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Oral Presentations

Boxy/peanut bulges and disc-like bulges
E. Athanassoula
Observatoire Astronomique de Marseille, Marseille, France

I discuss the formation, evolution and properties of boxy/peanut bulges and of disc-like bulges. The former are parts of bars seen edge-on, thus their formation and evolution is very closely connected to that of bars. The latter are disc-like objects in the central parts of disc galaxies, formed from gas inflow and subsequent star formation.

3D Accretion Discs Dynamics: Numerical simulations
Dmitri V. Bisikalo
Institute of Astronomy, Russian Academy of Sciences, Moscow, Russia

In this report the main attention is paid at physics of accretion discs in binary systems and particularly at formation of waves in discs. The characteristic features and possible observational manifestations of the “hot line” wave formed due to interaction between the circumdisc halo and the stream, two arms of the tidal shock, and precessional spiral density wave are discussed. The analysis of the numerical model and comparison of computational results with observational data allow us to define more exactly the physical processes resulting in formation of structures in accretion discs in binary stars.

Moving Groups in the Solar Neighbourhood - Signature of Chaos?
Dalia Chakrabarty
Univ. of Nottingham, U.K.

The origin of the clumpiness of the local phase space is a challenge to stellar dynamists. Though other schools of thoughts exist, a dynamical explanation for this observed structure is sufficient - the dynamical influence in question is that due to the non-axisymmetric features of the Galaxy. The pronounced degree of non-linearity in our local phase space, as indicated by velocity measurements of the close-by stars is understood on the basis of simple simulations in which the central bar and the outer spiral pattern in the Galaxy, act in concert to stir a background system of disc stars. The increased inducement of chaos, as brought about by increasing the intrinsic disk velocity dispersion and the perturbation strengths is noted. This is correlated to the prevalence of phase space structure, in a phenomenological way. Moreover, the relatively superior ability of the spiral, over the bar, in invoking chaos is also marked. The meaningfulness of model parameter estimation from the resulting phase space structure is discussed. Given this situation, better ways of achieving constraints on the relevant models is discussed. A test of the dynamical origin of the moving groups will also be considered.

Ordered and Chaotic Orbits in Spiral Galaxies
George Contopoulos
Research Center for Astronomy and Applied Mathematics, Academy of Athens

The spiral arms in normal spirals produce perturbations of the order of 2-10% of the axisymmetric background. Chaos is in general quite limited, appearing mainly near corotation and near the center. Thus, the spiral arms are composed mainly of regular orbits. On the other hand, barred galaxies produce large perturbations of the order of 100% and generate a large degree of chaos. Chaos is most important around and outside corotation and also in an outer envelope of the bar. The bar consists mainly of ordered orbits, while the outer spiral arms consist mainly of chaotic orbits. The structure of the spiral arms depends on the unstable asymptotic curves from the unstable periodic orbits around the Lagrangian points L1, L2 at the end of the bar. Stars, starting near these Lagrangian points, move close to the unstable asymptotic manifolds. Orbits, starting close to other unstable periodic orbits inside and outside corotation, follow similar manifolds. Various recent developments on this topic are discussed.

Distribution functions for galaxies with quadratic programming
H. Dejonghe
Sterrenkundig Observatorium, Universiteit Ghent, Belgium

One way to characterize the amount of chaos in real stellar systems is the construction of distribution functions that depend on as many integrals as possible. Deviations of such models with the data can give a measure of the amount of chaos present. In this talk, I review a program based on quadratic programming that is designed to be as flexible as possible in the task of producing distribution functions for real stellar systems. I also present some recent theoretical modeling work

Chaos in the motion of planetary system
Rudolf Dvorak
Astro Dynamics Group, Institute of Astronomy, University of Vienna

We compare the orbits of our planetary system with the motions of planets in extrasolar planetary systems. In the known more than 200 systems in the Solar environment the orbits of the mostly Jupiter-like planets are very eccentric, but stable. We review what we know about the dynamics of these systems and critically discuss the use of chaos indicators to understand their structure.

Normalization and reduction of perturbed Keplerian problems Constantinos Efstathiou
Instituut voor Wiskunde en Informatica, Rijksuniversiteit Groningen, Groningen, The Netherlands

We study the hydrogen atom in weak and homogeneous electric and magnetic fields (Stark and Zeeman effect respectively) and we construct in steps an integrable approximation of the 3DOF system.

The first step is the normalization and reduction of the approximate Keplerian symmetry. The result is a (in general not integrable) 2DOF Hamiltonian system. The reduced system has a second approximate 2- oscillator symmetry and we observe that different angles formed by the two fields correspond to different resonances. The second step is the normalization and reduction of the second approximate symmetry.

In this way we construct an one-parameter family of integrable 1DOF Hamiltonian systems that can be studied analytically. We describe the basic qualitative features and the bifurcations of the integrable systems near the major resonances 1:1 and 1:2. Finally, we compare our results to numerical simulations of the dynamics.

Invariant manifolds and spiral arms in barred Galaxies
Christos Efthymiopoulos
Research Center for Astronomy and Applied Mathematics, Academy of Athens

The unstable invariant manifolds of many different families of unstable periodic orbits near or beyond corotation generate a pattern in configuration space which essentially reproduces the observed spiral pattern of barred - spiral galaxies. Examples of this mechanism are given in an N-Body simulation and in a self-consistent model of the galaxy NGC3992. Despite the fact that the bar potential is dominant, in general, we find that the morphology induced by the invariant manifolds of the short period family around L1 or L2 is quite different when, besides the bar, the self-consistent potential of the spiral is taken or not taken into account. Starting from the bar only, an iterative simulation is proposed that reconstructs the spiral pattern, and its physical implications are discussed.

A method to measure intermittent chaos in N-body simulations Nicolas Faber
University of Strasbourg, Strasbourg, France

We present an efficient method to quantify intermittent chaos in gravitational N-body simulations. The technique, which we name CWaTIE, is based on a continuous wavelet transform of space- and velocity orbital time series. The wavelet power-spectrum is used to define an entropy which tracks trends in chaoticity continuously in time. We recover the time-dependent complexity of the dynamics accurately for small-N (2 < N <10) as well as for large-N (N >= 10) problems. Punctual events, such as e.g., the occurrence of core-collapse in star cluster dynamics can be singled-out by the technique.

Fully developed turbulence in the accretion disk of the IP Peg.: problems of frequency spectrum
A. Fridman 1, L. Pustil'nik, Yu. Torgashin, D. Bisikalo 1, A. Boyarchuk
1.Institute for Astronomy, Russian Academy of Sciences
2. Tel Aviv University, Israel Cosmic Ray & Space Weather Center, Qazrin, Israel

An analysis of the observed frequency spectrum of the IP Peg shows the existence of a fully developed turbulence in the accretion disk.

Barred Galaxies: An Observer's Perspective
Dimitri Gadotti
Max Planck Institute for Astrophysics, Garching bei München, Germany

Galactic bars are one of the most fascinating natural phenomena in the extragalactic domain. In particular, bars challenge dynamicists working towards an appropriate theoretical description of galaxies, mainly due to their intrinsic complexity and vivid relationship with other galactic components. In fact, a growing body of observational results acknowledge the major role played by bars in the evolution of galaxies. To properly describe this multifaceted richness in a theoretical framework, models must have an observational counterpart. In this talk, I will review our current understanding of the formation and evolution of bars, stemming from photometric and kinematical observations, in order to provide observable physical quantities which can either be used as a solid basis on which realistic models can be built, or be compared against more fundamental theoretical results. Finally, I will briefly discuss opens questions and directions for future research.

Statistical thermodynamics for dynamical systems in meta-equilibrium: recent advances in the FPU problem. The role of retarded electromagnetic interactions, and some comments on the role of retardation in gravitational forces.
L. Galgani
University of Milano, Milano, Italy

Up to now, most studies in the FPU problem were concerned with the consideration of  few initial conditions out of equilibrium. A review is given of some recent results in which the problem of the approach to equilibrium is studied instead  from the point of view of ergodic theory, in which in principle one follows the orbits for all initial data, weighted with an invariant measure. A metaequilibrium state is found also in such a situation. It is also discussed how the corresponding thermodynamics cannot fit the general Wien's law, if the role of the electromagnetic retarded field in the interatomic forces is neglected.
Some comments on the role of retardation in the gravitational forces are also mentioned.

The Kolmogorov-Sinai Entropy of selfconsistent galaxy models
Alejandro Gonzalez and William de la Cruz
Division Academica de Ciencias Basicas, UJAT,Mexico

Till recently the conventional wisdom has ignored the possibility of systematic dynamical changes in galaxies, apart of those produced by the obvious close galaxy-galaxy encounters. Thus, complex equilibria, including cuspy and nonaxisymmetric equilibria, may not exist or may be hard for a real galaxy to find; and this is even more relevant in high density environments trough mergers so triggering systematic changes in the bulk structure of a galaxy. In all these cases, the stochastic behaviour of an important fraction of orbits plays an important role. In this contribution we develop a new technique to quantify the relevance of the chaotic orbits along the Hubble sequence and analyze the transition of the orbital structure as the bulge of a central condensation is included. The novel feature of our mothod is that we treat the sampling of orbits as a Monte Carlo integration problem, with integral equal to the Kolmogorov-Sinai entropy of the system. So our method preferentially sample those orbits with non-zero Liapunov exponents. By using TAXON we clasify these orbits for each model.

Order and chaos in spiral galaxies seen through their morphology
Preben Grosbøl
European Southern Observatory, Garching , Germany

It is clear from dynamical considerations and N-body models that both order and chaos are important for spiral galaxies. To study the relative importance of order and chaos in real galaxies, one needs accurate kinematic information for the stellar population. Unfortunately, such data are still very difficult to obtain outside the very central parts of spiral galaxies due to the relative low surface brightness of their disks. As a second option, one can consider the morphological features in the disks (e.g. bars, spiral arms, and spurs) and interpret them as indicators of the underlying dynamics. The current paper reviews the morphology of spiral galaxies and discusses possible relation to order and chaos in them, both using optical and near-infrared data.

Orbital structure in barred galaxies and the role of chaos
M. Harsoula, N. Voglis, C. Contopoulos and C. Kalapotharakos
Research Center for Astronomy and Applied Mathematics, Academy of Athens

We study the orbital structure of a self-consistent N-body equilibrium configuration of a barred galaxy constructed from cosmological initial conditions, having a value of spin parameter near the one of our Galaxy. We classify the orbits in regular and chaotic ones using a combination of two different methods and find 60% of them to be chaotic. We examine the phase space using projections of the 4D surfaces of section for test particles as well as for real N-body particles. The real particles are not uniformly distributed in the whole phase space but they avoid orbits that do not support the bar. We use frequency analysis for the regular as well as for the chaotic orbits to classify certain types of orbits of our self-consistent system. We find the main resonant orbits and their statistical weight in supporting the shape of the bar and we emphasize the role of weakly chaotic orbits in supporting the boxiness at the end of the bar. Finally we show that, due to the fact that our system contains only weakly chaotic orbits, the effectiveness of Arnold diffusion is restricted and therefore a lot of chaotic orbits stay located in restricted areas outside corotation limited by tori and cantori on the 2-D projection of the surface of section.

On the stability of extrasolar planetary systems: Ordered and chaotic motion
John D. Hadjidemetriou and George Voyatzis
University of Thessaloniki, Thessaloniki, Greece

We consider extrasolar planetary systems with a sun and two planets, with small but non negligible masses, moving in the same plane. We study possible stable configurations, for several parameters which determine the orbit, as the eccenticities of the two planets, the sum and the ratio of their masses, the position of the line of apsides, (alignment or antialignments for symmentric orbits, or angle of the line of apsides for non symmetric orbits), phase of the two planets (position at perihelion or aphelion at t=0).

The stability depends on the topology of the phase space, and we study the stability by considering the basic families of periodic orbits, which are the backbone of the phase space. It is known that there exist families of periodic orbits, along which the eccentricities of the two planets vary. The orbits are periodic in a rotating frame, which implies that the relative configuration is repeated after one period. These families correspond to a mean motion resonance between the two planets. This means that the ratio of the two semi major axes is almost constant along the family.

A large part of these resonant families is stable. This means that a small perturbation may result to a quasi periodic motion, where the variation of the elements of the planetary orbits is bounded. We study the stable region around these stable resonant periodic orbits and we also study the factors that generate instability as a perturbation increases.

Orbits in N-Body Merger Remnants and Observational Properties of Ellipticals
Roland Jesseit
University Observatory of Munich, Munich, Germany

We determine orbital classes in a large sample of N-body merger remnants. We connect the orbital content with the 3-dimensional Shape of the remnants and show their paramount importance for understanding projected properties of Ellipticals. Some disagreements with observations can be understood by the population or depopulation of the different orbit classes alone. The Influence of different merging mechanisms on the orbital content is discussed.

Galaxy Spin: Using new techniques in data analysis to understand what is going on.
Bernard Jones
Kapteyn Institute, Groningen, Netherlands

One of Nikos Voglis' favourite topics was the origin of cosmic  angular monentum: as long ago as the late 1970's he thought that cosmic tidal fields might organise galaxy spins and  orientations.  However it was difficult then to test such a  hypothesis since neither the data sets nor the analysis techniques  were available.  That situation has now changed and we are able to  show convincingly that indeed there are significant such alignments between spin axes and large scale structure, and moreover we are  able to demonstrate that it is the large scale tidal fields that  are responsible for imposing this order on what would otherwise be  a random distribution.  We have the observation, we know the cause - all we need now is a theoretical model.

Orbital distributions and self-consistency in elliptical galaxies Constantinos Kalapotharakos
Research Center for Astronomy and Applied Mathematics, Academy of Athens

The orbital distributions (as regards regular and chaotic orbits) of non-rotating elliptical galaxies with smooth centres are studied. The focus is on examining how are these distributions connected with the self-consistency and the stability of these systems. A series of N-body models were produced that cover all possible observed morphological types of elliptical galaxies. These models were evolved towards their equilibrium state by a SCF code which is a modified version of Hernquist & Ostriker (1992) code. An orbital analysis is made in these models (as regards the real particles). Using a number of criteria for the classification of the particles’ orbits (e.g. Lyapunov exponents, Alignment index, Frequency analysis) we identify the types of orbits which are mostly relevant to the various morphologies. The orbital distributions do not necessarily reflect the volume of the phase space available to each type of orbit. The self-consistency imposes restrictions rendering some regions preferential to the motions of the real particles.

The Chaotic Light Curves of Accreting Black Holes
Demos Kazanas
NASA, Goddard Space Flight Center, U.S.A.

We review the time series analysis of the X-ray light curves of accreting Black Hole Candidate sources, both galactic and extragalactic.

These are chaotic with a Power Density Spectrum of power law form with several breaks at time scales not obviously related to any of the expected dynamical time scales of sources of this nature.

Attempts to uncover a low dimensionality strange attractor responsible for their chaotic character have been negative, indicating that these light curves are quite likely stochastic. Additional information on the underlying character of these light curves comes about from the fact that the X-ray emission is likely due to Comptonization of soft photons by hot electrons in the black hole vicinity; in particular, the dependence of the lags between the hard and soft X-ray photons as a function of the Fourier frequency suggests that the underlying hot electron cloud extends to ~1000$ or so Schwarzschild radii R, contrary to theoretical expectations that postulate it to be only ~10 R S. A simple model that attributes most of the observed variability to the stochastic character of Compton scattering appears to be in agreement with these observations.

More recent analysis using the novel technique of Fourier Resolved Spectroscopy indicate a stratification of the X-ray emitting region consistent with that of the simple Comptonization model, while at the same time revealing the presence of other components such as an accretion disk consistent with the theoretical expectations.

The Chaotic, Early Evolution of the Outer Solar System
Harold F. Levison
Southern Research Institute, Boulder, U.S.A.

It has been known for many years that the orbits of the planets are formally chaotic. However, it is only recently been recognized that the orbits of the giant planets have changed significantly, and perhaps violently, since the planets formed. This is made clear by the complex and excited dynamical state of the small body reservoirs. In this talk, I will discuss a new model for the early evolution of the outer Solar System. This model explains, for the first time, many of the observed characteristics, including 1) the orbits of the giant planets, particularly the eccentricities of Jupiter and Saturn, 2) the structure of the trans-Neptunian region, 3) the origin of the Trojan asteroids and other primitive asteroids, 4) the irregular satellites of the giant planets, and 5) the lunar late heavy bombardment.

Ansae in barred galaxies, simulations and observations
Inma Martinez-Valpuesta
Instituto de Astrofisica de Canarias, Tenerife , Spain

Many barred galaxies show a set of symmetric enhancements at the ends of the stellar bar, called ansae, or the “handles” of the bar. In this work, we make a quantitative analysis of the occurrence of ansae in barred galaxies, making use of The de Vaucouleurs Atlas of Galaxies by Buta and collaborators. We find that ~40% of the SB0's present ansae in their bars, thus confirming that ansae are common features in barred lenticulars. We compare these results with pure N-body numerical simulations in terms of bar strength, dark matter halos and orbital analysis.

Exploring the distribution of regular and chaotic orbits in barred galaxies
Thanos Manos
University of Patras, Center for Research and Applications of Nonlinear Systems (C.R.A.N.S) & Université de Provence, Observatoire Astronomique de Marseille - Provence (OAMP)

The dynamical evolution of galactic models is related to and influenced by the chaotic or regular nature of the orbits. For this reason, characterizing the nature of the orbit constitutes an important issue in the field of the galactic dynamics. There are lots of methods derived from the non - linear science that have been applied for the distinction between regular and chaotic motion. In our study we use the Smaller ALigment Index (SALI), which has been proved a very powerful tool for this goal. We briefly describe this method and its advantages and then we apply it to 2D and 3D barred galaxy potentials. Furthermore, we find the fraction of chaotic and regular orbits in such potentials and present how this fraction changes when the main parameters of the model are varied. In additional, we explore how the chaotic and regular orbits are distributed in the configuration and phase space of our models. For this, we consider models with different bar mass, bar thickness of the intermediate or short bar axis or pattern speed. Varying only one parameter at a time, we find that bars that are more massive, or thinner, or faster, have a larger fraction of chaotic orbits. We also try different packages of giving the initial conditions, in order to check these results.

The Galactic Corotation Gap: from Stellar Chaos to Gas Turbulence.
Marco A. Martos
Instituto de Astronomia, Universidad Nacional Autónoma de México, Morelia, México

Spiral density waves, propagating in a gaseous, magnetized Galactic-like disk, have a secular effect that yields a gap at the location of corotation of the spiral pattern (assumed rigidly rotating) and the gas differentially rotating. Eventually, some the energy deposited at corotation from the large orbital reservoir increases magnetic energy to the point breakdown of the radial equilibrium occurs in a violent manner, suddenly converting an initially orderly disk into a turbulent gas disk. There are indications of a connection of the onset of gas turbulence and the overlap of stellar resonances, such as that established by Contopoulos between such overlap and stellar chaos.

Regular and Chaotic Motion in Elliptical Galaxies
J.C. Muzzio
Facultad de Ciencias Astronomicas y Geofisicas de la Universidad Nacional de La Plata and Instituto de Astrofisica de La Plata (UNLP-CONICET)

The first self-consistent models of triaxial stellar systems were obtained through the collisionless collapse of, initially cold, N-body systems. Later on, methods like the one of Schwarzschild, where a large library of orbits is computed in the potential produced by a certain density distribution and then used to estimate the fraction of each kind of orbit needed to produce the original density distribution, became popular. More recently several authors, with N. Voglis and his co-authors among them, obtained triaxial models with the N-body approach, smoothed and fixed the potential of the N-bodies and were able to investigate the orbital structure of those models with interesting results; for example, perfectly stable systems obtained with these methods were shown to include high fractions of chaotic orbits. Both methods should be regarded as complementary: with the N-body method one can get systems that do not arise from mathematical convenience (e.g., with the same axial ratios from center to border), while the other method allows one to more precisely specify the systems to be studied. Here, we review the results obtained with these methods, by other authors and ourselves, on the regular and chaotic orbital structure of self-consistent triaxial systems that aim to represent elliptical galaxies.

The flow of material through the spiral arms
Panos A. Patsis
Research Center for Astronomy and Applied Mathematics, Academy of Athens

We examine the stellar flow through the arms of spiral galaxies from a qualitative point of view. In particular, we investigate the properties of ordered and chaotic flows. The study attempts to associate morphological and kinematical features with the one or other type of flow and establish a basis of criteria for future observational projects.

We present the features that are related with a flow typical for a density wave of stars rotating around the center of a spiral galaxy. For this purpose we use response models with a normal (non-barred) spiral perturbation. We do the same for the case of chaotic spirals in models with strong forcing at corotation. We underline the differences that are expected in the morphology of the galaxies in normal and barred spiral cases.

Relationship between global and microscopic chaos in galaxies. Pfenniger D., Brunetti, M.
Geneva Observatory, University of Geneva, Geneva, Switzerland

The exponential sensitivity of gravitational N-body systems to small changes in the initial conditions or to perturbations is a well known fact which still poses problems from the practical and theoretical point of views. In particular we are interested to better understand the link between this microscopic chaos existing in almost all N-body systems, and the global chaos that follows in large scale, collective instabilities leading to violent relaxation in case of fast global changes, or mild relaxation resulting from instabilities like bars of spiral arms. Do large scale instabilities change the local particle rates of divergence? Or do microscopic chaos stabilize or destabilize a galaxy?

For studying such questions we integrate with a symplectic integrator full N-body models of galaxies, as well as the "diagonal" variational equations allowing to compute 6 exponential rates of divergence of each particles due to their interaction with all the other particles. This is therefore a reduced variational problem neglecting the effect of higher order particle correlations on the rates of divergence. With periodic Gram-Schmidt normalizations of the variational vectors we ensure the numerical stability of the variational computations. Such N-body models up to about 10 5 particles including reduced variational vectors can nowadays be calculated even on a moderate-sized parallel computer cluster.

This is an ongoing work, therefore partial results will be presented.

Secular instabilities of stellar systems: slow mode approach
Evgeny Polyachenko, Valerij Polyachenko, Ilia Shukhman
Institute of Astronomy, Russian Academy of Sciences, Moscow, Russia
Stellar systems of two different scales are discussed.

The first ones are disks and spheres of ~1pc in size located around massive black holes in centers of galaxies. They exhibit secular evolution on slow timescales of the order of inverse precession frequency. A new instability due to the presence of a loss cone caused by the black hole is found. The mechanism of AGN fuel support based on this instability must be more efficient than one based on the conventional collisional relaxation.

The second systems are galactic stellar disks. It is argued that the usual (fast) bar mode can be treated as a slow mode in a disk of stellar orbits. The enhancement of the mode is produced due to interaction with stars at Corotation and Outer Lindblad Resonances.

Normal form approach to galactic potentials
Giuseppe Pucacco
University of Rome “Tor Vergata”, Rome, Italy

We apply detuned resonant normal forms, computed via Lie transform normalization, to investigate the phase-space structure of potentials suitable for elliptical galaxies. We focus the attention on the stability of the main periodic orbits and on the phase-space fraction occupied by boxlets and compare the results with numerical investigations.

Stochastic and self-organization in solar flares - critical analysis of the current approach.
Lev Pustil'nik
Israel Space Weather and Cosmic Ray Center, Tel Aviv University & Israel Space Agency, Israel, levpust@post.tau.ac.il

We review current approach to solar flare origin with critical analysis of different aspects of this problem:

1. Pre-flare equilibrium state: role of observed numerous ultra fine structure of the force-free magnetic fields of active region with elements of fractal dimensions and dynamical equilibrium of stochastic ensemble of magnetic ropes

2. Loss of pre-flare equilibrium: absence of adequate mechanisms for description of this catastrophic transition from previous force free equilibrium to singular state with concentration of dispersed currents into very thin current sheet & string, where anomalous magnetic dissipation and reconnection.

3. Flare energy release itself: problem of “survival”; of flare’s current sheet caused by its strong (threshold-like) sensitivity to current density, by effects of current’s region overheating and by “splitting” of current sheet.

We present some possible solution of demonstrated problem on the base of “percolation approach” to current propagation in unstable turbulent current sheet. We show that current percolation in stochastic resistor’s network with feed-back of resistors to current level and show that this approach is able to explain naturally main observed properties of the solar flares (“threshold-like” start of flare energy release, power-like frequency-amplitude spectra of flare bursts and power-like energetic spectra of accelerated particles) as manifestation of self-organization in percolated system.

Resonance overlap in circumstellar and galactic disks
Alice C Quillen
University of Rochester, Rochester, U.S.A

I will introduce 3 topics where estimates of resonance overlap have been useful in interpreting extrasolar circumstellar disks and the dynamics of stars in the solar neighborhood. The chaotic zone boundary near corotation of a low mass planet may impact morphology of dust particles observed in circumstellar disks such as Fomalhaut that have eccentric clearings. I will describe generalization of the chaotic zone boundary size estimate to systems with low eccentricity planets. Resonance capture theory has recently been generalized to cover the non-adiabatic limit. However, the phenomenology of resonance capture for resonances with multiple components is rich and yet to be fully explored. In the solar neighborhood chaotic motion could be caused by multiple perturbing patterns (spiral + bar for example) resulting in heating of star motions. A new heating mechanism is proposed. If this mechanism operates in the Galaxy then we would predict variations in the velocity dispersion as a function of radius in the galaxy. These variations could be revealed by radial velocity pencil beam surveys.

The formation of spiral arms and rings in barred galaxies
M. Romero-Gomez 1, E. Athanassoula 1, J.J. Masdemont, C. Garcia-Gomez 2
1.Observatoire Astronomique de Marseille, Marseille, France
2. Instituto de Astronomia, Universidad Nacional Autónoma de México, Morelia, México

We propose a new theory to explain the formation of spiral arms and of all types of outer rings in barred galaxies. We have extended and applied the technique used in celestial mechanics to compute transfer orbits. Thus, our theory is based on the chaotic orbital motion driven by the invariant manifolds associated to the periodic orbits around the hyperbolic equilibrium points. In particular, spiral arms and outer rings are related to the presence of heteroclinic or homoclinic orbits. Thus, R1 rings are associated to the presence of heteroclinic orbits, while R1R2 rings are associated to the presence of homoclinic orbits. Spiral arms and R2 rings, however, appear when there exist neither heteroclinic nor homoclinic orbits. We examine the parameter space of three realistic, yet simple, barred galaxy models and discuss the formation of the different morphologies according to the properties of the galaxy model. The different morphologies arise from differences in the dynamical parameters of the galaxy.

Adiabatic chaos in the Prometheus-Pandora system
Ivan I. Shevchenko
Pulkovo Observatory, St. Petersburg, Russia

The main object of the presented study is the adiabatic chaotic regime in the orbital dynamics of Prometheus and Pandora, the 16th and 17th satellites of Saturn. The chaos in their orbital motion, as found by Renner and Sicardy (2003) and Goldreich and Rappaport (2003), is due to interaction of resonances in the resonance multiplet corresponding to the 121/118 commensurability of the mean motions of the satellites. A method of analytical estimation of the maximum Lyapunov exponent of the chaotic motion near separatrix of nonlinear resonance (Shevchenko, 2000, 2006) is applied. The method is based on the separatrix map theory. In the case of slow perturbation (i.e., the case of adiabatic chaos) the method allows one to calculate the maximum Lyapunov exponent of the original Hamiltonian system almost exactly, if the perturbed pendulum model is valid for given nonlinear resonance. The Lyapunov time (the ”time horizon of predictability”; of the motion) is calculated analytically and compared to estimates obtained in numerical experiments. The observed chaotic regime in the motion of the Prometheus-Pandora system can play an essential role in the long-term orbital evolution of the system. It is shown that parametric proximity of a slowly chaotic satellite system to low-order secondary resonance drastically modifies global properties of the chaotic layer, i.e., its global structure, relative measure of inner regular component, and the value of the maximum Lyapunov exponent.

A Measure of Stickiness and Chaos Strength
Ioannis Sideris
Institute for Theoretical Physics, University of Zurich

A new method of chaos characterization is presented. This method is based on the new approach of recognition of orbital patterns, and focuses on local, epochal characterization of orbits as opposed to global characterization usually employed by most established measures. Thus, it has the advantage to provide information about the chaos strength of sticky epochs of chaotic orbits, but it can also treat time- dependent orbits (which may experience both regular and chaotic epochs).

The net result is that it produces extremely detailed pictures of the phase space of a system, as well as characterizations (regular versus chaotic) early in the evolution of orbits. Moreover, the method applies generally; all it requires is a signal, of which an orbit is merely an example.

Velocity Distribution in Stellar Disks
Christian Theis 1 & Eduard Vorobyov 2
1 Inst. of Astronomy, Vienna
2 Dept. of Physics and Astronomy, Univ. of Western Ontario, Canada

Problems of stellar dynamics often have been tackled either by N-body simulations or by stellar-hydrodynamics, e.g. the Jeans equations. Though N-body simulations are generally more flexible, stellar-hydrodynamics is often the method of choice when deriving analytical results (like stability criteria) or when the evolution of slowly growing small perturbations is considered, e.g. the formation of spiral arms.

Here we present the results of our recently developed numerical code solving the Boltzmann moment equations up to second order. This code considers flat (2D) self-gravitating disks embedded in a dark matter halo. Our treatment allows for a calculation of the velocity distribution in the disk, i.e. we determine locally the anisotropy and the vertex deviation in galactic disks. Different to previous analyses by linear perturbation theory, we can also study the non-linear evolutionary stages. Our results show that there is a global correlation between the non-axisymmetric structures (spiral arms and bar) and the velocity distribution (in both, vertex deviation and anisotropy). However, this correlation does not hold very well locally: there is only a weak correlation with the local mass distribution, potential or potential gradient, but a strong correlation with the local mean velocity field. We discuss possible implications for actual and future data interpretation of the Galactic velocity structure.

Effects of Resonance-Crossing in Planetary Migration
Kleomenis Tsiganis
Aristotle University of Thessaloniki, Thessaloniki, Greece

Planetesimals-driven planetary migration is the last episode of primordial evolution of planetary systems. During this phase, the outer planets of our solar system achieved their current orbital configuration. Recent results (Nice model) support the occurrence of a short phase of orbital instability during migration, associated to the crossing of the 1/2 Jupiter-Saturn mean motion resonance. However, different initial configurations may lead to the crossing of different low-order mean motion resonances, as the latter are closely spaced for initially "compact" 4-planet systems. We will present new results, for different initial planetary configurations. We will show that the essential features of the aforementioned instability mechanism are preserved: the crossing of low-order resonances increases the orbital eccentricities and leads to a short phase of planet-planet scattering, which is slowly suppressed by planet-disc interactions. Although the probability of destroying the system (by "losing" a planet) increases as the initial system becomes more "compact", the final orbital architecture of the "surviving" systems matches closely the one of our solar system.

Chaotic diffusion in Hamiltonian systems and applications to planetary dynamics
H. Varvoglis
Aristotle University of Thessaloniki, Thessaloniki, Greece

In the last decades it became evident that many non-integrable dynamical systems describing systems of astronomical interest show non-trivial chaotic behaviour. Many scientists however, in particular astronomers working in Celestial Mechanics, have grown up in the belief that most dynamical systems show regular behaviour. Unfortunately, methods devised for the study of regular systems cannot be applied to chaotic ones. In this respect we present here a statistical approach for the study of chaotic dynamical systems using the Fokker-Planck equation, a partial differential equation describing diffusion of an ensemble of chaotic trajectories. We give some simple solutions of the above equation. Then we apply the diffusive approach to two typical problems pertaining to the dynamics of our planetary system:

  • We estimate, through the simplest solution of the F-P equation, the age of the Veritas asteroid family, a group of minor planets moving in the border between ordered and chaotic regions of the asteroid belt.
  • We confirm earlier results that a large number of Trojans asteroids follow chaotic orbits and are slowly dispersing in configuration space. We find that almost 1/5 of numbered Trojans are unstable over the age of the Solar System. We also show the existence of a statistical relation, similar to the one found by Lecar et al., between the Lyapunov time and the escape time of Trojans.

Chaos in the Mergers of Galaxies and in Other Interactions of Stellar Systems
Peter O. Vandervoort
Department of Astronomy and Astrophysics, The University of Chicago, Illinois, U.S.A

After a brief, general survey of chaotic behavior in interacting stellar systems, this review concentrates on three particular examples. The first example is chaos in a wide binary star that results from a resonant coupling of its Keplerian motion and its orbital motion in the Galaxy. In this example, we revisit an investigation in which it was concluded that most of the possible orbits of the putative solar companion star Nemesis would be chaotic.

Our second example is chaos in the merger of two galaxies modeled in terms of a nonlinear oscillator with just a few degrees of freedom. The model is governed by an equation of motion for the separation of the centers of mass of the two galaxies, systems of tensor virial equations describing the internal dynamics of the galaxies, energy and angular momentum integrals, and additional model conditions that suffice to close the system of governing equations. The equations of the model are derived from the Fokker-Planck equation in order to include the effects of dynamical friction. The galaxies are represented as density distributions stratified on similar and similarly situated ellipsoids, and the stratification is required to evolve homologously. Chaotic behavior is exhibited in solutions of the governing equations that model head-on collisions of the two galaxies. This is one of many examples in dynamics in which chaotic behavior in a system with many degrees of freedom manifests itself in a model with only a few degrees of freedom. Finally, we describe a conjecture concerning chaotic behavior in a galaxy in a state of stationary oscillation. Stationary oscillations are nonlinear oscillations of galaxies, which are stellar-dynamical counterparts of nonlinear BGK waves in homogeneous, electrostatic plasmas. Such oscillations are sustained by stars trapped in suitable families of resonant orbits. The present conjecture is that the motions of the resonant stars would become chaotic as a consequence of the overlap of resonances or as a consequence of external perturbations of the system. The latter case would be one of transient chaos in the sense of Henry Kandrup. It is envisaged that such oscillations would consequently decay and that such systems would approach stationary equilibria. In particular, this is envisaged as a scenario for the approach to equilibrium in a galaxy formed in a state of stationary oscillation by mergers.

Planets in Multiple Star Systems: a Symplectic Approach
Patricia Verrier & Wyn Evans
Institute of Astronomy, University of Cambridge, UK

In recent years exoplanets have been discovered in both binary and triple star systems, and in some cases the stars can significantly affect the dynamics of the planetary system. A study of the Gamma Cephei system, an eccentric giant planet in a relatively close binary, reveals that the stability of small bodies in the system is extremely complex. Developing a symplectic integration scheme specifically for planetary orbits in hierarchical triple stellar systems permits the dynamics of particles to be investigated in these more complicated environments. A general study of the circumbinary region shows that, although often well modelled by the overlay of the effects of two decoupled binary systems, there is a regime when the stars are relatively close and eccentric where the stability is far more complex, as the combined effect of all three stars acts to destabilise test particles.

Complexity in Solar and Stellar Active Regions
Loukas Vlahos
Department of Physics, University of Thessaloniki

Solar active regions are driven dissipative dynamical systems. The convection zone drives the magnetic fields above the active region which responds with impulsive releases of energy in the form of nano-, micro-, flares and large scale coronal mass ejections. It has been documented that the energy release follow a specific energy distribution law f(E_T)~E_T^-a, where a~1.6-1.8 and E_T is the total energy released. It has been shown that a possible explanation for the statistical properties of the energy release by the active region reach a self organized critical state (SOC). The implications of these findings on the analysis of the existing observations will be discussed.


The dynamics of non-symmetrically collapsing stars.
Gennady S. Bisnovatyi-Kogan & Oleg Yu. Tsupko
Space research Institute, Russian Academy of Science, Moscow, Russia

The collapse and regular and chaotic dynamics of non-symmetrically collapsing stars are investigated. The equations of motion of self-gravitating ellipsoidal objects are derived by variation method. The equations are solved numerically for wide set of initial conditions. Regular and chaotic types of motion are revealed, and the structure of phase space is described.

N-body simulations of thick disk formation
Maura Brunetti and Daniel Pfenniger
Geneva Observatory, University of Geneva, Geneva, Switzerland

The formation mechanisms of thick disks in spiral galaxies are investigated. The effects of bar heating and merging are studied by means of N-body simulations of a disk galaxy.

Stellar trajectories are analyzed in order to estimate the number of stars on chaotic orbits which are heated up in the bar regions and migrate in the outer disk contributing to populate a thick disk.

Stabilization of chaotic behavior and capture in the restricted three body problem
Arsen Dzhanoev, Alexander Loskutov, Miguel A. F. Sanjuan
Universidad Rey Juan Carlos, Madrid, Spain

A new type of orbits in the restricted three-body problem is constructed. It is analytically shown that along with well-known chaotic and regular orbits in three-body problem there also exists a qualitatively different type of orbits which can be named as a stabilized. We illustrate that the stabilized orbit of the small mass (particle) body is a result of influence of “additional bodies” that placed in the triangular Lagrange points. A numerical analysis confirms the result and makes it clear that the stabilized orbits are captured orbits.

Gas orbits in a spiral potential.
Gilberto Gomez & Marco Martos
Instituto de Astronomia, Universidad Nacional Autónoma de México, Morelia, México

We performed MHD simulations of the response of a gaseous galactic disk to a spiral perturbation in the background potential. In this poster, as a complement to M. Martos oral presentation, we present the results of our analysis of the orbits of the gas and their interaction with the resonances expected from stellar orbit theory.

Acceleration of Particles in Solar Reconnecting Current Sheets
C. Gontikakis, C. Efthymiopoulos, A. Anastasiadis
Research Center for Astronomy and Applied Mathematics, Academy of Athens

We investigate the orbits of charged particles (electrons and protons) that interact with a solar current sheet. Taking advantage of the translational symmetries of the magnetic and electric fields of current sheets the orbits computation is simplified to integration of the equations of motion in a Hamiltonian of two degrees of freedom. Orbits as well as kinetic energy distributions of the accelerated particles are presented.

On the Topology of the Regions of 3-D Particle Motions in Annular Configurations of N Bodies with a Central Post-Newtonian Potential Tilemahos Kalvouridis
National Technical University of Athens, Athens, Greece

We study the topology of the regions where three-dimensional motions of a mass-less particle can take place in a field created by n big bodies with equal masses situated at the vertices of a regular polygon centered in the (n+1)-th body with a mass that is generally different from the previous ones. Maxwell in 1859 used this configuration with gravitational potentials in order to describe the rings of Saturn. Mioc and Stavinschi in 1998 and 1999 used post-Newtonian potentials and studied the dynamics of this configuration. Here, we assume that the potential created by each peripheral mass is Newtonian, while that of the central mass is a post-Newtonian one: either a Manev-type potential of the form or a Schwarzschild-type one of the form , where r0 is the distance of the particle from the central primary. With this assumption the symmetry of the resultant field with respect to the central mass is preserved. These potentials model several situations belonging mainly to Astronomy, such as the case of a spheroid central primary, either oblate or prolate (Arribas and Elipe, 2004), the case of a central radiating source (Kalvouridis, 2001), as well as various relativistic fields like those proposed by Fock and Reissner-Nordstr&#246;m. The system is characterized by four parameters, that is the number of the peripheral primaries n, the mass parameter β=m0/m and the two parameters A and B that can be zero or non-zero, negative or positive. The topology of the aforementioned regions of the existing solutions is primarily defined by a Jacobian-type integral of motion. Furthermore, the size and shape of these parts of space, as well as their exact way of evolution are determined by the particular values of the four parameters. When the Jacobian constant reaches the values that characterize some particle’s equilibrium positions, then remarkable changes and bifurcations take place in the zero-velocity surfaces that bound these territories.

The structure of phase space in galactic potentials of three degrees of freedom
M. Katsanikas & P.A. Patsis
Research Center of Astronomy and Applied Mathematics, Academy of Athens

The purpose of this paper is the study of the structure of phase space in galactic potentials of three degrees of freedom. A basic problem we have to overcome, is to visualize the 4D space of section of the 6D phase space in a 3D Hamiltonian systems. A method used is the method of color and rotation (Patsis & Zachilas 1994). We apply this method to some cases of families of simple periodic orbits in a 3D potential, which describes the potential of the Milky Way (Miyamoto & Nagai 1975). We describe the differences in the orbital behavior in the neighborhood of stable, simple unstable, double unstable and complex unstable periodic orbits.

Regular and chaotic orbits in narrow 2D bar models
David E. Kaufmann1 & Panos A. Patsis 2
1.Southwest Research Institute, San Antonio, U.S.A
2.Research Center of Astronomy and Applied Mathematics, Academy of Athens

We study orbits in a sequence of narrow 2D n=2 Ferrers bar models with bar axial ratios a/c ranging from 2.5 to 7.5. We find that the central, or x1, family of periodic orbits is the most important one in models with lower values of a/c. However, in models with a/c >= ~6, we find that a new family of stable orbits having propeller shapes plays the dominant role. In our models this propeller family is in fact a distant relative of the x1 family.

We also find intermediate cases in which both families are important. The dominance of one family over the other may have direct consequences on the morphological properties of the bars that can be constructed from them. Finally, we confirm that the general level of chaos increases with increasing bar axial ratio, ultimately capping the value that the bar axial ratio can reach.

The sunspot as an autonomous dynamical system the growth and decay phases of the sunspot evolution
George Livadiotis, Xenophon Moussas
Physics Department, University of Athens, Athens, Greece

Α physical model for describing the evolution of the sunspot as an autonomous dynamical system both in its growth and decay phases, is introduced. The model consists of a two-dimensional system of ordinary differential equations of first order with respect to time. The two time-depended functions are the sunspot area on the photosphere, A(t), and the mean spatial value of the magnetic field strength inside the sunspot, B(t). The model reproduces both the sunspot growth and decay phases. Each one of the three main decay laws (the linear, the parabolic and the exponential) that are supported by the observational data, is reproduced for different times of the decay phase. The experimental log-normal distribution of the maximum sunspot areas is satisfactorily derived, and an upper limit for the sunspot area is also predicted. Finally, we consider a non-integrable extension of the model, exhibiting chaotic behavior.

The production of Tsallis entropy in the limit of weak chaos and a new indicator of chaoticity
G. Lukes-Gerakopoulos, N. Voglis, C. Efthymiopoulos
Research Center of Astronomy and Applied Mathematics, Academy of Athens

We study the connection between the appearance of a `metastable' behavior of weakly chaotic orbits, characterized by a constant rate of increase of the Tsallis q-entropy (Tsallis 1988), and the solutions of the variational equations of motion for the same orbits. We demonstrate that the variational equations yield transient solutions, lasting for long time intervals, during which the length of deviation vectors of nearby orbits grows in time almost as a power-law.

The associated power exponent can be simply related to the entropic exponent for which the q-entropy exhibits a constant rate of increase. This analysis leads to the definition of a new sensitive indicator distinguishing regular from weakly chaotic orbits, that we call `Average Power Law Exponent' (APLE).

We compare the APLE with other established indicators of the literature.

In particular, we give examples of application of the APLE in a) a thin separatrix layer of the standard map, b) the stickiness region around an island of stability in the same map, and c) the web of resonances of a 4D symplectic map. In all these cases we identify weakly chaotic orbits exhibiting the `metastable' behavior associated with the Tsallis q-entropy.

The effect of supersonic turbulence on star formation
Spyridon Kitsionas, Ralf Klessen
Astrophysikalisches Institut Potsdam, Potsdman, Germany

We will review recent theories/models on the effect of supersonic turbulence on star formation in clusters. We will also present the results of a recent comparison of the treatment of astrophysical supersonic turbulence by different numerical codes. Finally, we will present the results of recent hydrodynamic calculations of star and planet formation.

Application of the Generalized Alignment Index Method to the Dynamics of Multi-dimensional Symplectic Maps
Thanos Manos 1, Charis Skokos 2 & Tassos Bountis 3
1 University of Patras, Center for Research and Applications of Nonlinear Systems (C.R.A.N.S) & Université de Provence, Observatoire Astronomique de Marseille - Provence (OAMP)
2 Astronomie et Systemes Dynamiques, Institut de Mécanique Céleste et de Calcul des Ephémérides (IMCCE), Observatoire de Paris
3 University of Patras, Center for Research and Applications of Nonlinear Systems (C.R.A.N.S)

We study the phase space dynamics of multi-dimensional symplectic maps, using the method of the Generalized ALignment Index (GALI). In particular, we investigate the behavior of the GALI for a system of N coupled standard map and we show that it provides an efficient criterion for rapidly distinguishing chaos from order. We also present examples of its ability to identify the dimensions of tori and predict slow diffusion.

Orbital structure of Self-consistent Triaxial Stellar Systems
Juan Carlos Muzzio, Hugo D. Navone 2 And Alejandra Zorzi 2
1 Facultad de Ciencias Astronomicas y Geofisicas de la Universidad Nacional de La Plata and Instituto de Astrofisica de La Plata (UNLP-CONICET)
2 Instituto de Fisica de Rosario (CONICET-UNR), Observatorio Astronomico Municipal de Rosario and Facultad de Ciencias Exactas, Ingenieria y Agrimensura de la Universidad Nacional de Rosario.

We created five self--consistent triaxial stellar systems through the cold disipationless collapse of 1,000,000 particles whose evolution was followed with a multipolar code. Three of them have semiaxis ratios corresponding to those of E4 through E6 elliptical galaxies, while the other two are equivalent to the E4 and E6 models but with greater central concentration, i.e., they represent cuspy models. The E6 models have significant rotation velocities although their total angular momentum is zero, that is, they exhibit figure rotation; the rotation of the E5 model is barely significant and that of the E4 models is essentially zero. Except for the unavoidable relaxation effects, the systems remain stable over periods of 1,000 crossing times. The potential of each system was subsequently approximated with interpolating formulae yielding smooth potentials, stationary for the non-rotating models and stationary in the rotating frame for the rotating ones. The Lyapunov exponents could then be computed for randomly selected samples of the bodies that make up the different systems, allowing the recognition of regular and partially and fully chaotic orbits. Finally, the regular orbits were Fourier analyzed and classified using their locations on the frequency map. As it could be expected, the percentages of chaotic orbits increase with the flattening and the concentration of the systems. Triaxiality diminishes from E6 through E4, the latter systems being almost axially symmetric; as a result, as one goes from E6 through E4, the number of partially chaotic orbits relative to that of fully chaotic ones increases, probably due to the presence of a pseudo integral of motion similar to the axial component of the angular momentum and, besides, there is a significant increase of the tubes around the axis of "almost" symmetry among the regular orbits.

Variational equations for the self-consistent field method of Hernquist and Ostriker
Juan Carlos Muzzio 1, Hugo D. Navone 2 & Alejandra Zorzi 2
1 Facultad de Ciencias Astronomicas y Geofisicas de la Universidad Nacional de La Plata and Instituto de Astrofisica de La Plata (UNLP-CONICET)
2 Instituto de Fisica de Rosario (CONICET-UNR), Observatorio Astronomico Municipal de Rosario and Facultad de Ciencias Exactas, Ingenieria y Agrimensura de la Universidad Nacional de Rosario.

L. Hernquist and J.P. Ostriker (Ap.J. 386, 375, 1992) devised a self-consistent field method which is very convenient for the integration of N-body problems. We have found the variational equations that correspond to that method and we have written a computational procedure to obtain them. We used the Hernquist and Ostriker code to create self-consistent models of triaxial stellar systems and the LIAMAG code of S. Udry and D.

Pfenniger (Astron. Astroph. 198, 135, 1988) together with our routine to compute the Lyapunov exponents of random samples of orbits in those systems. We compare our results with others that we obtained before for similar systems using a different method.

Invariant manifolds, chaotic orbits, and the spiral structure of barred galaxies
P. Tsoutsis & C. Efthymiopoulos
Research Center for Astronomy and Applied Mathematics, Academy of Athens

We demonstrate how the unstable asymptotic manifolds of the short period family of unstable periodic orbits around L 1 or L 2 create correlations among the phases (angles of the apocenters) of a number of chaotic orbits which are strong enough so as to support a spiral structure in an N-Body model of a galaxy with a strong bar. We furthermore present evidence that the unstable manifolds of all the families of unstable periodic orbits near and outside corotation contribute to the same phenomenon. The invariant manifolds of all the families produce a phenomenon of `resonant stickiness' that slows down the rate of chaotic escape of the stellar orbits. We estimate the stickiness time as several times 10t dynamical which is enough to explain the observed spiral structure.

Unstable periodic orbits and chaos in planetary dynamics
George Voyatzis
Aristotle University of Thessaloniki, Thessaloniki, Greece

The dynamics of planetary systems composed by two planets is studied in the framework of the three body problem. Periodic orbits, which are obtained in a rotating frame of reference, are essential for the structure of the phase space and the distribution of regular and chaotic orbits. Linear stability of periodic orbits is examined through the computation of stability indices while nonlinear stability is examined by using the method of Fast Lyapunov Indicator. It has been shown that linearly stable periodic orbits are surrounded by quasiperiodic orbits in phase space and correspond to apsidal corotation. In the present work we show that also apsidal corotation can be found near unstable periodic orbits which correspond to small planetary eccentricities. In these cases chaotic motion is practically not detectable and diffusion is not apparent even for long-term evolution. As planetary eccentricities increase, linearly unstable periodic orbits are associated with chaotic motion which dominates in phase space.