Springer chaotic dynamics and transport in classical and quantum systems (2005) YYePG lotb

Chaotic Dynamics and Transport in Classical and Quantum Systems NATO Science Series A Series presenting the results of scientific meetings supported under the NATO Science Programme The Series is published by IOS Press, Amsterdam, and Kluwer Academic Publishers in conjunction with the NATO Scientific Affairs Division Sub-Series I II III IV V Life and Behavioural Sciences Mathematics, Physics and Chemistry Computer and Systems Science Earth and Environmental Sciences Science and Technology Policy IOS Press Kluwer Academic Publishers IOS Press Kluwer Academic Publishers IOS Press The NATO Science Series continues the series of books published formerly as the NATO ASI Series The NATO Science Programme offers support for collaboration in civil science between scientists of countries of the Euro-Atlantic Partnership Council The types of scientific meeting generally supported are “Advanced Study Institutes” and “Advanced Research Workshops”, although other types of meeting are supported from time to time The NATO Science Series collects together the results of these meetings The meetings are co-organized bij scientists from NATO countries and scientists from NATO’s Partner countries – countries of the CIS and Central and Eastern Europe Advanced Study Institutes are high-level tutorial courses offering in-depth study of latest advances in a field Advanced Research Workshops are expert meetings aimed at critical assessment of a field, and identification of directions for future action As a consequence of the restructuring of the NATO Science Programme in 1999, the NATO Science Series has been re-organised and there are currently Five Sub-series as noted above Please consult the following web sites for information on previous volumes published in the Series, as well as details of earlier Sub-series http://www.nato.int/science http://www.wkap.nl http://www.iospress.nl http://www.wtv-books.de/nato-pco.htm Series II: Mathematics, Physics and Chemistry – Vol 182 Chaotic Dynamics and Transport in Classical and Quantum Systems edited by P Collet Ecole Polytechnique, Paris, France M Courbage Université Paris 7-Denis Diderot, France S Métens Université Paris 7-Denis Diderot, France A Neishtadt Space Research Institute, Moscow, Russia and G Zaslavsky New-York University, New York, NY, U.S.A KLUWER ACADEMIC PUBLISHERS NEW YORK, BOSTON, DORDRECHT, LONDON, MOSCOW eBook ISBN: Print ISBN: 1-4020-2947-0 1-4020-2946-2 ©2005 Springer Science + Business Media, Inc Print ©2005 Kluwer Academic Publishers Dordrecht All rights reserved No part of this eBook may be reproduced or transmitted in any form or by any means, electronic, mechanical, recording, or otherwise, without written consent from the Publisher Created in the United States of America Visit Springer's eBookstore at: and the Springer Global Website Online at: http://ebooks.springerlink.com http://www.springeronline.com Contents Preface vii Participants ix Part I : Theory P Collet : A SHORT ERGODIC THEORY REFRESHER M Courbage: NOTES ON SPECTRAL THEORY, MIXING AND TRANSPORT 15 Valentin Affraimovich, Lev Glebsky: COMPLEXITY, FRACTAL DIMENSIONS AND TOPOLOGICAL ENTROPY IN DYNAMICAL SYSTEMS 35 G.M Zaslavsky, V Afraimovich: WORKING WITH COMPLEXITY FUNCTIONS 73 Giovanni Gallavotti: SRB DISTRIBUTION FOR ANOSOV MAPS 87 P Gaspard : DYNAMICAL SYSTEMS THEORY OF IRREVERSIBILITY 107 Walter T Strunz: ASPECTS OF OPEN QUANTUM SYSTEM DYNAMICS 159 Eli Shlizerman, Vered Rom Kedar: ENERGY SURFACES AND HIERARCHIES OF BIFURCATIONS 189 Monique Combescure: PHASE-SPACE SEMICLASSICAL ANALYSIS AROUND SEMICLASSICAL TRACE FORMULAE 225 Part II : Applications Ariel Kaplan, Mikkel Andersen, Nir Friedman and Nir Davidson: ATOM-OPTICS BILLIARDS 239 Fereydoon Family, C Miguel Arizmendi, Hilda A Larrondo: CONTROL OF CHAOS AND SEPARATION OF PARTICLES IN INERTIA RATCHETS 269 v vi F Bardou: FRACTAL TIME RANDOM WALK AND SUBRECOIL LASER COOLING CONSIDERED AS RENEWAL PROCESSES WITH INFINITE MEAN WAITING TIMES 281 Xavier Leoncini, Olivier Agullo, Sadruddin Benkadda, George M Zaslavsky: ANOMALOUS TRANSPORT IN TWO-DIMENSIONAL PLASMA TURBULENCE 303 Edward Ott,Paul So,Ernest Barreto,Thomas Antonsen: THE ONSET OF SYNCHRONISM IN GLOBALLY COUPLED ENSEMBLES OF CHAOTIC AND PERIODIC DYNAMICAL UNITS 321 A Iomin, G.M Zaslavsky: QUANTUM BREAKING TIME FOR CHAOTIC SYSTEMS WITH PHASE SPACE STRUCTURES 333 S.V.Prants: HAMILTONIAN CHAOS AND FRACTALS IN CAVITY QUANTUM ELECTRODYNAMICS 349 M Cencini,D Vergni, A Vulpiani: INERT AND REACTING TRANSPORT 365 Michael A Zaks: ANOMALOUS TRANSPORT IN STEADY PLANE FLOWS OF VISCOUS FLUIDS 401 J Le Sommer, V Zeitlin: TRACER TRANSPORT DURING THE GEOSTROPHIC ADJUSTMENT IN THE EQUATORIAL OCEAN 413 Antonio Ponno: THE FERMI-PASTA-ULAM PROBLEM IN THE THERMODYNAMIC LIMIT 431 Lectures 441 Preface From the 18th to the 30th August 2003 , a NATO Advanced Study Institute (ASI) was held in Cargèse, Corsica, France Cargèse is a nice small village situated by the mediterranean sea and the Institut d'Etudes Scientifiques de Cargese provides ∗ a traditional place to organize Theoretical Physics Summer Schools and Workshops in a closed and well equiped place.The ASI was an International Summer School* on "Chaotic Dynamics and Transport in Classical and Quantum Systems" The main goal of the school was to develop the mutual interaction between Physics and Mathematics concerning statistical properties of classical and quantum dynamical systems Various experimental and numerical observations have shown new phenomena of chaotic and anomalous transport, fractal structures, chaos in physics accelerators and in cooled atoms inside atom-optics billiards, space-time chaos, fluctuations far from equilibrium, quantum decoherence etc New theoretical methods have been developed in order to modelize and to understand these phenomena (volume preserving and ergodic dynamical systems, non-equilibrium statistical dynamics, fractional kinetics, coupled maps, space-time entropy, quantum dissipative processes etc) The school gathered a team of specialists from several horizons lecturing and discussing on the achievements, perspectives and open problems (both fundamental and applied) The school, aimed at the postdoctoral level scientists, non excluding PhD students and senior scientists, provided lectures devoted to the following topics : Statistical properties of Dynamics and Ergodic Theory Chaos in Smooth and Hamiltonian Dynamical Systems Anomalous transport, fluctuations and strange kinetics Quantum Chaos and Quantum decoherence Lagrangian turbulence and fluid flows Particle accelerators and solar systems More than 70 lecturers and students from 17 countries have participated to the ASl The school has provided optimal conditions to stimulate contacts between young and senior scientists All of the young scientists have also received the opportunity to present their works and to discuss them with the lecturers during two posters sessions that were organized during the School The proceedings are divided into two parts as follows: I Theory This part contains the lectures given on basic concepts and tools of modern dynamical systems theory and their physical implications Concepts of ergodicity and mixing, complexity and entropy functions, SRB measures, fractal dimensions and bifurcations in hamiltonian systems have been thoroughly developed Then, ∗ http://www.ccr.jussieu.fr/lptmc/Cargese/CargeseMainPage.htm vii viii models of dynamical evolutions of transport processes in classical and quantum systems have been largly explained II Applications In this part, many specific applications in physical systems have been presented It concerns transport in fluids, plasmas and reacting media On the other hand, new experiments of cold optically trapped atoms and electrodynamics cavity have been thoroughly presented Finally, several papers bears on synchronism and control of chaos We also provide most recent references of the other given lectures at the school These lecture notes represent, in our views, the vitality and the diversity of the research on Chaos and Physics, both on fundamental f and applied levels, and we hope that this summer-school will be followed by similar meetings The summer-school was mainly supported by NATO and the staff of the University Paris M Courbage was a coordinator off the Summer-School, G Zaslavsky was a director of the NATO-ASI and A Neishtadt was a co-director We would like to thank all the institutions who provided support and encouragements, namely, NATO ASI programm, the European Science Foundation through Prodyn programm, the Collectivités Territoriale Corse, the Laboratoire de Physique Théorique de La Matière Condensée (LPTMC) and the Présidence of the University Paris Thanks also to the Centre de Physique Théorique de l'Ecole Polytechnique de Paris The meeting was an occasion for a warm interactive atmosphere beside the scientific exchanges We want to thank those who contributed to its success: the director and the staff of the Institut d'Etudes Scientifiques de Cargèse and, the team of the Université Paris 7, especially Evelyne Authier, Secretary of the LPTMC, who provided essential help to organize this ASI P Collet, M Courbage, S Metens, A Neishtadt, G.W Zaslavsky 436 estimate, for a given k, is the ratio (in a sense to be specified below) of Ωk zk to ∂H /∂zk∗ In order to this, we suppose the system to be in a “typical” configuration, characterized by partial equipartition between the first kc modes, all of them having approximately the same value of modal energy Ec , while the other ones (for k > kc ) are essentially at rest Further, we suppose kc /N small, so that ωk πk/(N + 1) for all k ≤ kc Such a configuration is given by ζk = Ec iφk e , k = 1, , kc ωk (15) ζk = , k = kc + 1, , N , where the phases φ1 , , φkc are considered as independent random variables, each of them being uniformly distributed over [0, 2π] We come back to the hypothesis (15) in Remark in the last section For the moment one can accept it as the simplest possible one Now, by inserting (15) into (13) we get a first relation linking Ec and kc , namely kc Ec = J ≈ N ε (16) As an estimate for the ratio of the dispersive to the nonlinear term appearing in the right hand side of equation (14) we use the ratio |Ωk ζk |2 R(k) = ∂H ∂ζk∗ φ , (17) φ where both the numerator and the denominator are calculated on the configuration (15), φ denoting average over the random phase distribution Notice that the simple phase average of both Ωk ζk and ∂H /∂ζk∗ is zero Of course, the function R is defined for values of k smaller than kc An easy computation yields k π2 N +1 , R(k) = √ α E h kc − k N +1 N +1 (18) showing that R(k) is in fact a monotonically increasing function of k/(N + 1) Thus, the higher k/(N + 1) is, the greater is the effect of dispersion with respect to that of nonlinearity for the corresponding 437 The Fermi-Pasta-Ulam problem in the thermodynamic limit mode k Our criterion to determine kc consists in requiring R(kc ) = 1, which, by exploiting (13), for large N , yields √ kc √ 1/4 = αε (19) N +1 π The meaning of the relation R(kc ) = is almost obvious: it means that kc is the first mode for which dispersion compensates nonlinearity, so that kc represents the highest mode of the inertial (nonlinear) scale Remembering that the wavelength of the k-th mode is λk = 2(N + 1)/k, one then has that energy is injected on spatial scales larger than c = λkc ∼ α−1/2 ε−1/4 By inserting (19) into (13) one gets (for large N ) π ε3/4 Ec = √ √ , α (20) which gives the scaling of the partial equipartition level Finally, the characteristic time scale over which such a state of partial relaxation sets in can be estimated through the formula |ζk |2 |ζ˙k |2 T (k) = φ , (21) φ where ζ˙k stands for the right hand side of equation (14) The dimensional meaning of the above formula is obvious: T (k) represents the time scale characterizing the dynamics of the k-th mode Here too, both the numerator and the denominator have to be calculated on the configuration (15) Noting that |ζ˙k |2 φ = |Ωk ζk |2 φ + |∂H /∂ζk∗ |2 φ , an easy computation yields for (21) 24 T (k) = πk N +1 + 36α2 E c πk N +1 c Nk+1 − Nk+1 1/2 , (22) showing that T (k) too depends in fact on k/(N + 1) Anyway, at variance with R(k), the quantity T (k) is not a monotonic function of its argument, at least if the specific energy is low enough The time scale ˆ the mode of interest here corresponds to τc ≡ lim1≤k≤kc T (k) = T (k); kˆ thus determined is the quickest one to reach the ”plateau” of partial equipartition at Ec It can be easily shown that τc has the same order of magnitude of T (kc ), which can be easily computed, yielding τc ≈ T (kc ) = 33/2 α3/2 ε3/4 (23) 438 It can also be immediately checked that for those low modes such that k/(N + 1) → as N → ∞, one has T (k) ∼ (N + 1)/(kαε1/2 ), a time which is much longer than τc and diverges with N This represents the characteristic time scale of those first few modes that approach the partially relaxed state on times diverging with the size of the system What we can conclude is the following: if the specific energy of the 1, the system is expected to relax to system is low enough, say α2 ε a state of partial equipartition involving only a small fraction of long wavelength modes Such a state is reached in times larger than a first characteristic time scale which depends only on intensive quantities and, in practical (numerical) computations, might be quite large One can expect that this first stage of the cascade be followed by a second stage, which takes place on much lomger times t τc , possibly leading to full equipartition The existence of extremely long time scales of such a type (stretched exponentials of 1/ε) is supported by numerical evidence is available (see e.g [15, 16]) An analytical description of such phase of the relaxation process is beyond the scope of the present work Concluding remarks Remark - In the quoted paper of Shepelyansky [10] use is made of the resonance overlap criterion along the lines of Chirikov [11, 12] In applying such a principle, one compares the so called nonlinear frequency shift of mode k with the difference of frequencies of modes k and k + In our opinion, such a criterion cannot be applied to the FPU models for two reasons The first one is that no resonance of the form Ωk+1 − Ωk appears in FPU system (12) For such a system dangerous (quasi-)resonances (appearing as small denominators in the first step of canonical perturbation theory) have the form Ωp+q − Ωp − Ωq The second problem is due to the use of action-angle variables (I, ϕ) in the theory, which involves quantities such as the frequency of a given mode (ϕ˙ k ) and its nonlinear correction When one considers a nearly unexcited high mode k, the value of its action Ik almost vanishes and the fact the action-angle coordinates become singular This reflects in √ that, the cubic part of the Hamiltonian being dependent on Ik , its derivative with respect to Ik (which yields the nonlinear frequency correction) becomes artificially large That is why, we think, the results of Shepelyansky seem not to persist in the thermodynamic limit But in our opinion this is an artifact of the method, and not a property of the system Remark - Our result is essentially based on the hypothesis that the state of the system at a given time t > has the form (15) The assump- The Fermi-Pasta-Ulam problem in the thermodynamic limit 439 tion of complete equipartition between the first kc modes has been made here just for the sake of simplicity One could relax it and obtain the ratio R(k) (17) as a functional of the unknown energies E1 , , Ekc The minimization of R constarined to the conservation of the second integral J k Ek would then yield a more realistic distribution of the modal energies (Ek vs k) This is deserved to further research At present, an estimate of the spectral distribution of modal energies, based on soliton theory, is given in [9] The connection between the present approach and that given in [9], namely the rigorous justification of the Korteweg-de Vries equation as a resonant Hamiltonian normal form of system (1) is the object of a joint work in progress with Dario Bambusi Remark - It is sometimes objected that the cubic potential in the Hamiltonian (1) is not bounded from below and the chain could, in principle, break-down (blow-up of some coordinates and momenta in a finite time) Anyway, break-down is caused by possible highly localized excitations on the chain, which requires very short wavelengths having a consistent amount of energy, i.e the energy cascade to be really effective The results of the present paper confirm that, under suitable conditions, small scale motions are frozen over rather long times, which implies absence of break-down In references [2, 17] numerical runs over large times display no pathology, even when the system approaches equipartition Acknowledgments The present contribution is the result of many discussions made on the problem within the Mathematical Physics Group, University of Milano, to which I presently belong Special thanks go to Dario Bambusi, for useful discussions on the subject, and to Luigi Galgani, who introduced me to the ”core” of the FPU problem as meant here References [1] Fermi E., Pasta J., Ulam S (1954), reprinted in ”Non Linear Wave Motion”, ed by A Newell, Lectures in Applied Mathematics 15, 1974, pp 143-156 [2] Benettin G (1986), in ”Molecular dynamics simulation in classical statistical mechanical systems”, Proceedings of the E Fermi school of Varenna, 1986, ed by G Ciccotti and W G Hoover [3] Ford J (1992), Phys Rep 213, 271-310 [4] Carati A., Galgani L., Ponno A., Giorgilli A (2002), Il Nuovo Cimento B 117, 1017-1026 [5] Bocchieri P., Scotti A., Bearzi B., Loinger A (1970), Phys Rev A 2, 2013-2019 [6] Biello J A., Kramer P R., Lvov Y (2002), in ”Proceedings of the 4th International Conference on Dynamical Systems and Differential Equations”, Wilmington, NC, USA, 2002 440 [7] Berchialla L., Galgani L., Giorgilli A (2003), DCDS-B, in print [8] Zabusky N J., Kruskal M D (1965), Phys Rev Lett 15, 240-243 [9] Ponno A (2003), Europhys Lett 64, in print [10] Shepelyansky D L (1997), Nonlinearity 10, 1331-1338 [11] Izrailev F M., Chirikov B V (1966), Sov Phys Dokl 11, 30-32 [12] Chirikov B V (1969), reprinted and translated in CERN Trans 71-40, Geneva, 1971, pp 1-237 [13] Frisch U (1995), Turbulence, Cambridge University Press, 1995 [14] Ford J (1961), J Math Phys 2, 387-393 [15] Pettini M., Landolfi M (1990), Phys Rev A 41, 768-783 [16] Berchialla L., Giorgilli A., Paleari S (2003), preprint [17] Livi R., Pettini M., Ruffo S., Vulpiani A (1985), Phys Rev A 31, 2740-2742 Some references for other lectures and posters presented during the school LECTURES Fluctuation-Dissipation Dispersion Theorem for Slow Processes V.V Belyi IZMIRAN, Russian Academy of Sciences V.V Belyi, Fluctuation-Dissipation Relation for a Nonlocal Plasma, Phys Rev Let., 2002, 88, N 25, pp 255001-4 V Belyi, Fluctuation-Dissipation Dispersion Relation and Quality Factor for Slow Processes, Phys Rev E 69, N1, p 017104, 2004 441 442 Control of chaotic diffusion in Hamiltonian dynamiques Guido Ciraolo Centre de Physique Théorique CNRS, Case 907 13288 Marseille Cedex 9, France ciraolo@cpt.univ-mrs.fr 1) M Vittot, Perturbation theory and control in classical or quantum mechanics by an inversion formula, J Phys A: Math Gen 37, 6337 (2004) 2) G.Ciraolo, F Briolle, C.Chandre, E.Floriani, R Lima, M Vittot, M Pettini, C Figarella, Ph Ghendrih, Control of Hamiltonian Chaos as a possible tool to control anomalous transport in fusion plasmas, Phys Rev E 69, 056213 (2004) 3) G Ciraolo, C Chandre, R Lima, M Vittot, M Pettini, C Figarella, Ph Ghendrih, Controlling chaotic transport in a Hamiltonian model of interest to magnetized plasmas, J Phys A: Math Gen 37, 3589 (2004) 4) G Ciraolo, C Chandre, R Lima, M Vittot, M Pettini, Control of Chaos in Hamiltonian Systems, Cel Mech & Dyn Astr to appear, (2004), archived in arXiv.org/nlin.CD/0311009 443 The directional entropy of Z^d-actions on a Lebesgue space Brunon Kami\'nski Faculty of Mathematics and Computer Science Nicholas Copernicus University, 87-100 Toru\'n, Poland M COURBAGE and B KAMINSKI : On the directional entropy of Z^2-actions generated by permutative cellular automata , Studia Mathematica, vol 153 (3), 2002 Kami\'nski, B.; Park, K K On the directional entropy for Z^2-actions on a Lebesgue space Studia Math 133 (1999), no 1, 39 51 J.MILNOR (1986) Directional entropies of cellular automaton - maps in: Disordered Systems and Biological Organization, NATO Adv Sci Inst Ser F20, Springer : 113-115 Park, Kyewon Koh : On directional entropy functions Israel J Math 113 (1999), 243—267 444 Destruction of adiabatic invariance and transport phenomena in systems with slow and fast motions Anatoly Neishtadt Space Research Institute Profsoyuznaya 84/32 Moscow 117997, Russia Some references 1.Arnold, V I Mathematical methods of classical mechanics Springer-Verlag, New York, 1989 2.Arnold, V I.; Kozlov, V V.; Neishtadt, A I Mathematical aspects of classical and celestial mechanics Springer-Verlag, Berlin, 1997 3.Itin, A P Resonant phenomena in classical dynamics of three-body Coulomb systems Phys Rev E (3) 67 (2003), no 2, 026601, pp 70F10 Itin, A P.; Neishtadt, A I Resonant phenomena in slowly perturbed elliptic billiards Regul Chaotic Dyn (2003), no 1, 59–66 5.Itin, A P.; Neishtadt, A I.; Vasiliev, A A Captures into resonance and scattering on resonance in dynamics of a charged relativistic particle in magnetic field and electrostatic wave Phys D 141 (2000), no 3-4, 281–296 Itin, A.P.; Neishtadt, A.I.; Vasiliev, A.A Resonant phenomena in slowly perturbed rectangular billiards Phys Lett., A 291 (2001), No.2-3, 133-138 Neishtadt, A.I Passage through a resonance in the two-frequency problem Sov Phys., Dokl 20 (1975), 189-191 Neishtadt, A I On destruction of adiabatic invariants in multi-frequency systems International Conference on Differential Equations, Vol 1, (Barcelona, 1991), 195–207, World Sci Publishing, River Edge, NJ, 1993 Neishtadt, A.I Scattering by resonances Celestial Mech Dynam Astronom 65 (1996/97), no 1-2, 1–20 10.Neishtadt, A.I On adiabatic invariance in two-frequency systems Hamiltonian systems with three or more degrees of freedom (S'Agaro, 1995), 193–212, NATO Adv Sci Inst Ser C Math Phys Sci., 533, Kluwer Acad Publ., Dordrecht, 1999 445 I Levy flights and Walks in Physics II Pitfalls and paradoxes in the history of probability theory Dr Michael F Shlesinger Chief Scientist for Nonlinear Science Physical Sciences Division Office of Naval Research 800 N Quincy St Arlington VA 22217 USA Sources of Exponents (John T Bendler, John J Fontanella, and Michael F Shlesinger) Physica D 193 67-72 (2004) A New Vogel-like Law: Ionic Conductivity, Dielectric Relaxation and Viscosity Near the Glass Transition (John T Bendler, John J Fontanella, and Michael F Shlesinger) Physical Review Letters 87 195503 - (1-4) (2001) ABOVE, BELOW AND BEYOND BROWNIAN MOTION (J Klafter, M F Shlesinger, and G Zumofen) Am J Phys 67, 1253-1259, (1999) BEYOND BROWNIAN MOTION" (J Klafter, M F Shlesinger, and G Zumofen) Physics Today Feb., pp.33-39 (1996) TIME SCALE INVARIANCE IN TRANSPORT AND RELAXATION (H Scher, M F Shlesinger, and J Bendler) Physics Today January, pp 26-34 (1991) ON THE WONDERFUL WORLD OF RANDOM WAL K (E W Montroll and M F Shlesinger) in Studies in Statistical Mechanics, Eds J L Lebowitz and E W Montroll, 11 (North-Holland Pub.), pp1-121 (1984) 446 The effects of Lagrangian chaos on multiphase processes and network dynamics Tom Solomon Department of Physics, Bucknell University, Lewisburg, PA 17837 USA Some references: "Uniform resonant chaotic mixing in fluid flows," T.H Solomon and Igor Mezic, Nature 425, 376-380 (2003) "Lagrangian chaos and multiphase processes in vortex flows," Thomas H Solomon, Brian R Wallace, Nathan S Miller and Courtney J L Spohn, Comm Nonlin Sci and Numer Sim 8, 239-252 (2003) "Lagrangian chaos: Transport, coupling and phase separation," Thomas H Solomon, Nathan S Miller, Courtney J L Spohn, and Jeffrey P Moeur, Proceedings of 7th Experimental Chaos Conference, ed V In, L Kocarev, T.L Carroll, B.J Gluckman, S Boccaletti and J Kurths, 195-206 (AIP, New York, 2003) "Resonant flights and transient superdiffusion in a time-periodic, two-dimensional flow," T.H Solomon, A.T Lee, and M.A Fogleman, Physica D 157, 40-53 (2001) "Lagrangian chaos and correlated Levy flights in a non-Beltrami flow: transient versus long-term transport," M.A Fogleman, M.J Fawcett, and T.H Solomon, Phys Rev E 63 020101(R), (February, 2001) "The Role of Lobes in Chaotic Mixing of Miscible and Immiscible Impurities," T.H Solomon, S Tomas, and J.L Warner, Phys Rev Lett 77, 2682-2685 (1996) 447 POSTERS "Model of Ejection of Matter from Dense Stellar Cluster and Chaotic Motion of Gravitating Shells" authors: M.V Barkov barmv@sai.msu.ru G.S Bisnovatyi-Kogan gkogan@mx.iki.rssi.ru A.I Neishtadt aneishta@mx.iki.rssi.ru affiliation: Space Research Institute, 84/32 Profsoyuznaya Str , Moscow, Russia, 117997 and V.A Belinski National Institute of Nuclear Physics (INFN) and International Center of Relativistic Astrophysics (ICRA), Dip di Fisica - Universita` degli Studi di Roma "La Sapienza" P.le Aldo Moro, - 00185 Roma, Italy belinski@icra.it M V Barkov, V A Belinski and G S Bisnovatyi-Kogan, Journal of Experimental and Theoretical Physics 95, 3, 371 (2002) M V Barkov, V A Belinski and G S Bisnovatyi-Kogan, Mon Not R Astron Soc 334, 338 (2002) M.V Barkov, G.S Bisnovatyi-Kogan, V.A Belinski and A.I Neistadt Model of Ejection of Matter from Dense Stellar Cluster and Chaotic Motion of Gravitating Shells // Series: Lecture Notes in Physics 2003 V 626 Galaxies and Chaos / Eds Contopoulos G., Voglis N 448 Periodic orbit theory of strongly anomalous transport Giampaolo Cristadoro Center for Nonlinear and Complex Systems, Dipartimento di Scienze Chimiche, Fisiche e Matematiche, Universita' dell'Insubria and Istituto Nazionale di Fisica della Materia, Unita' di Como, Via Valleggio 11, 22100 Como, Italy REFERENCES: 1) R.artuso, G Cristadoro, Phys Rev Lett 90, 244101 (2003) 2) P.Castiglione, A.Mazzino, P.Muratore-Ginanneschi and A.Vulpiani, Physica D, 134, 75 (1999) 3) P.Cvitanovi\'c , R.Artuso, P.Dahlqvist , R Mainieri ,G.Tanner , G.Vattay , N.Whelan and A Wirzba ; Chaos: classical and quantum (www.nbi.dk/ChaosBook/) (2003) 449 "Weak chaos and Information » Stefano Galatolo Dipartimento di Matematica Applicata Universita di Pisa Via bonanno 24 Pisa / Italy \item V.Benci, C.Bonanno, S.Galatolo, G.Menconi, M.Virgilio {\em Dynamical systems and computable information} to appear on Disc Cont Dyn Sys-B (autumn 2004) \item GalatoloS \emph{Complexity, initial condition sensitivity, di\-men\-sion and weak chaos in dynamical systems}, { Nonlinearity num 4, vol 16, pp 1219-1238 (2003) } \item Galatolo S {\em Global and local Complexity in weakly chaotic dynamical systems} Disc Cont Dyn Sys num 6, vol 9, pp 1607-1624 (2003) \item Bonanno, C.; Galatolo S; Isola S {\em Recurrence and algorithmic information.} Nonlinearity 17 No ( 2004) 1057-1074 450 Existence of Invariant Measures for Transcendental Subexpanding Functions Janina Kotus Department of Mathematics Warsaw University of Technology 00-661 Warsaw, Poland and Mariusz Urbanski, University of North Texas Denton, TX 76203-1430, USA [1] Mathematische Annalen 324, 2002, 619-656 [2] Mathematische Zeitschrift 243, 2003, 25-36 ... Summer Schools and Workshops in a closed and well equiped place.The ASI was an International Summer School* on "Chaotic Dynamics and Transport in Classical and Quantum Systems" The main goal of the... give a short introduction to ideas, results and machinery of this part of modern nonlinear dynamics 35 P Collet et al (eds.), Chaotic Dynamics and Transport in Classical and Quantum Systems, 35–72... Chaotic Dynamics and Transport in Classical and Quantum Systems, 15–33 © 2005 Kluwer Academic Publishers Printed in the Netherlands 16 of L2 type [18] This led von Neumann to publish in 1932