R. Artuso
(University of Cômo, Italy) 
Deterministic ratchets and beyond: zeta functions techniques. We illustrate the use of non perturbative techniques in
the study of deterministic ratchets by working out an exactly solvable model,
inspired by the socalled Parrondo games.
P. H. Chavanis
(University Paul Sabatier, France) 
Selfgravitating Brownian particles
We discuss the dynamics and thermodynamics of a gas of
selfgravitating Brownian particles. We consider a strong friction limit and a
mean field approximation where the problem reduces to the study of the
SmoluchowskiPoisson (SP) system. We show the existence of a canonical phase
transition between a gaseous state and a condensed state below a critical
temperature T_c. For T<T_c the system undergoes an isothermal collapse. The
evolution is selfsimilar and leads to a finite time singularity. However, ``the
singularity contains no mass''. In fact, the collapse continues after the
singularity has arisen and a Dirac peak is formed in the postcollapse regime by
accreting the mass of the halo. We mention different extensions of this problem:
selfgravitating fermions, multispecies systems, generalized thermodynamics,
influence of the dimension of space, etc. We also discuss the analogy between
the collapse of selfgravitating Brownian particles and the chemotactic
aggregation of bacterial populations in biology.
M. Courbage
(University Paris VII, France) 
Complexity and Entropy in Colliding Particle Systems
We develop quantitative measures of entropy evolution for
particle systems undergoing collision process in relation with various
unstability properties.
A. Luo (Southern Illimois University, USA)
 On global transversality and chaos in twodimensional nonlinear
dynamical systems
In this paper, the global transversality and tangency in
twodimensional nonlinear dynamical systems are discussed, and the exact energy
increment function (Lfunction) for such nonlinear dynamical systems is
presented. The Melnikov function is an approximate expression of the exact
energy increment. A periodically forced, damped Duffing oscillator with a
separatrix is investigated as a sampled problem. The corresponding analytical
conditions for the global transversality and tangency to the separatrix are
derived. Numerical simulations are carried out for illustrations of the
analytical conditions. From analytical and numerical results, the simple zero of
the energy increment (or the Melnikov function) may not imply chaos exists. The
conditions for the global transversality and tangency to the separatrix may be
independent of the Melnikov function. Therefore, the analytical criteria for
chaotic motions in nonlinear dynamical systems need to be further developed. The
methodology presented in this paper is applicable to nonlinear dynamical systems
without any separatrix.
R. S. MacKay
(University of Warwick, UK) 
Manifestations of chaos
I will describe various systems for which good forms of
chaos have been proved relatively recently and discuss their robustness:
1. Two charges of opposite sign in a uniform magnetic field and unequal
gyrofrequencies exhibit "second species" chaos;
2. An exact areapreserving tilt map of the cylinder which is mixing and has
diffusion constant a certain expression involving 2+sqrt{3}, approximately
0.14434;
3. Three mixing volumepreserving vector fields in 3D containers with noslip
boundaries.
G. Morfill
(MaxPlanck Institute, Germany) 
Nonlinear Processes in Complex Plasmas
"Complex Plasmas" are strongly coupled systems, consisting
of ions, electrons, charged microparticles and some neutral gas. The dynamically
dominant microparticles are individually visualised and due to their
comparatively large mass (many billions atomic masses) characteristic dynamical
time scales are in the 10’s of msec range, easily resolved with CCD cameras.
Thus complex plasmas are ideal systems for studying some fundamental physics
questions, such as the onset of cooperative phenomena, the kinetics of phase
transitions and other critical phenomena, nanofluidic properties as well as many
other hydrodynamic processes – all at the most elementary kinetic level. After a
brief introduction into the properties of this interesting state of matter, some
examples of recent research results in the abovementioned fields will be
presented.
O. Piro (IMEDEA, Spain)
 Dissipative embedding of conservative dynamics
In a number of apparently unrelated problems, Hamiltonian
or volume preserving dynamics takes place on an invariant, eventually attracting,
submanifold of a dimensionally larger dissipative system. Examples of such
situations are a) the motion of neutrally bouyant finite size particles, b)
controlled Hamiltonian systems, c) tidally synchronized celestial bodies, etc. I
will try to present an integrated view of this class of problems and show a
number of phenomena that are likely to appear in all its instances.
S. Rica (ENS Paris, France)
 Condensation of classical nonlinear waves
We study the formation of a largescale coherent structure
(a condensate) in classical wave equations by considering the defocusing
nonlinear Schr"odinger equation as a representative model. We formulate a
thermodynamic description of the condensation process by using a wave turbulence
theory with ultraviolet cutoff. In 3 dimensions the equilibrium state undergoes
a phase transition for sufficiently low energy density, while no transition
occurs in 2 dimensions, in analogy with standard BoseEinstein condensation in
quantum systems.
Numerical simulations show that the thermodynamic limit is reached for systems
with ^3$ computational modes and greater.
On the basis of a modified wave turbulence theory, we show that the nonlinear
interaction makes the transition to condensation subcritical. The theory is in
quantitative agreement with the simulations.
V. RomKedar (Institut Weizmann, Israël)
 Stable motion in highdimensional steep repelling potentials
The appearance of elliptic periodic orbits in families of
ndimensional smooth repelling billiardlike potentials that are arbitrarily
steep and close to dispersing billiards is established for any finite n. The
width of the stability regions in the parameter space scales as a powerlaw in
1/n and in the steepness parameter. Thus, it is shown that even though these
systems have a uniformly hyperbolic (albeit singular) limit, the ergodicity
properties of this limit system are destroyed in the more realistic smooth
setting. The considered example is highly symmetric and is not directly linked
to the smooth many particle problem. Nonetheless, the possibility of explicitly
constructing stable motion in smooth ndegrees of freedom systems limiting to
strictly dispersing billiards is now established. Thus, it is shown that the
billiards instability mechanism cannot justify by itself the observed ergodicity
of smooth many particle systems.
R. Sanchez
(Oak Ridge National Laboratory, USA) 
Nondiffusive modeling of strange transport in magneticallyconfined, turbulent
plasmas
The most promising route for the successful production of
large amounts of energy from fusion requires the containment of a hot plasma by
magnetic fields with the topology of nested, toroidal magnetic surfaces.
Understanding the transport properties of these configurations is essential to
the success of the fusion energy program, since they can help us to control the
leak of particles and energy out of the system. However, multiple experiments
carried out during the last two decades in this type of magnetic trap have
revealed that the confined plasma is in a strong turbulent state in which
transport phenomena exhibit a strange, nondiffusive nature. The observation of
a degradation of confinement with increasing external power suggests that the
plasma profiles tend to stay close to some critical threshold for the onset of
instabilities. In addition, the observed scaling of the energy/particle
confinement times and the detection of "avalanchelike" transport events suggest
that the dominant transport mechanism may lack a characteristic length scale.
Similar evidence also suggests that a characteristic temporal scale may also be
missing.
In this focus talk, I will discuss some recent work that aims at improving our
understanding of the dynamics of transport in these plasmas as much as at
constructing mathematical models better suited to capturing them then the ones
currently available. The basic tools used are continuoustimerandom walks (CTRWs)
and/or fractional differential equations (FDEs), both of which can accommodate
this lack of characteristic scales. Numerous examples obtained from numerical
simulations and experiments will be used to illustrate the main ideas.
D. L. Shepelyansky
(Univ. Paul Sabatier,
France)  Directed transport and chaos in
asymmetric nanostructures
A board of rigid disks on a triangular lattice has been
invented by Galton in 1889 to demonstrate the appearance of statistical laws
from dynamical motion. This system, also called periodic Lorentz gas, has been
proved to be completely chaotic by Sinai in 60th. The dynamics remains chaotic
also for semidisks scatterers oriented in one direction. In this case the
inversion symmetry is broken but the directed transport remains forbidden by the
detailed balance principle. This remains true also in presence of polarized
monochromatic force produced by microwave radiation. However, when dissipation
is present, a new stationary state is born from chaos as shown in [1]. It is
characterized by a directed transport which can be efficiently controlled by the
microwave polarization even if mean force is zero. Being universal this effect
exists for Maxwell or FermiDirac thermostatted gas moving between semidisks in
presence of a microwave field [25]. Nowadays technology allows to realize the
semidisk Galton board with a twodimensional electron gas in a superlattice of
micron size antidots. In this case the theory predicts appearance of strong
currents induced by microwave fields. This opens new possibilities for creation
of room temperature detectors of terahertz radiation.
T. H. Solomon
(Bucknell College, USA) 
Experimental studies of advectionreactiondiffusion systems
We present the results of experiments on the effects of
chaotic fluid mixing on the dynamics or reacting systems. The flows studied
include a blinking vortex flow and a chain of alternating vortices. The reaction
is the oscillatory or excitable state of the wellknown BelousovZhabotinsky
chemical reaction. Three sets of experiments are described. (1) Experiments in a
blinking vortex flow demonstrate that chemical patterns for the oscillatory
reaction reflect the structures predicted by numerical models of the mixing
fields for the flow. (2) Synchronization of oscillating reactions in an extended
flow (a chain of vortices) is found to be enhanced significantly by the presence
of superdiffusive transport characterized by Levy flights that connect different
parts of the flow. (3) Front propagation is investigated in a chain of
alternating and oscillating vortices that are coupled by chaotic mixing. It is
found that the fronts modelock onto the frequency of the oscillation of the
vortices. The effects of superdiffusion on the modelocking is also investigated.
V. Tarasov
(State University of Moscow, Russia) 
Fractional Dynamics of Systems with LongRange Space Interaction and Temporal
Memory
Field equations with time and coordinates derivatives of
noninteger order are derived from stationary action principle for the cases of
powerlaw memory function and longrange interaction in systems. The method is
applied to obtain a fractional generalization of the GinzburgLandau equation
and nonlinear Schrodinger equation, and dynamical equations for particles chain
with powerlaw interaction and memory. In more details, we consider two
different applications of the action principle: generalized Noether's theorem
and Hamiltonian type equations. In the first case, we derive conservation laws
in the form of continuity equations that consist of fractional timespace
derivatives. Among applications of these results, we consider a chain of coupled
oscillators with a powerwise memory function and powerwise interaction between
oscillators. In the second case, we consider an example of fractional action and
find the corresponding Hamiltonian type equations. The obtained fractional
equations and conservation laws can be applied to complex media with/without
random parameters or processes.
J. L. Thiffeault
(Imperial College, UK)  A
Topological Theory of Rod Stirring
In a fluid, stirring is usually necessary to overcome the
slow diffusion of most substances. This is important in a wide range of
applications, from industry to geophysics. Here I focus on a prototypical
application, the stirring of a twodimensional viscous fluid with rods.
Mathematically, stirring rods can be viewed as
'punctures' in a twodimensional surface: they present topological obstructions
to material lines in the fluid. The theory developed by Thurston and Nielsen to
classify the possible periodic rod motions in such a system can be used to
decide which stirring methods are best, in the sense of generating chaotic
trajectoris that lead to chaotic advection. Global aspects of the concentration
field of a substance in the stirring device can then be determined via 'train
tracks', which are skeletons of an important underlying structure called the
unstable foliation. Since there are only a limited number of possible train
tracks, all stirring protocols can be classified, as well as their properties.
This can be extended to completely general situations where rods are replaced by
periodic orbits. I will present experimental and numerical examples of all these
concepts. Finally, I will introduce a rod stirring device designed and optimized
using these topological principles.
A. Torcini
(CNR Florence, Italy) 
Synchronization in spatially extended systems
The mechanisms responsible for the synchronization of
chaotic extended systems with shortrange interactions are revised. Moreover
recent results concerning powerlaw coupled systems are presented. In particular,
chaotic synchronization of replica of coupled map lattices is considered. The
spatial extension allows to interpret the synchronization transitions (STs) as
nonequilibrium critical phenomena: namely, as absorbing phase transitions.
Within this framework two different kind of continuous transitions have been
identified for nearestneighbour coupling : one ruled by linear mechanisms and
one by nonlinear effects. In the first case the ST belongs to the multiplicative
noise universality class, while if nonlinear effects prevail the critical
exponents coincide with those measured for directed percolation (DP). More
recently spatially extended chaotic systems with algebraically decaying
interactions have been considered.
Also in this situation the STs appear to be continuous, while the critical
indexes vary with continuity with the power law exponent characterizing the
interaction. Moreover, strong numerical evidences indicate that the transition
belongs to the "anomalous directed percolation" family of universality classes
previously found for Levyflight spreading of epidemic processes.
D. Treschev
(Steklov Mathematical Inst.,
Russie) Travelling waves in onedimensional
lattices
We construct a travelling wave in a onedimensional
lattice of identical classical particles, with potential interaction. Some
possibilities concerning lattices with several different kinds of particles are
discussed.
