Aarseth Sverre Institute of Astronomy, Cambridge NBODY6: A new star cluster simulation code This new N-body code is based on the old workhorse NBODY5 but has been converted to the modern Hermite integration method. It resembles closely the code NBODY4 which has proved successful on the special-purpose HARP computers, and like in NBODY5 the Ahmad-Cohen neighbour scheme has been added. Hermite integration is also employed for treating binaries by the KS regularization method, where the so-called Stumpff formulation has now been adopted. The standard Population I synthetic stellar evolution is replaced by a newly developed scheme (Hurley & Tout 1999) for a wide range of metallicity. The code is intended for workstations and laptops, and is more efficient and better behaved numerically than NBODY5. It is avaliable for general use by public ftp.

 Allahverdyan Armen University of Amsterdam Information theory approach to cosmological problems

 Beciu M. Ovidius University, Constanta A retroaction mechanism to solve the cosmological constant problem We present a mechanism that brings, in a natural way, the effectivecosmological constant(c.c) arbitrary close to zero. Our scenario dampsthe c.c to a vanishing small value, whatever its initial value wouldhave been. We consider the potential energy of the Higgs field in thetrue vacuum, with a temperature dependent mass term and theinfluence this term may exert on the cosmological dynamics. With aninitial value equal to or smaller than the Higgs potential in the true vacuum, we show that the effective c.c (the initial arbitrary constant and the contribution of the Higgs field) tends to zero, roughly,exponentially fast.

 Beckman John Instituto de Astrofisica de Canarias, Tenerife Chaotic behaviour in the star formation rate near the sun Using observational data on the number of stars formed as a function of age , using cromospheric activity as the indicator , we show that the star formation rate in the solar neighbourhood must have been chaotic on timescales less than 1Gyr. We also offer evidence that the system of gas and stars in the local spiral arm shows behaviour analogous to that of the logistic map.

 Belinski Volodia ICRA On the gravitational turbulence The talk represents a review on the past development, contemporary status and future prospects of the chaoticity phenomenon in the evolution of the Universe near the cosmological singularity.

 Bini Donato, Jantzen Robert Universita' di Napoli Accelerated circular orbits in Kerr spacetime The Frenet-Serret approach is applied in several ways to a familiar but still not geometrically well understood example: circular orbits in black hole spacetimes. An invariant spacetime Frenet-Serret frame approach is useful in understanding the properties of these orbits and of Fermi-Walker transport along them, and provides a visual interpretation of the geometry of this family of orbits. Closely related to the spacetime frame for these special curves are the relative Frenet-Serret frames that may be defined with respect to a family of test observers on the spacetime. The latter connect more directly to our 3-dimensional intuition about the tangent, normal, and binormal to a curve in ordinary space. These absolute and relative frames together help interpret the effects of space curvature, and the gravitoelectric and gravitomagnetic effects on circular orbiting test particles and on their gyroscopic frames of reference.

 Capaccioli Massimo Osservatorio astronomico di Napoli The discovery of an intracluster stellar population

 Capozziello Salvatore, Illuminati Fabrizio Dipartimento di Scienze Fisiche "E. R. Caianiello", Universita' di Salerno Quantum signature of large scale cosmological structures Planck's action constant is recovered, as the minimum action, in large scale cosmological structures, where gravitation is the only relevant interaction. This feature can be connected to the onset of chaotic behaviour.

 Chen Fusheng Nanchang University The criterion for chaotic pulsation in pulsating stars The criterion of Period-doubling bifurcation in pulsating stars is obtained by the Chaotic Theory. It is found that for stellar pulsating mapping the bifurcation from a fixed point to a 2-cycle occurs at a parameter lambda=-0.63, i.e. adiabatic exponent Gamma=1.331.

 Chernin Arthur D. Nanchang University Kolmogorov-Sinai entopy, intermittency and strange attractor in three-body chaos Kolmogorov-Sinai entropy is estimated for three body systems with use of homology mapping. Type III intermittency is discovered in the time series generated from these systems. Strange attractor with fractor dimension slightly above 2 is found in the structure of three body chaos.

 Cincotta Pablo Universidad de Barcelona Conditional entropy: an efficient technique to study the phase space. In two recent papers (Cincotta & Sismo, 1998, a,b) the conditional entropy of nearby orbits, computed using the arc-lenght parameter along a given orbit, was introduced as a tool to investigate the phase space associated with a given Hamiltonian.

 Cipriani Piero Universita' Tor Vergata, Roma Sources of chaos in afew degrees of freedom dynamicaal systems Here we present some recent results we obtaine of few-dimensional systems wich show the existence of clear, though not obvious, correlations between local quantities and qualitative asymptotic behaviour of dynamics.

 Combes Francoise Observatorie de Paris Fractals and self-gravity: from interstellar medium to galaxies The interstellar medium, and to some extent galaxies in the Universe,are distributed in hierarchical structures that appear self-similar over a large range of scales. They can be described by large density fluctuations, that obey power-law density distributions and correlation functions. These hierarchical structures appear chaotic and turbulent,but are also self-organizing. They can be described by a fractal with a Haussdorf dimension of the order of 1.5-2. We discuss the theories advanced to describe these fractal structures, and in particular a new theory of the self-gravity thermodynamics, in the grand-canonical ensemble,that could explain their existence, and predict their fractal dimension. The media obeying scaling laws can be considered critical, as in second order phase transitions, and the renormalization group theory can be applied.

 Dabrowski Mariusz P. University of Szczecin Problems with chaos in string cosmology We investigate possible ways of generating chaos in cosmological models based on low-energy-effective-action for strings. We discuss different approaches to establish chaos in general relativity and string theory. In particular we show that Bianchi IX models within low-energy bosonic string theory in both the Einstein and the string frames are not chaotic and emphasize the role of duality symmetry of string theory in that (J.D.Barrow and M.P.Dabrowski, Phys.Rev. D 57, 1998 - also hep-th/9711049 ) . We suggest the admission of other string modes together with different compactification schemes in order to recover chaos

 de Oliveira, Henrique Pereira Instituto de Fisica/ Universidade do Estado do Rio de Janeiro Fractal and statistical properties of phase space sets in FRW Preinflationary cosmologies In the dynamics preinflationary FRW universes with a minimally coupled massive scalar field, we study the fractal structure of sets of initial conditions. The chaotic dynamics and the fractal structure are consequence of the transversal crossings of homoclinic cylinders emanating from unstable periodic orbits present in the neighborhood of the saddle-center. The fractal dimension of the chaotic set is determined. We further show that the partition of energy into the approximately conseved rotational energy and hyperbolic energy - in a neighborhood of the saddle-center - has an statistical distribution; the statistical distribution function $N(E_{rot})$ or $N(E_{hyp})$ obtained numerically depends on the values of the scalar field mass, and in the average it can be approximately a Maxwell-Boltzmann distribution.

 De Paolis Francesco Universita di Lecce A gamma-ray halo around the Milky Way. A recent re-analysis of the EGRET data by Dixon et al. has shown the existence of a statistically significant diffure gamma-ray emission from the galactic halo.    This emission can naturally be explained within a previously-proposed model for baryonic dark matter in which gamma-rays are produced through the interaction of high-energy cosmic-ray protons with gas clouds clumed into dark clusters populating the outer galactic halo.

 Di Bari MariaTeresa Universita' di Parma Ergodic properties and structures of phase space of many-body systems. We show that customary as well more recent attempted explanations of the approach towards equilibrium of many degrees of freedom systems have their counterpart in the stucture of the ambient space where their dynamics lives. These conjectures will be illustrated through analytical and numerical application to systems of astrophysical and cosmlogical interest.

 Eckmann J-P Geneva university On the relation between porosity and dimension of fractals We show that customary as well more recent attempted explanations of the approach towards equilibrium of many degrees of freedom systems have their counterpart in the stucture of the ambient space where their dynamics lives. These conjectures will be illustrated through analytical and numerical application to systems of astrophysical and cosmlogical interest.

 Einasto Jaan Toravere Observatory, Tartu Distribution of clusters and power spectrum of matter Distribution of clusters of galaxies is quasi-regular. New data on the distribution and regularity are presented. The power spectrum of matter is calculated and initial post-inflational power spectrum found. The initial power spectrum is broken which suggests that the inflation was more complicated as assumed earlier.

 Fang Li Zhi University of Arizona Structure formation on scales of clusters and beyond; I will discuss in this review the cosmic gravitational clustering on scales equal to and larger than galaxy clusters. The emphasis here is on the dynamical features of structure formation of clusters and beyond, and their constraints on cosmological parameters. 2. Searching for scale-scale correlations of CMB

 Galgani Luigi Universita' di Milano A classical approach to the black body problem. in the light of modern Mathematical results on dynamical systems. The modern studies on the problem of equipartition of energy in classical mechanics (the Fermi-Pasta-Ulam problem) show that the relaxation times to equilibrium increase exponentially fast with the frequency, so that in concrete situations there is no equipartition, and one has effects qualitatively similar to those of glasses and spin glasses. The expected distribution of energy turns out to be qualitatively of Planck's type for the high energies, with equipartition for the low frequencies. It is discussed whether such qualitative modification of Planck's distribution might be of interest for the black body theory.

 Gevorkian Zhyrair Institute of Radiophysics and Electronics, Ashtarak Diffusion radiation in the random media Diffusion radiation originates when a relativistic charged particle passes through a medium with randomly inhomogeneous dielectric constant. Interstellar medium with randomly spaced dust grains can serve as an example of a random system. We have calculated the spectral indices and luminosites in the X-ray region 1-10kev when the diffusion mechanism occurs.It turns out that these quantites well agree with the observational data, for example, for AGN.

 Gurzadyan Vahe ICRA and Yerevan Physics Institute 1.Revealing the curvature of the universe The effect of geodesic mixing on negatively curved Riemannian manifolds can have essential physical consequences while studying the anisotropic properties of the Cosmic Microwave Background radiation. This can provide direct information on the curvature of the Universe. The analysis of the COBE 4 year data shows evidence for the predicted physical consequences of geodesic mixing, thus most probably indicating the negative curvature of the Universe. The problem of CMB properties in an hyperbolic Universe can be studied also from the point of view of information theory, namely using Kolmogorov complexity as a descriptor of CMB data expected from forthcoming CMB experiments. 2.Chaos in stellar systems: is the computer image correct ? The studies of chaos in stellar systems by means of various methods should be discussed.

 Hafizi Mimoza Tirana University The relativistic Emden sphere In this work we make use of the relativistic Emden sphere as a model for galaxies, which are considered to be configurations of isothermic relativistic kinetic gas taken in the conditions of the general relativity. We find that the profile for thermodynamic quantities like presure, density etc. can be reduced to a non—parametric form, indipendent from the radius of the sphere.    With the Runge--Kuta method we find some theoretic configurations with very high values of masses concentrated in small radius.

 Holz Daniel Erwin Albert Einstein Institut, Potsdam Lensing and high red-shift supernovae We analyze the effects of an inhomogeneous universe on the brightness of standard candles at high redshift. Consequences for the results from recent supernova surveys are analyzed.

 Kamenshchik Alexander, Khalatnikov Isaak Landau Institute of Theoretical Physics, Russian Academy of Sciences, Moscow Singularity, initial conditions and quantum tunneling in modern cosmology The key problems in modern cosmology, such as the cosmological singularity, initial conditions, and the quantum tunneling hypothesis, are discussed. The relationship between the latest cosmological trends and Landau's oldest ideas is anlyzed. Particular attention is given to the oscillatory approach to singularity; quantum tunneling processes determining wave function of the Universe in the presence of a scalar field; and the role of quantum corrections in these processes. The classical dynamics of closed models with a real scalar field is investigated from the standpoint of chaotic, fractal, and singularity-avoiding properties.

 Keheyan Elisabetta Istituto di Chimica Nucleare-Monterotondo Interstellar chemistry: radiative association reactions Radiative association is important in interstellar chemistry, and recently, many investigations have been carried out in this field. Radiative association is a clustering together of an ion and a neutral molecule in the gas phase with stabilization of the collision complex by photon emission.    Ion-molecule radiative association reactions are considered to be of relevant importance in the interstellar medium 1,2. First results on radiative association rate coefficients were obtained with ion cyclotron resonance (ICR) technique3,4. In an ICR apparatus the pressure can be varied over a wide range, typically between 10-8 and 10-4 Torr, and direct information on the radiative and ternary association reactions can be obtained. Most of the laboratory studies have been concerned primarily with collisional stabilization of the intermediate (AB+)* complex whose important parameter is lifetime with respect to unimolecular dissociation. By ICR technique it is possible also to measure the distribution of the lifetimes of these complexes.    We report here the results of ion-molecule reactions between polycyclic aromatic hydrocarbon (PAHs) cations and H20, NH3, H2, CO studied by the ICR technique. Previously these reactions were studied by the SIFT technique at 0.5 Torr and the reaction of dehydrogenated naphthalene cations with hydrogen molecules was also studied with ICR5,6 . The ionized PAHs have been implicated as potential carriers of diffuse interstellar bands7,8.    The radical cations of PAHs are unreactive with H2, CO, H20 and NH3, whilst dehydrogenated PAHs readily associates with H2 and other molecules. These studies suggest that the protonated ions are likely to be the dominant forms of PAH cations in the interstellar medium.

 Kostuchenko Irina Karpov inst.phys.chem.Moscow The comparative analysis of dynamic charateristics of solar terrestrial processes The search of the links between different solar terrestriat processes is usually based on simultaneous analysis of time series, describing various manifestations of solar activity, solar wind parameters and characteristics of some terrestrial processes. But when one deals with complex dynamic processes, the calculation of the correlation coefficients for this purpose can occur not sufficient because of nonlinearity of links. In this case the comparison of dynamic characteristics of studied processes provide additional helpful information.    We developed a method for extracting of a set of such dynamic characteristics. It is based on the idea, that main information about dynamic properties of studied process is cintained in irregularities of various types (such as bursts and jumps of relatively hight amplitude in value of the observational variable on the background of its relatively insignificant changes) and distribution of irregularities in time. This reflects the most general way of dynamic systems evolution with storage and releasing of energy. The applied mathematics allowed to associate these peculiarities in time series with characteristics of its power spectrum and difference moments of the second order.    Using the presented approach we determined the dynamic characteristics of all mentioned processes and compared them. In case of solar activity the dynamic characteristics are almost the same for the processes which are observed at the level of solar photosphere and are gradually modified for more remote layers. This can occur when the variations of conditions responsible for variations of observational variables ia all solar layers are governeed by a single global source. It is natural to suggest that this source is the turbulent convection in the convective zone. From this point of view all Sun layers is a single dinamic system. Its typical time of memory turned out to be not less than 5 years. We find that the sort term variations of the solar neutrino capture rate are not random but are caused by some dynamic process. The analysis of solar total irradiance time series allowes us to suggest that the mentioned dynamic process exsits in the solar core.    Dynamic characteristics of variations of the solar wind plasma parameters and the interplanetary magnetic field value differs for two different timescales. Variations with a typical scale of few days can be associated with the a spatial inhomogeneity in the region of solar wind source in solar corona and with interaction of neigbours streams. The similarity of dynamic characteristics of variations of solar activity and solar wind parameters on the time scale from 100 days up to solar activity cycle allow us to conclude that the temporal variations of solar activity modulates the variations of solar wind parameters. The dynamic parameters of global geomagnetic disturbances (Dst, Kp-indexes) on a large timescale demonstrate also the influence of solar activity and solar wind temporal variations.

 Kovacs Zoltan Eotvos University, Budapest Chaotic scattering in the three-dimensional Hill's problem Chaotic scattering is studied in the three-dimensional version of Hill's problem. It is shown that the underlying invariant set governing scattering is formed by the stable and unstable manifolds of the halo orbits around the Lagrangean points L1 and L2. The topological properties of this set are also discussed.

 Laskar Jacques Bureau des Longitudes, Paris TBA

 Le Bourlot Jacques Observatoire de Meudon, Universite Paris 7 Lattice dynamical system modelling of molecular clouds

 Levin Janna Sussex University Chaos in curved space In Einstein's theory of curved space, chaos may be the rule rather than the exception. Chaos has been found to influence the creation of a universe in a generic big bang as well as its demise in a big crunch. Similarly the arbitrary collapse of matter has a chaotic approach to the formation of a black hole. There are ways to search the skies for evidence of chaos in curved space. We consider two examples. (1) A chaotic history in our own universe may have left observable records of a fractal web of motions. (2) Around black holes, the chaotic scattering of light could aid in their direct detection.

 Lopac Vjera University of Zagreb Chaotic dynamics of some selected Hamiltonian systems Classical and quantal dynamical behavior is investigated for two classes of lemon-shaped billiards. Dynamics of these systems exhibits the mixed behavior where regular and chaotic components coexist in the sense of the KAM theorem. Degree of chaos is analysed in dependence on the shape parameter. The validity of the Bohigas-Giannoni-Schmit conjecture for these systems is tested by comparing classical and quantal measures of chaos. Another typical Hamiltonian system, which is essentially a gravitationally driven Fermi oscillator but is also close to the everyday experience, is presented and discussed as a useful didactic example for chaotic behavior(2). (1) V. Lopac, I. Mrkonjiæ and D. Radiæ, accepted for publ. in Phys. Rev. E (2) V. Lopac and V. Dananiæ, Am. J. Phys. 66, 892-902 (1998)

 Magliocchetti Manuela Institute of Astronomy, Cambridge Clustering properties of radio sources at high redshifts We investigate the large-scale clustering of radio sources in the FIRST 1.4-GHz survey by analysing their distribution function. We select a reliable sample from the the FIRST catalogue, paying particular attention to the problem of how to define single radio sources from the multiple components listed and we also consider the incompleteness of the catalogue. We estimate the angular two-point correlation function $w(\theta)$, the variance $\Psi_2$, and skewness $\Psi_3$ of the distribution; both $w(\theta)$ and $\Psi_2$ show power-law behaviour with an amplitude corresponding a spatial correlation length of $r_0 \sim 10 h^{-1}$Mpc. We detect significant skewness in the distribution, the first such detection in radio surveys. This skewness is found to be related to the variance through $\Psi_3=S_3(\Psi_2)^{\alpha}$, with $\alpha=1.9\pm 0.1$, consistent with the non-linear gravitational growth of perturbations from primordial Gaussian initial conditions.    We also investigate how different theoretical predictions for $w(\theta)$ can match our measurements at different flux limits. The models have been worked out for three different functional forms of the redshift distribution N(z), and by allowing for evolution of bias b(z) with redshift. Models with either biasing strongly evolving with epoch or with an N(z) not showing a low-redshift component are ruled out. Models where the bias varies linearly as a function of redshift could be accepted only in the case of a redshift distribution strongly dominated by the low-redshift spike, even though the resulting values for the correlation length $r_0\sim 3-4h^{-1}$Mpc are somewhat too small. The best fit is provided by models with constant biasing and a redshift distribution characterised by a lower-amplitude spike, both in the case of open/flat Universe. The correlation lengths corresponding to these models are $r_0\sim 9-11h^{-1}$Mpc.

 Makino Jun Tokyo University Stellar Dynamics and special-purpose computer: GRAPE project Ill overview the recent developments in GRAPE project, wich include: (a) GRAPE-5 (b) GRAPE-6 (C) a new high-accuracy tree algorithm, and (d) some important scientific results.

 Mardling Rosemary Monash University, Melbourne Chaotic tides in binary stars and the three-body problem It has been known since the days of George Darwin that a binary is unstable to mass transfer if its component stars approach each other within a certain distance. Recently, I showed that a second type of tidal instability exists in close binaries when their orbits are eccentric (Mardling 1995, ApJ, 450,722). This instability involves a chaotic interaction between the orbit and the tides of at least one of the stars with the possibility of huge tides being raised, in contrast to normal tidal interactions where tidal energies are relatively small. A quite separate problem which has received much attention in the last 200 years is the celebrated 3-body problem. The rich behaviour exhibited by this simple configuration has led people to attempt to qualify and quantify its stability: for what configurations can it exist as a bound system? In this talk I will demonstrate that these two problems are intimately connected. The chaotic behaviour exhibited by both systems may be understood in terms of the modes of oscil

 Melkonian Anahit Yerevan Physics Institute Chaos in galaxies with double nuclei The relative chaos in N-body gravitating systems with double massive nuclei are studied using the Ricci curvature criterion.

 Melnikov Vitaly Center for Gravitational and Fundamental Metrology VNIIMS, Moscow Chaotic behavior in multidimensional cosmological models Cosmology-Billiard behavior of multidimensional cosmological models with arbitrary number of dimensions is analyzed with different sources like perfect fluid,scalar fields and fields of forms in classical and quantum cases.

 Merafina Marco Universita' di Roma "La Sapienza" Thermodynamic stability of relativistic stellar systems Thermodynamic stability in relativistic regime is analysed for spherically symmetric selfgravitating systems. The equilibrium configurations are developed by using a distribution function with a cutoff in the energy as a function of Schwarzschild metrics. The stability analysis was considered in several cases in Newtonian regime where the results are known in literature. The situation is different in relativistic regime where unclear and not coherent results are existing. In this work the problem is clarified by reconstructing the critical curve of the onset of instability in the plane zc-T (central redshift-temperature) of the equilibrium configurations. This curve under particular conditions coincides with the one connected to dynamical instability while, for "cooled" systems, separates. The dynamical instability curve, never reaches Newtonian regime while the thermodynamic one reaches it, giving a critical value of the central gravitational potential in complete accordance to the classical results in Newto

 Merritt David Rutgers University Chaos in elliptical galaxies

 Meylan Georges ESO , Garching Internal dynamics of globular clusters Galactic globular clusters, which are ancient building blocks of our Galaxy, represent a very interesting family of stellar systems in which some fundamental dynamical processes have taken place on time scales shorter than the age of the universe. In contrast with galaxies, these clusters represent unique laboratories for learning about two-body relaxation, mass segregation from equipartition of energy, stellar collisions, stellar mergers, and core collapse. This review will summarize the tremendous developments, as much theoretical as observational, that have taken place during the last two decades, and which have led to a quantum jump in our understanding of these beautiful dynamical systems.

 Montani Giovanni ICRA Aspects of chaotic cosmology.

 Muzzio Juan Carlos La Plata Observatory, Argentina Chaotic motion inside galactic satellites Galactic satellites cannot be perfectly spherical since, even if no other effects (such as rotation) are present, the tidal forces of the galactic field make them triaxial. Besides, due to the tidal forces, the star orbits within the satellite are not restricted by the energy integral. As a result, chaotic motions play a significant role in the structure and evolution of galactic satellites.

 Nieuwenhuizen Th.M. Van der Waals-Zeeman Instituut -University of Amsterdam Thermodynamics of black holes, stellar systems and spin glasses The present equilibrium formulation of thermodynamics for black holes has several drawbacks, such as assuming the same temperature for black hole and heat bath. Recently this author formulated non equilibrium thermodynamics for glassy systems. This approach is applied to black holes, with the cosmic background temperaturebeing the bath temperature, and the Hawking temperature the internal temperature. Both Hawking evaporation and absorption of background radiation are taken into account. It is argued that black holes did not form in the very early universe.

 Nucita Achille Andrea Dipartimento di Fisica, Universita di Lecce Gravitational waves and evolution timescales of relativistic clusters Relativistics clusters may exist within galactic centers and mimic the presence of massive black holes. The main evolution timescales of these kind of clusters are investigated with particular emphasis to the gravitational wave damping time.The possibility to study the relative abundance of hard binaries to binary stars interacting on hyperbolic orbits is also explored, by using the capabilities of the VIRGO detector.

 Nuritdinov Salakhutdin Tashkent University 1.Non-linearly non-stationary model of disk galaxy and its stability We have constructed a seriese of analytically solvable models of the evolution early stages of the disk galaxy. The stability problem of non-linearly unequilibrium model is studied in the framework of the galaxy dissipationless collapse theory. 2.On the identification problem of probable double open star clusters A problem of search for probable double and multiple open star clusters is discussed. The preliminary list of 25 candidates of such type of objects is presented.

 Ossenkopf Volker Universitot zu Koeln Characterization of molecular cloud structures We have tested different approaches for the simulation of three dimensional structure of molecular clouds –fractal models and MHD simulations. For a comparison to the observations, we solved the full radiative tranfer problem and computed the emrgent line profiles and maps. By the iterative comparison of model fits and observed cloud chaateristics it was possible to find important constraints on the Physical structure and processes governing molecular clouds.

 Patsis Panos A. Max-Planck-Institut fuer Astronomie, Heidelberg Chaos and the morphology of disk galaxies The role of non-linear and chaotic phenomena for the observed morphology of disk galaxies is investigated. A comparison is made between models where the galaxy is treated as a Hamiltonian system withwhat is found in N-body simulations. The constraints on the dynamical modelling imposed by recent observations in the Near-IR made by the author are discussed.

 Perez Jerome ENSTA-observatoire de Paris, Meudon Analytical solution of the classical gravitational collapse I present an analitical solution of the N body classical gravitating problem in the limit where N goes to infinity. The method used is based on the sympletctic formulation of collisionless Boltzmann-Poisson system.

 Pesin Yakov Penn state university Lyapunov exponents, multifractals and recent advance in dimension theory.

 Petrovskaya Irina St.Petersburg University Time evolution of orbit integrals distribution in gravitating systems For investigation on evolution of orbits in gravitating systems the probabilities of given variations of orbit integrals are found for the case of couple interactions in plane and 3-D systems. The probabilities are the kernels of the kinetic equation. The method takes into account not only small but also large variations of orbit integrals(close encounters). The evolution of distribution of orbit integrals, including the changing of degree of the radial orbit stretch, in spherical clusters is investigated.

 Prigogine Ilya Instituts Internationaux de Physique et de Chimie, Bruxelles Thermodynamic universe - dynamics and the arrow of time.

 Pucacco Giuseppe Universitá di Tor Vergata , Roma Stochasticity of elliptical galaxies Relativistics clusters may exist within galactic centers and mimic the presence of massive black holes. The main evolution timescales of these kind of clusters are investigated with particular emphasis to the gravitational wave damping time.The possibility to study the relative abundance of hard binaries to binary stars interacting on hyperbolic orbits is also explored, by using the capabilities of the VIRGO detector.

 Raikov Alexander Scientific-Educational Union "Earth and Universe"Saint Petersburg Gravitational lenses as an instrument in studies of large scale statistical properties of matter distribution in Universe The distribution of ratio of lensing object/lensed object redshif among the known sample is examined. It is shown that it differs from that of expected in the case of pure uniform large scale matter distridution, tending to be ruffly uniform. Though gravitational lens samples is still not complete for detailed statistical implications,both its statistics and large scale matter distribution models are discussed and observational tests are suggested.

 Roya Mohayaee ICTP, Trieste Hierarchical perturbation in standard cosmology We use de Vaucoleurs power law desity-distance relation, to study a hierarchical perturbation of the Friedmann universe. We solve the Einstein equation and obtain the density contrast and the amplification factor for the perturbations. It is shown that scale-invariant inhomogeneities decay in the Einstein-de Sitter universe. On the contrary in a open dust universe, the inhomogeneities grow. Remarkably, for low values of Omega by up to a factor of 10^13 from the recombination to a present time.

 Ruffo Stefano Universitá di Firenze Ensemble equivalence, weak violations of extensivity and chaos in Hamiltonian models with long-range attractive interactions. We introduce a class of mean field hamiltonians wich results from a Fourier series development of the well known gravitational sheet model. The HMF model can be easily solved in the canonical ensemble (CE). Microcanonical simulations show disagreements with canonical ensemble of two kinds: 1- Near first-order phase transitions equilibrium results are different in CE and ME. 2- Near second-order phase transitions quasi-stationary chaotic metastable states appear, whose life-time grows with the number N of degrees of freedom. Some universal properties of the Lyapunov spectra are also investigated.

 Semadeni Enrique Vazquez UNAM, Mexico 1. Turbulent formation of interstellar structures . A principal component analysis comparison between observations and simulations of the ISM. We present three-dimensional numerical simulations of turbulence in two different regimes: a) Full models of the ISM including the magnetic field, parameterized cooling and diffuse and stellar heating, self-gravity and rotation. b) Barotropic flows ($P \propto \rho^\gamma$), with $\gamma$ smaller, equal and larger than 1. We compare the physical-space density distribution with its representation in position-position-velocity space, noting that the two may give very different views, the latter introducing features of the velocity structure into that of the density. We compare the morphologies of both regimes with that of spectral map observational data. Finally, we apply the technique of Principal Component Analysis (PCA) to the simulations, discussthe interpretation of the PCA results, emphasizing the ambiguities and capabilities of the method, and compare with similar analyses of observational data existing in the literature.

 Shevchenko Ivan Pulkovo observatory, St.Petersburg Orbital resonances and the separatrix algorithmic map The chaotic motion near separatrices of orbital resonances in planetary satellite systems is considered. A separatrix map is constructed, describing the motion in a vicinity of the separatrix of a model orbital resonance. This is done within the framework of the "perturbed pendulum"model for a primary resonance.The map is two dimensional and has two parameters determined by a set of original parameters of the dynamical problem. The derived separatrix map is alorithmic: it contains conditional tranfer statements. Hence it is called sepratrix algorithmic map. In order to show its performance, it is applied to the particular problem of orbital eolution of a system of two uranian satellites, Miranda and Umbriel

 Soares, Ivano Damiao Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro 1)Chaos in pre-inflationary FRW universe. 2)Chaos and a resonance mechanism for structure formation in inflationary models. 1)The dynamics of a preinflationary phase of the universe, and its exit to inflation, is discussed.This phase is modeled by a closed Friedmann-Robertson-Walker geometry, the matter content of which is radiation plus a scalar field minimally coupled to the gravitational field. The phase space presents critical points of saddle-center type and saddle type (each of the critical points is determined by an extremum of the scalar field potential). As a consequence, the topology of the phase space is characterized by homoclinic cylinders emanating from unstable periodic orbits,and the transversal crossings of the cylinders, due to the non-integrability of the system, results in a chaotic dynamics. The model exhibits one or more exits to inflation, associated to one or more strong asymptotic de Sitter attractors present in phase space, but the way out from the initial singularity into any of the inflationary exits is chaotic.

 Sota Yasuhide Waseda university, Tokyo Renormalization group approach in newtonian cosmology We apply the renormalization group method to examine the observable scaling properties in newtonian cosmology. The original scaling properties of the equation of motion in our model are modified for the averaged observables on costant tme slices. In the RG flow diagram we find three robust fixed point Einstein-de Sitter, Milne and Quiescent fixed points. Their stability properties do not change under the effect of fluctuations. Inspired by the inflation scenario we set the Einstein-de Sitter fixed point with small fluctuations, as the boundary condition at the horizon scale. Solving the RG equations under this boundary condition toward the smaller scales, we find generic behaviour of observables such that the density parameter Omega decreases while the Hubble parameter H increases for smaller averaging volume. The quantitative scaling properties are analyzed by calculating the charateristic exponents around each fixed point. Finally we argue the possible fractal structure of the universe beyond the horizon sc

 Szidlowski Marek Jagiellonian university, Krakow Integrability and non-integrabilty in planar indefinite Hamiltonian systems of cosmology We study the problem of nonintegrability (integrability) of cosmological dynamical systems in Hamiltonian form with indefinite kinetic energy form $T = \frac{1}{2} g(v,v)$, where $g$ is a pseudoriemannian metric which has a Lorentzian signature $(+,-)$, $v \in T_{q} M$ is a tangent vector to $M$ and $M$ is 2-dimensional configuration space. The examples of the Liouville integrable cosmological systems are given. The Ziglin theorem about nonintegrability of planar cosmological systems is formulated and the Yoshida criterion of nonintegrability for homogeneous potentials is generalized. We also find interconnections between the Yoshida criterion and the strong Morales-Ramis theorem concerning the nonexistence of additional mereomorphic first integrals. The consequences of this theorem for nonintegrability of the Friedman-Robertson-Walker cosmology with massive scalar fields are analyzed in details. 2.Dynamical systems in general relativity.

 Toporensky Alexey Sternberg Astronomical Institute, Moscow Chaos in closed FRW Universe with a scalar field The dynamics of closed Fiedmann-Robertson-Walker cosmological model with scalar field is investigated for various scalar field potentials.It was shown that well known chaotic behaviour of the model with massive scalar field is not a generic feature for wider class of potentials. In paticular, sufficiently large constant term in the potential of the scalar field makes the dynamics regular. The conditions for the scalar field potential which allow the chaotic behaviour of corresponding cosmological model is discussed.

 Torrente Emilio Instituto de Fisica Corpuscular -C.S.I.C., Departament de Fisica Teorica Universitat de Valencia, Valencia. Neutrino conversions in Solar random magnetic fields We consider the effect of a random magnetic field in the convective zone of the Sun superimposed to a regular magnetic field on resonant neutrino spin-flavour oscillations. We argue for the existence of a field of strongly chaotic nature at the bottom of the convective zone. In contrast to previous attempts we employ a model motivated regular magnetic field profile: it is a static field, solution to the solar equilibrium hydromagnetic equations. These solutions has been known for a long time in the literature, we show for the first time that in addition they are twisting solutions. The expected signals in the different experiments (SK,GALLEX-SAGE,Homestake) are obtained as a function of the level of noise, regular magnetic field and neutrino mixing parameters. Previous results obtained for small mixing and ad-hoc regular magnetic profiles are reobtained. We find that MSW regions are stable up to very large levels of noise (P=0.7-0.8) but they are acceptable from the point of view of antineutrino production.

 Tsuchiya Toshio Kyoto University Relaxation of one-body distribution function and KS time in gravitating systems The relation betweeen relaxation, the time-scale of Lyapunov instability and the Kolmogorov-Sinai time in a one-dimensional gravitating sheet system is studied. The Lyapunov spectrum approximately converges with increasing the number of sheets, N. Both the maximum Lyapunov exponent and the Kolmogorov-Sinai entropy decrease as proportional to N^(-1/5). These times evidently differ from any relaxation times found previously. The microscopic relaxation, wich is the time of mixing among the energies of individual sheets, is found to coincide with the inverse of the minimum positive Lyapunov exponent.

 Umehara Hiroaki Kashima Space Research Center Periodic solutions in the three body problem. Periodic orbits are searched for using the annealing Monte-Carlo method in the three body problem with zero angular momentum.

 Valluri Monica Rutgers University Regular and stochastic motion in triaxial galaxies We use Laskar's frequency mapping technique to study the dynamics of a family of triaxial galaxies with realistic density profiles with and without slow figure rotation. The objects of fundamental importance in structuring phase space are found to be the resonant tori, regions in phase space satisfying a relation between the fundamental frequencies of the form $0=l\omega_1+m\omega_2+n\omega_3$ with integer $l,m,n$. When stable these resonant tori generate regions of phase space in which the motion is regular; these regions are not necessarily associated with a stable period orbit. They are infact associated with a family of orbits we call thin box orbits - multiply connected two dimensional surfaces in 3-space. Models with high central concentrations -- steep central cusps or massive black holes -- exhibit the most stochasticity. Even a modest black hole, with a mass $\sim 0.3\%$ the mass of the galaxy, is as effective as the steepest central density cusp at inducing stochastic diffusion. There is a transition to global stochasticity in box-like phase space when the mass of a central black hole exceeds $\sim 2\%$ the galaxy mass. Figure rotation substantially increases the fraction of box orbits which are stochastic particularly in systems with shallow cusps. We trace the reason for this to the destruction of thin box orbits. Figure rotation also destroys a larger fraction of the long-axis tube orbits making them behave more like stochastic box orbits. While the majority of short-axis tubes are unaffected by rotation, the fractions of stochastic tube orbits of all three kinds increases especially in the transition layers between the orbit families. These results have important consequences for the evolution of slowly rotating triaxial ellipticals and the bulges of spirals. The presence of a large fraction of stochastic orbits leads to a slow evolution in the shape of a triaxial galaxy to an axisymmetric one. This process is accelerated by figure rotation. These results support those of our earlier studies which suggested that the stochastic diffusion rates in low luminosity spheroids (including the bulges of spiral and SO galaxies) were sufficiently short that they must be largely axisymmetric.

 Vitali David Universita di Camerino Chaos, thermodynamics and quantum mechanics: an application to celestial dynamics. We address the issue of the quantum-classical correspondence in chaotic systems using, as recently done by Zurek, the solar system as a whole as a case study: this author shows that the classicality of the planetary motion is ensured by the environment-induced decoherence. We show that equivalent results are provided by the theories of spontaneous fluctuations and that these latter theories, in some cases, result in a still faster process of decoherence. We show that, as an additional benefit, the assumtion of spontaneous fluctuation makes it possible to genuinely derive thermodynamics from mechanics, namely, without implicitly assuming thermodynamics.

 Zakharov Alexander Inst Theoretical and Experimental Physics, Moscow "Bad"and "nice"models for gravitational lenses Using two model of gravitational lensing we analyze stucturally stable and unstable properties of the models

 Zalaledtinov Roustam Tashkent University Approximate Symmetry, Inhomogeneous Spaces and Gravitational Entropy The notion and existing definitions of an approximate, or almost, symmetry proposed earlier by Matzner and Spero and Baierlein are reviewed and discussed in some details. Particular attentionis paid to the geometrical content of the proposed definitions. Specifically, their relevance and usefulness for the description of inhomogeneous manifolds, a definition of measure of degree of inhomogeneity, and a geometric definition of the notion of background are pointed out. Physical interpretation of such symmetries is also discussed. It is shown that approximate symmetry invariants are related to a kind of averaged energy of inhomogeneities. Some generalisations and specifications of approximate symmetries are considered. Possible utilisation of the notion of an approximate symmetry for a definition of the concept of geometrical entropy as measuring the disorder/order degree of a manifold with respect to a given 'symmetric' (i.e. ordered) one is discussed. Using the entropy as a candidate to serve as gravitational entropy is also discussed.

 Zurek Wojciech Los Alamos Chaos, Decoherence and the Second Law