@cims.nyu.edu\\[0.2cm]} {\bf
Chaotic Advection in Three-point-vortex system \\[0.2cm]} Advection in the FLOW
PRODUCED BY THREE MOVING POINT VORTIVES WITH IDENTICAL strength is analyzed in
detail. Methods of chaotic dynamics are applied to find the border between
regular and chaotic motion of the tracers, numerical simulations reveal various
topological structures in the advection pattern in particular, strong stickness
of the vortex cores was observed for some parameter values implications for the
transport are briefly discussed.\\

{\bf \large Jacques  LASKAR\\} {\it  Bureau des Longitudes Astronomie et Syst�mes
Dynamiques 77 avenue Denfert-Rochereau 75014 Paris \\  laskar@bdl.fr\\[0.2cm]}
{\bf  \\[0.2cm]} \\

{\bf \large Jos� Tom�s  LAZ\`ARO\\} {\it Dept. Matem�tica Aplicada i Telem�tica
Escola Universit�ria Polit�cnica Av. Victor Balaguer, s/n E-08800 Vilanova i la
Geltru (Barcelona) Espagne \\ lazaro@mat.upc.es\\[0.2cm]} {\bf Splitting in
Reversible Systems (with Amadeu Delshams) \\[0.2cm]} We deal with reversible
systems, i.e. systems invariant under a linear involution in the phase space and
a reversion in the time parametrization. Our aim is to study the measure of the
splitting of the separatrices of a paradigmatic example: a pendulum forced by a
quasiperiodic, non-Hamiltonian, reversible perturbation. For this purpose, we
present a convergent iterative scheme leading the perturbed reversible system to
its Normal Form, by means of a sequence of suitable transformations, preserving
reversibility and providing good estimates of the loss of domain in the angular
variables. \\

{\bf \large Lev  LERMAN\\} {\it Research Institute for Applied Mathematics \&
Cybern 10 Ul'yanovSt., Nizhny Novgorod  603005 Russia \\
lerman@focus.nnov.ru\\[0.2cm]} {\bf  Hyperbolic and elliptic behavior in
Hamiltonian systems with saddle-foci or saddle-centers \\[0.2cm]} Last results
concerning orbit structure in Hamiltonian systems with two and more degrees of
freedom having homoclinic orbits and heteroclinic contours to saddle-focus or
saddle-center equilibriums will be discussed. This includes hyperbolic subsets
and their description by means of symbolic dynamics continuously depending on
Hamiltonian levels and parameters, tangencies of separatrix manifolds and related
creation of elliptic periodic orbits, other bifurcational phenomena in such
systems. Applications of results obtained to systems from different  branches of
physics will be pointed out.\\



{\bf \large Rafael  LLAVE  (de la)\\} {\it Dept. of Mathematics University of
Texas Austin TX 78712-1802 USA \\ \\[0.2cm]} {\bf Some geometric approaches to
problems in diffusion \\[0.2cm]} I plan to discuss haw some recent results can be
formulated in a more geometric manner and, hence obtain a different understanding
on them.\\


{\bf \large Ugo  LOCATELLI\\} {\it Observatoire de la Cote d'Azur, Bd de
l'Observatoire, BP 4229, 06304 Nice France \\ ugo@obs-nice.fr\\[0.2cm]} {\bf
\\[0.2cm]} \\

{\bf \large Pierre  LOCHAK\\} {\it \\ Pierre.Lochak@ens.fr\\[0.2cm]} {\bf
\\[0.2cm]} \\



{\bf \large Robert  MACKAY\\} {\it Silver Street Cambridge CB3 9EW UK \\
rm214@damtp.cam.ac.uk\\[0.2cm]} {\bf Hamiltonian Dynamics \\[0.2cm]} Poincar�'s
<<orbites � chocs>> In Chapter XXXII of �Les m�thodes nouvelles de la m�canique
c�leste�, Poincar� claimed existence of orbits of the three-body problem which in
the limit as two of the masses go to zero (but with well defined ratio) tend to
piecewise ellipses with a collision at each point of change of direction. His
proof looks to me like wishful thinking, but the idea is good and I will provide
a proof.\\

{\bf \large Jean-Pierre  MARCO\\} {\it Institut de Math�matiques de Jussieu.
Universit� Paris 6 Case 247 4 place Jussieu 75252 Paris cedex 05 \\
marco@math.jussieu.fr\\[0.2cm]} {\bf Dynamics in the vicinity of double
resonances. \\[0.2cm]} We consider a near-integrable hamiltonian system and
examine its behaviour near double resonances. An averaging allows to consider the
system as a perturbation of the product of  natural system on the cotangent
bundle of the 2 torus and a trivial system. One can prove, under natural
conditions, the existence of chains hyperbolic periodic orbits for the first
system, and as a consequence after suspension and perturbation, the existence of
orbits circulating around the double resonance. \\

{\bf \large Stefano  MARMI\\} {\it Dipartimento di Matematica Ulisse Dini Viale
Morgagni 67/A 50134 Firenze Italy \\ marmi@udini.math.unifi.it\\[0.2cm]} {\bf
Materialization of resonances in the linear homological equation and in the
semistandard map \\[0.2cm]} The problem of the materialization of resonances in
perturbation series of KAM theory has been first addressed by V. I. Arnol'd in
his study of the diffeomorphisms of the circle. In this case the analytical and
the dynamical origin of the divergence of the series is quite weel understood
thanks to the fundamental contributions of Herman, Yoccoz and Perez-Marco. Much
less is known is more general situations including Hamiltonian systems. In the
first part of this lecture I will  briefly describe a result on the quasianalytic
monogenic properties of the solutions of one dimensional linear homological
equations obtained in collaboration with D. Sauzin. Our results give a partial
answer to a question of M. Herman.  In the second part I will describe a result
on the behaviour at resonances of the linearization of the semi--standard map and
state some open problems  on the Borel summability and quasianalytic properties
of the linearization. This second part is joint work with A. Berretti and D.
Sauzin.\\

{\bf \large Pau  MARTIN\\} {\it Departament de Matematica Aplicada i Telematica
Ed-C3, Campus Nord, Universitat Politechnica de Catalunya, c/ Jordi Girona, 1-3
08034 BARCELONA Spain \\ martin@mat.upc.es\\[0.2cm]} {\bf Arnold diffusion in
perturbations of analytic integrable Hamiltonian systems. \\[0.2cm]} 
Given an analytic integrable Hamiltonian with three or more degrees of freedom, we
construct, arbitrarily close to it, an analytic perturbation with transition chains
whose lengths only depend on the unperturbed Hamiltonian. Then we deduce that the
perturbed system has orbits shadowing the chains.\\

{\bf \large Alfredo  MARTINEZ\\} {\it 222 S. El Molino Ave, 11 Pasadena Ca 91101
USA \\ alfredo@cds.caltech.edu                \\[0.2cm]} {\bf  \\[0.2cm]} \\



{\bf \large Vieri  MASTROPIETRO\\} {\it Dip. di Matematica, II Universita di Roma
Tor Vergata Via Della Ricerca Scientifica 20 ROMA \\
vieri@ipparco.roma1.infn.it\\[0.2cm]} {\bf  Lower bounds for homoclinic
splitting. Some examples. II \\[0.2cm]} In a three time scales model, coming from
a problem of celestial mechanics, a lower bound for the homoclinic splitting is
found, by exploiting suitable cancellations. From such bound the existence of
heteroclinic chains can be proved.\\


{\bf \large Juan Jose  MORALES-RUIZ\\} {\it Dep. Matematica Aplicada II
Universitat Politecnica de Catalunya c/Pau Gargallo 5 E-08028, Barcelona Espagne
\\ morales@ma2.upc.es\\[0.2cm]} {\bf Non-integrability of Hamiltonian systems by
means of the variational equations\\[0.2cm]} It is  proved that a necessary
condition for complete integrability (in Liouville sense and with several degrees
of regularity of the firts integrals) is that the identity component of the
Galois group of the variational equation (in the complex domain) must be abelian.
We test the efficacy of these results on several classical systems: Homogeneous
Potentials, Three Body Problem, Bianchi IX Cosmological Model, etc... \\

{\bf \large Alessandro  MORBIDELLI\\} {\it CNRS, Observatoire de Nice B.P. 4229
06304 Nice Cedex 4 \\ morby@obs-nice.fr\\[0.2cm]} {\bf  \\[0.2cm]} \\

{\bf \large Pierre  MOUSSA\\} {\it Service de Physique Th�orique, C. E. A- Saclay
F-91191 Gif Sur Yvette Cedex France \\ moussa@spht.saclay.cea.fr\\[0.2cm]} {\bf
Real and complex Brjuno functions (work in collaboration with S. Marmi
(Florence), and J.-C. Yoccoz, (College de France, Paris) \\[0.2cm]} We discuss
the properties of the Brjuno functions, introduced by Yoccoz to describe the size
of the stability domains in  discrete holomorphical dynamical systems close to a
rotation. We analyse in particular the complex extension of these  functions, and
try to characterize their singularity on the real axis. This is a non trivial
model where one can describe  in a global way the singularities generated by
small divisors  problems.\\

{\bf \large David  NESVORNY\\} {\it CNRS, Observatoire de Nice B.P. 4229 06304
Nice Cedex 4 \\ david@obs-nice.fr\\[0.2cm]} {\bf  \\[0.2cm]} \\

{\bf \large Laurent  NIEDERMAN\\} {\it Astronomie et Syst�mes dynamiques 77
avenue Denfert-Rochereau 75014 PARIS \\ laurent@bdl.fr\\[0.2cm]} {\bf Nonlinear
stability around an elliptic equilibrium point in a Hamiltonian system \\[0.2cm]}
Using a scheme given by Lochak, we derive a result of stability  over
exponentially long times with respect to the inverse of the distance to an
elliptic equilibrium point which has a definite torsion. At the price of this
assumption, our study is valid without arithmetical properties of the linearized
system while the previous theorems of this kind rely on a Diophantine condition
on the linear spectrum. Actually, under the latter condition and a  definite
torsion, a result of stability over superexponentially long times can be proved.
Finally, the same kind of theorems are also valid for an elliptic lower
dimensional invariant torus.\\


{\bf \large Merce  OLLE\\} {\it Dept. Matematica Aplicada I, ETSEIB Diagonal 647,
08028 Barcelona Espagne \\ olle@ma1.upc.es\\[0.2cm]} {\bf Dynamics and
bifurcation near the transition from stability to complex instability : An
analytical approach using normal forms. \\[0.2cm]} We consider a Hamiltonian of
three degrees of freedom and a family of periodic orbits with a transition from
stability  to complex instability, such that there is an irrational collision of
the Floquet eigenvalues of opposite sign. We analyse the local dynamics and the
bifurcation phenomena linked to this transition. We study the resulting Hopf-like
bifurcation from an analytical point of view by means of normal forms. The
existence of a bifurcating family of 2D tori is derived and both cases (direct
and inverse bifurcation) are described.\\

{\bf \large Juan Ramon  PACHA ANDUJAR\\} {\it DPTO. de Matematica Aplicada I,
ETSEIB. Diagonal 647, 08028 Barcelona Espagne \\ joanr@tere.upc.es\\[0.2cm]} {\bf
DDynamics and bifurcation near the transition from stability to complex
instability : An analytical approach using normal forms. \\[0.2cm]} We consider a
Hamiltonian of three degrees of freedom and a family of periodic orbits with a
transition from stability to complex instability, such that there is an
irrational collision of the Floquet eigenvalues of opposite sign. We analyse the
local dynamics and the bifurcation phenomena linked to this transition. We study
the resulting Hopf-like bifurcation from  an analytical point of view by means of
normal forms. The  existence of a bifurcating family of 2d tori is derived and
both cases (direct and inverse bifurcation) are described.\\

{\bf \large Jesus  PALACIAN\\} {\it Departamento de Matematica e Informatica
Universidad Publica de Navarra Campus de Arosadia s/n 31006 Pamplona Spain \\
\\[0.2cm]} {\bf  Reduction of harmonic oscillators in 1-1-1 resonance: relative
equilibria and bifurcations of quartic perturbations \\[0.2cm]} This
communication deals with the dynamics of a Hamiltonian system with three equal
modes around an elliptic equilibrium. The Taylor expansion is truncated at fourth
order. The process of the reduction is performed in two steps. First, a regular
reduction, due to the oscillator symmetry, brings the system to the orbit space
$\mbox{\bf CP}2$. In systems with axial symmetry a singular reduction can be
carried out. This permits to make a global study of a one-degree-of-freedom
system which lies on a double reduced space. As an application we analyze a
parametric family modelling an elliptical galaxy, showing its relative
equilibria, bifurcations and stability. \\

{\bf \large Yannis  PAPAPHILIPPOU\\} {\it CERN SL Division (AP) CH-1211 Geneva 23
SUISSE \\ yannis@mail.cern.ch         (or
ioannis.papaphilippou@cern.ch)\\[0.2cm]} {\bf Study of the non-linear behaviour
of the Large Hadron Collider through dynamical systems methods \\[0.2cm]}
Dynamical system methods are used in view of predicting the causes which limit
the dynamic aperture in the LHC models. To this end, analytical one- turn maps
have been produced  and analysed by means of Lie algebraic methods for all models
and 60 different realisations of the accelerator at injection energy. It has been
of particular interest to establish the  link between the individual magnetic
multipoles and their higher order contributions to a specific resonance.  \\

{\bf \large Paolo  PERFETTI\\} {\it Dept. of Mathematics University of Roma  2
Via della Ricerca Scientifica 00133 ROMA Italy \\
perfetti@axp.mat.uniroma2.it\\[0.2cm]} {\bf Fixed point method in the Arnol�d
model about the instability of the action variables in phase space \\[0.2cm]} A
proof of the existence of the stable and unstable manifolds of invariants tori is
presented using the fixed point theorem as suggested by Arnol�d himself in his
famous paper �Instability of systems with several degrees of freedom�.\\

{\bf \large Jurgen  P\"0SCHEL\\} {\it Mathematisches Institut A Universit�t
Stuttgart Pfaffenwaldring 57 D-70569 Stuttgart Allemagne \\
poschel@mathematik.uni-stuttgart.de\\[0.2cm]} {\bf  \\[0.2cm]} \\

{\bf \large Rafael  RAMIREZ\\} {\it Departament d'Enginyeria Inform\`atica Escola
Tecnica Superior d'Enginyeria C  Carretera de Salou, s/n 43006 Tarragona Espagne
\\ rramirez@etse.urv.es\\[0.2cm]} {\bf On the inverse problems in dynamics
\\[0.2cm]} The problem of finding the forces which generate a given orbit or
family of orbits has always been considered of prime importance in dynamics, in
the Newton's time as well as at present. In fact, Book One of Newton's
"Philosophi� Naturalis Principia Mathematica" (1686) is dominated by the idea of
determining the force capable of generating given orbits.\\

{\bf \large Philippe  ROBUTEL\\} {\it  Bureau des Longitudes Astronomie et
Syst�mes Dynamiques 77 avenue Denfert-Rochereau 75014 Paris \\
robutel@bdl.fr\\[0.2cm]} {\bf  \\[0.2cm]} \\



{\bf \large Vered  ROM-KEDAR\\} {\it Department of Applied Mathematics and
Computer Science Weizmann institute of Science P.O.B. 26, Rehovot, 76100 Isra�l
\\ vered@wisdom.weizmann.ac.il\\[0.2cm]} {\bf Big islands appearing in
near-ergodic flows \\[0.2cm]} We prove the existence and derive a rigorous
estimate of the size of islands (in both phase space and parameter space)
appearing in smooth Hamiltonian approximations of scattering billiards. The
derivation includes the construction of a local return map near singular periodic
orbits for an arbitrary scattering billiard and for the  general smooth billiard
potentials. Thus, {\it universality} classes for the local behavior are found.
Moreover, for all scattering geometries and for many types of natural potentials
which limit to the billiard flow as a parameter $\epsilon \longrightarrow 0$,
islands of {\it polynomial} size in $\epsilon$ appear. This suggests that the
loss of ergodicity via the introduction of the physically relevant effect of
smoothening of the potential in modeling, for example, scattering molecules, may
be of physically noticeable effect. Generalizations to higher dimensions will be
discussed.\\



{\bf \large Michael  RUDNEV\\} {\it Dept. of Mathematics /C 1200 UT at Austin
Austin, TX  78712 USA \\ mru@math.utexas.edu\\[0.2cm]} {\bf A geometric method
for the splitting of separatrices \\[0.2cm]} A new method to study the splitting
of separatrices in  near-integrable Hamiltonian systems is presented, which
allows to prove exponentially small upper bounds for the  splitting for a generic
Hamiltonian with two time scales. The issues of obtaining the lower bounds in
exponentially small case are addressed, in particular how one can come up  with
exact asymptotics in a model problem with three degrees  of freedom.\\

{\bf \large Stefano  RUFFO\\} {\it Dipartimento di Energetica Via S. Marta 3
50139 Firenze Italie \\ ruffo@ingfi1.ing.unifi.it\\[0.2cm]} {\bf Time scales for
the relaxation to thermodynamical equilibrium \\[0.2cm]} We have studied two
physical examples where the time-scale for the relaxation to thermodynamical
equilibrium of an isolated system started from a non-equilibrium initial
condition becomes exceedingly large: a) The Fermi-Pasta-Ulam (FPU) systems at
small energy density e. Here a power-law increase of the time-scale as $e- a (a
\approx 3$ for the FPU-b system) is shown numerically and justified analytically.
This sharp increase was previously interpreted as a ``freezing" of the
non-equilibrium initial state (a packet of Fourier modes with the same energy),
while a slow �diffusion� to the energy equipartition state is now exhibited. b)
The Hamiltonian Mean-Field model. $N$ particles moving on a circle are all
coupled via an attractive or repulsive cosine interaction. At equilibrium the
attractive case has a clustering-declustering phase transition. When started in a
�water-bag� initial condition near the phase transition the systems stays far
from equilibrium for a time which increses with $N$. The repulsive case also
shows time-scales which increase with $N$ in the high- energy region. Both
effects are related to a slow diffusion of the orbit in the phase-space, which
remains to be fully characterized.\\



{\bf \large Manuele  SANTOPRETE\\} {\it Via Paradisa  7 56124 PISA Italy \\
df170999@CERD1.UNIPI.DIFI.IT\\[0.2cm]} {\bf  \\[0.2cm]} \\

{\bf \large David  SAUZIN\\} {\it Bureau des Longitudes Astronomie et Syst�mes
Dynamiques 77 avenue Denfert-Rochereau 75014 Paris \\ sauzin@bdl.fr\\[0.2cm]}
{\bf Averaging and Borel transformation \\[0.2cm]} We are interested in analytic
dynamical systems ${dx\over dt} = \varepsilon f(x,t) ,$ where $x$ is a
multi-dimensional variable, and $f$ is $2\pi$-periodic in $t$: the time plays the
role of a rapid phase. The goal is to find a close to identity change of
variables $ x= y + \varepsilon u(\varepsilon,y,t) $ ($2\pi$-periodic in $t$)
which reduces the time-dependence in the equation, {\em i.e.} which gives to it
the form $ {dy\over dt} = \varepsilon (g(\varepsilon,y) + h(\varepsilon,y,t)) $
with a function $h$ as small as possible and of zero mean. On one hand,
Neishtadt's Theorem (1984) ensures that the purely oscillating part

of the equation can always be made exponentially small: one can obtain $ |h| \leq
\mbox{const}\, \exp(-{a\over\varepsilon}). $ But what value may reach the
positive constant $a$? On the other hand, one can find formal series $\tilde
u(\varepsilon,y,t)$ and $\tilde g(\varepsilon,y)$ (formal expansions in
$\varepsilon$ with coefficients analytic in $(y,t)$) which eliminate completely
the time-dependence: {\em formally} one can obtain $h=0$. The link between the
formal solution $(\tilde u,\tilde g)$ and Neishtadt's result lies in the {\em
Gevrey properties} of these formal series [Sauzin 1992], [Ramis-Sch\"afke 1996].
Their formal Borel transforms $\hat u(\zeta,y,t)$ and $\hat g(\zeta,y)$ converge,
and an incomplete Laplace transform (one truncates the path of integration so
that $\zeta$ does not leave the disk of convergence) allows to recover
Neishtadt's Theorem.  We present a new result on the domain of analyticity of
these Borel transforms, which allows to obtain simply from the domain of
analyticity of the flow of the mean field $<f>(y)$ some explicit constant $a$.\\

{\bf \large Luca  SBANO\\} {\it Via C.Dossi 45 00137- Roma Italy \\
L.Sbano@mclink.it  sbano@atmos.ifa.rm.cnr.it\\[0.2cm]} {\bf The topology of the
planar three-body problem with zero total angular momentum and the existence of
periodic orbits}

We consider the planar three-body problem (3BP) with a Newtonian-like potential
of the form $\sum_{i\neq j}m_im_j \|x_i-x_j\|^{-\alpha}$ with $\alpha\geq 2$ and
also the Newtonian case $\alpha=1$. We study the Least Action principle on the
manifold defined by the vanishing of the total angular momentum. Using the
topology of the reduced configuration space we are able to find a class of
trajectories on which a generalizationof the Poincar� inequality holds, then we
can apply the direct method of calculus of variations. We prove the existence of
a periodic solution: for the Newtonian potential ($\alpha=1$), this periodic
solution has at most a finitenumber of  collisions. For $\alpha\geq 2$ the
solution is in the homotopic class of the trajectories that go through at least
three different collinear configurations. This result is a direct consequence of
the first homotopy group of the reduced configuration space without coincidence
set, this group is isomorphic to the free-generated group $Z *Z$. {\it To appear
in NONLINEARITY} \\

{\bf \large Constance  SCHOBER\\} {\it Dept. of Mathematics and Statistics Old
Dominion University Norfolk Virginia 23529-0077 USA \\
schober@math.odu.edu\\[0.2cm]} {\bf Chaotic Dynamics for Symmetry-Breaking
Perturbations of Integrable Equations \\[0.2cm]} It is well known that for
certain parameter regimes the periodic focusing Non Linear Schr�dinger equation
(NLS) exhibits a chaotic response when the system is perturbed. When even
symmetry is imposed, the dynamics is characterized by irregular center-wing
flipping of the wave form and one can prove persistence of transversal
intersections of the invariant manifolds as the main source of homoclinic chaos.
In this talk  we will examine evenness breaking perturbations of the NLS. Novel
types of chaotic behavior are observed for the non-even case, including solutions
which exhibit random flipping between a left and right modulated traveling wave.
This behavior is related to an extra circle symmetry in the problem and to random
variations of the sign of the linear momentum. A Mel'nikov analysis
interpretation of this new phenomena is provided.\\



{\bf \large Carles  SIM\'O\\} 
{\it Departament de Matem\`atica Aplicada i An\`alisi, Universitat de Barcelona,
Gran Via de les Cortes Catalanes, 585,  08007 Barcelona, Espagne \\
carles@maia.ub.es\\[0.2cm]}  
{\bf Geometry of Hamiltonian Dynamics: Goals and
Methods \\[0.2cm]} 
Hamiltonian Dynamics appears in a natural way in many physical
problems. In this talk we shall present a quick review dealing with the following
topics:
\begin{itemize}
\item[-] The geometrical objects in the phase space: fixed points, periodic
orbits, invariant tori. Hyperbolicity and the related invariant manifolds.
Homoclinic phenomena.
\item[-] Dynamical consequences: Integrability, stability and practical
stability, capture and escape, diffusion, chaos. Their detection.
\item[-] The tools: Algebraic, analytical, variational, topological,
probabilistical, symbolical, numerical, graphical.
\item[-] Applications: Variants of the two body problem, the three body
problem, geodesics, billiards, etc.
\item[-] Extensions: Limit PDE, volume preserving flows, etc. 
\\
\end{itemize}




{\bf \large Sinisa  SLIJEPCEVIC\\} {\it Dept.of Applied Mathematics and
Theoretical Physics, Silver Street, Cambridge CB3 9EW, UK \\
ss265@damtp.cam.ac.uk\\[0.2cm]} {\bf  Monotone gradient dynamics and Mather's
shadowing \\[0.2cm]} 
We prove that the all the invariant measures for the
monotone gradient dynamics of the (non-tilted) Frenkel-Kontorova model are
supported on the set of stationary configurations, and that the $\omega$-limit
set of each configuration with monotone dynamics is either a single stationary
configuration or contains an invariant circle of stationary configurations.  A
consequence is a new proof of existence of Mather's shadowing orbits of twist
diffeomorphisms. We compare results with the dynamics of other dissipative
spatially extended systems, such as the real Ginzburg-Landau equation. Using our
results, we calculate the phase diagram of a second-order phase transition for
the non-conservative diffusion on 1D lattices. \\

{\bf \large Teresinha  de Jesus  STUCHI\\} {\it Departamento de Fisica Matematica
Instituto de Fisica Cidade Universitaria CT - Bloco A 21945-970 - Rio de Janeiro
- RJ Brasil \\ tstuchi@if.ufrj.br\\[0.2cm]} {\bf Center stable/unstable manifolds
and the destruction of KAM tori in the 2D Hill Problem \\[0.2cm]} The classical
Hill Problem (HP) is a simplified version of the Restricted Three Body Problem
(RTBP) where the distance of the two massive bodies is made infinite through the
use of Hill's variables and a limiting procedure. In this way Hill's problem is
obtained as zero--th order approximation of a power series in the mass ratio
$\mu^{1/3}$.  A Levi-Civita canonical regularization is used to obtain a
polynomial Hamiltonian which is very convenient for numerical calculations. Also
it conveys a general framework for many researchers of other areas than Celestial
Mechanics. The final form of the Hamiltonian is: \begin{eqnarray} \nonumber
H(Q_1,Q_2,P_1,P_2)=\frac {1} {2} (Q_1^2+Q_2^2+P_1^2+P_2^2)+ 2({Q_1}^2+
{Q_2}^2)(Q_2P_1-Q_1P_2) \cr
\quad-4({Q_1}^6-3{Q_1}^4{Q_2}^2-3{Q_1}^2{Q_2}^4+{Q_2}^6). \end{eqnarray}
\noindent Up to fourth degree we have the Kepler problem in a rotating frame,
which is integrable. The perturbation of degree six is due to the  massive body
and numerical evidence shows that it is non integrable. A rigorous proof of
non--integrability is easier to obtain in the $3D$ case, but it is skipped on
this report. In this approximation we retain only two of the three collinear
Lagrangian equilibria. These points are usually denoted as $L_1$ and $L_2$ and,
in this case, they are symmetric with respect to the origin. As it is well known
the $L_1$ and $L_2$ points are of saddle--center type. The centre manifold is
foliated by symmetric periodic orbits known as Lyapunov orbits. From them we
obtain, through numerical globalization, the stable/unstable manifolds of $L_1$
and $L_2$. A convenient Poincar� section exhibits the relation of these manifolds
and the remaining KAM structure. The geometrical relation between this structure
and the center- stable/unstable manifolds is our main concern in this report. An
interesting result is the persistence of KAM structure, even for large values of
the energy, confined by the stable and unstable manifolds.\\



{\bf \large Dmitry  TRESCHEV\\} {\it Department of Mechanics and Mathematics
Moscow State University Vorob�evy Gory Moscow 119899 Russia \\
dtresch@mech.math.msu.su\\[0.2cm]} {\bf Multidimensional separatrix maps
\\[0.2cm]} A map presenting the requrrence dynamics of a multidimensional
symplectic map in a neighborhood of surfaced asymptotic to a hyperbolic torus is
constructed. Some dynamical consequences are presented.\\

{\bf \large Hiroaki  UMEHARA\\} {\it Communications Research Laboratory 893-1
Hirai, Kashima, Ibaraki, 314-0012 Japan \\ ume@crl.go.jp\\[0.2cm]} {\bf Escape
and non-escape orbits close to triple collision in the planar three-body problem
\\[0.2cm]} The existence of escape and non-escape orbits arbitrarily close to the
triple-collision orbits of equilateral type is considered with the aid of the
blow-up analysis in the planar three-body problem with zero initial velocities
and equal masses. It is found that any binary-collision orbit sufficiently close
to triple collision leads to escape after the first triple encounter. In order
show this, it is proved that the escape criterion is formulated in the planar
isosceles problem and translated into the words of blow-up variables.\\




{\bf \large Enrico  VALDINOCI\\} {\it Piazza Enrico Fermi 37 00146 Roma Italy \\
valdino@ciop.mat.uniroma3.it\\[0.2cm]} {\bf Whiskered tori for �a priori� stable
and unstable Hamiltonian systems \\[0.2cm]} We revisited one of the results of
Chierchia and Gallavotti�s �Drifts and diffusion in phase space�, 1994, and gave
a detailed statement for a KAM theorem about the conservation of partially
hyperbolic tori on a fixed energy level. In particular, we emphasized  the
following items: sharp estimates on the measures of the surviving tori on the
energy level, an elementary proof of smooth embedding (without making use of the
Whitney Extension Theorem) and Cinfinity-closeness to identify of the
transformations involved, analyticity properties (including dependence upon
parameters), geometric structures. These results apply to both �a priori
unstable� and �a priori stable� systems.\\

{\bf \large Claudia  VALLS\\} {\it Departament de Matem�tica Aplicada i An�lisi
Universitat de Barcelona Gran Via de les Cortes catalanes, 585 08007 Barcelona
Espagne \\ claudia@maia.ub.es\\[0.2cm]} {\bf A generalization of the standard
separatrix map in the classical Arnold�s  example of diffusion with two equal
parameters \\[0.2cm]} Symplectic maps appear in a natural way as Poincar\'e
sections of Hamiltonian systems with three or more three degrees of freedom. Let
us consider the system with Hamiltonian $$ \begin{array}{c} \vspace*{2mm}
H(\vec{I},x,y,\vec{\phi},\varepsilon)= h_{0}(x,y,\vec{I},\varepsilon)+ \mu
\varepsilon h_{1}(x,y,I_{2},\vec{\phi}) = \\ \frac{1}{2} (x^{2} + I_{2}^{2})+
I_{3} + \varepsilon  (\cos  y -1)+ \varepsilon \mu (\cos y -1) (\cos \phi_{2} +
\cos \phi_{3}), \end{array} $$ where $(x,\vec{I}) \in G \subset R^{3}$,  and
$(y,\vec{\phi}) \in T^{3}.$ The above Hamiltonian is real analytic in the
parameter $\varepsilon$, it is an entire function of $x,\vec{I}$, $y,\vec{\phi}$
and we shall consider $\mu=\varepsilon$.

Taking suitable Poincar\'e sections, the resulting map is given as follows:\\
$(x,I_{2},\theta_{2},\theta_{3}) \stackrel{f}{\rightarrow}
(x',I_{2}',\theta_{2}', \theta_{3}'),$ where $$ \begin{array}{rcl} \vspace*{2mm}
x' &=& x + \Delta x, \\ \vspace*{2mm} I_{2}' &=& I_{2} + \Delta I_{2}, \\
\theta'_{2} &=& \theta_{2} + c_1 {I'}_2 + \vspace*{2mm}
\frac{I_{2}'}{\sqrt{\varepsilon}} \log \left |\frac{c_2}{x'}\right |, \\
\theta_{3}' &=&
\theta_{3} + c_1 +  \frac{1}{\sqrt{\varepsilon}} \log \left |\frac{c_2}{x'}\right | ,
\end{array} $$ where $\Delta x$ and $\Delta I_2$ are given by the related
splitting functions, $c_1=2\log (\pi/y_{0})/\sqrt{\varepsilon},$ $c_2=x_{0} y_{0}
\sqrt{\varepsilon}$ and $(x_{0},y_{0})$ are the initial values of $(x,y).$

Moreover, $f$ can be seen as a generalization to higher dimensions of the
standard separatrix map defined on the cilinder $R$ x $T.$ It is given by
$(x,y)\rightarrow (x',y')$ where $$ \begin{array}{rcl} \vspace*{2mm} y'&=& y
+\sin (2 \pi x), \\ x'&=& x + c +d \log \left |y'\right | , \end{array} $$ $c$ and
$d$ being two parameters. \\


{\bf \large Alexei  VASILIEV\\} {\it Space Research Institute, Profsoyuznaja
84/32, 117810 Moscow Russia \\ valex@iki.rssi.ru\\[0.2cm]} {\bf Changes of
adiabatic invariant in 3D volume-preserving reversible systems \\[0.2cm]} Let the
system possess a non-degenerate saddle point. Then because of the reversibility,
there is another saddle point, connected to the first with two families of
heteroclinic trajectories: one family consisting of two trajectories, and another
family filling a certain 2D surface (a separatrix of the unperturbed system). It
can be deduced that in a vicinity of these two families the system possesses two
independent integrals of motion. Under a small volume-preserving perturbation the
separatrix splits. The integrals of the unperturbed system slowly evolve in time.
An adiabatic invariant is introduced, well-preserved far from the separatrix of
the unperturbed system. However, if a trajectory of the perturbed system crosses
the separatrix (of the unperturbed system), the adiabatic invariant along this
trajectory changes strongly. An asymptotic formula for this change is obtained.
The change is quasi-random. If the geometry of the system allows multiple
separatrix crossing by a trajectory, this may result in diffusion of the
adiabatic invariant and development of chaos in the system. This latter
phenomenon is demonstrated in a particular example of such a system.\\

{\bf \large Luz-Vianey  VELA-AREVALO\\} {\it 370 S. Catalina Ave.  301 Pasadena
CA 91106 USA \\ luzvela@cds.caltech.edu\\[0.2cm]} {\bf  \\[0.2cm]} \\

{\bf \large Andrea  VENTURELLI\\} {\it Astronomie et Syst�mes dynamiques Bureau
des Longitudes 77 avenue Denfert-Rochereau 75014 Paris \\
venturel@bdl.fr\\[0.2cm]} {\bf  \\[0.2cm]} \\


{\bf \large Jordi  VILLANUEVA\\} {\it Departament de Matematica Aplicada I ETSEIB
Universitat Politecnica de Catalunya Diagonal 647 08041 Barcelona Espagne \\
jordi@tere.upc.es\\[0.2cm]} {\bf Bifurcations of lower-dimensional tori due to an
internal-normal resonance in quasi-periodically forced Hamiltonian systems.
\\[0.2cm]} Let us consider a quasi-periodic time-dependent perturbation of an
autonomous Hamiltonian system. Let us focus on an elliptic equilibrium point of
the unperturbed Hamiltonian. Under suitable hypotheses, this equilibrium point
becomes an invariant torus under the effect of the perturbation, with the same
frequencies as the perturbation. The normal frequencies of this torus are a
perturbation of the normal frequencies around the initial equilibrium point. In
this work we consider the interaction between these frequencies. We add a
parameter to the system such that the internal frequencies are always fixed
(hence, the torus is not destroyed because of an internal resonance) and the
normal ones move with the parameter. In the talk we will present a numerical
study about several bifurcations that take place when the normal and internal
frequencies are in resonance. We will also discuss the theoretical support for
those bifurcations.\\

{\bf \large Stephen  WIGGINS\\} {\it 107-81 Caltech Pasadena   CA 91125 USA \\
wiggins@cds.caltech.edu\\[0.2cm]} {\bf Exponentially small separatrix splittings
in Hamiltonian systems with three or moredegrees of freedom \\[0.2cm]} In this
talk I will describe recent results on this problem obtained in collaboration
with Michael Rudnev. I will describe our approach to the problem, as well as the
relation of our techniques and results with those of other groups (e.g.,
Delshams, Gelfreich, Jorba, Seara, Gallavotti, Gentile, Mastropietro).\\

{\bf \large Zhihong Jeff  XIA\\} {\it Department of Mathematics Northwestern
University Evanston IL 60208 USA \\ xia@math.nwu.edu\\[0.2cm]} {\bf Arnold
diffusion, variational method and Arnold-Mather mechanism \\[0.2cm]} We use
variational method to study the problem of Arnold diffusion and instabilities in
near-integrable Hamiltonian systems. With generic assumptions, verifiable by
Melnikov integrals in a priori unstable situation and perhaps other methods in a
priori stable situation, we construct transition chain of arbitrary length,
crossing gaps of any sizes between invariant tori. We also obtain shadowing
properties in the transition chain, non-integrability of the system and many
other interesting properties. Our method is strictly variational and applies to
cases where the normally hyperbolic (weakly) set is 2-D (for diffeos, 3-D for
flows). Its generalization to higher dimensional normally hyperbolic set is
currently being studied.\\

{\bf \large Yoshiyuki  YAMAGUCHI\\} {\it Department of Physics, School of
Science, Nagoya Univ., Furo-cho, Chikusa-ku, Nagoya, 464-01, Japan \\
yamaguchi@allegro.phys.nagoya-u.ac.jp\\[0.2cm]} {\bf Renormalization group
equation and integrability \\[0.2cm]} We investigate Hamiltonian systems with two
degrees of freedom by using renormalization group method, which vanishes secular
terms and reduces equation of motion  to renormalization group equation. We show
that the original Hamiltonian systems and the renormalization group equations are
integrable if the renormalization group equations are Hamiltonian systems up to
the second leading order of small parameter.by using renormalization group
method, which vanishes secular terms and reduces equation of motion  to
renormalization group equation. We show that the original Hamiltonian systems and
the renormalization group equations are integrable if the renormalization group
equations are Hamiltonian systems up to the second leading order of small
parameter.\\

{\bf \large Patricia  YANGUAS\\} {\it Departamento de Matematica e Informatica
Universidad Publica de Navarra Campus de Arosadia s/n 31006 Pamplona Spain \\
yanguas@upna.es\\[0.2cm]} {\bf Reduction of harmonic oscillators in 1-1-1
resonance: relative equilibria and bifurcations of quartic perturbations
\\[0.2cm]} This communication deals with the dynamics of a Hamiltonian system
with three equal modes around an elliptic equilibrium. The Taylor expansion is
truncated at fourth order. The process of the reduction is performed in two
steps. First, a regular reduction, due to the oscillator symmetry, brings the
system to the orbit space $\mbox{\bf CP}2$. In systems with axial symmetry a
singular reduction can be carried out. This permits to make a global study of a
one- degree-of-freedom system which lies on a double reduced space. As an
application we analyze a parametric family modelling an elliptical galaxy,
showing its relative equilibria, bifurcations and stability.\\

{\bf \large Haruo  YOSHIDA\\} {\it National Astronomical Observatory Mitaka Tokyo
181 Japan \\ yoshida@gauss.mtk.nao.ac.jp\\[0.2cm]} {\bf A new necessary condition
for the integrability of Hamiltonian systems with a two dimensional homogeneous
potential \\[0.2cm]} Recently, Morales-Ruiz and Ramis obtained a strong necessary
condition for integrability of Hamiltonian systems with a  homogeneous potential,
based on their own theorem on the  differential Galois theory (Picard-Vessiot
theory) for Hamiltonian systems. The theorem claims that if the original
Hamiltonian system is integrable, then the variational  equation around a
particular solution is solvable in the  sense of the differential Galois theory,
i.e., the solution is obtained only by a combination of quadratures, exponential
of quadratures and algebraic functions. In this paper, an elementary and
independent proof of this statement is given for Hamiltonian systems with a two
dimensional homogeneous  potential, which leads to a new necessary condition for
integrability. This new necessary condition well justifies  the so-called weak
Painlev� conjecture of Ramani et al. for the first time. \\

{\bf \large Loukas  ZACHILAS\\} {\it University of Thessalia Dept. of Civil
Engineering Pedio Aeros 38334 Volos Gr�ce \\ zachilas@rigas.uth.gr\\[0.2cm]} {\bf
Using color and rotation for visualizing 4D Poincar�  cross sections \\[0.2cm]}
The problems encountered in the study of three-dimensional Hamiltonian systems by
means of the Poincare cross-sections are reviewed. A new method to overcome these
problems is proposed. In order to visualize the 4D "space" of section we
introduce the use of color and rotation. We apply this method to the case of a
family of simple periodic orbits in a three-dimensional potential and we describe
the differences in the orbital behavior between regions close to stable and
unstable periodic orbits.\\

\end{document}
