长距离相互作用随机及分数维动力学

出版时间：2010-6  出版社：高等教育出版社  作者：罗朝俊，（墨）阿弗莱诺维奇　　编  页数：308  

前言

anomalous chaotic transport, plasma physics, and theory of chaos in waveguides. The book "Nonlinear Physics: from the Pendulum to Turbulence and Chaos" （Nauka, Moscow and Harwood, New York, 1988）, written with R. Sagdeev, is now a classical textbook for everybody who studies chaos theory. When studying interaction of a charged particle with a wave packet, George with colleagues from the Institute discovered that stochastic layers of different separatrices in degenerated Hamiltonian systems may merge producing a stochastic web. Unlike the famous Arnold diffusion in non-degenerated Hamiltonian systems, that appears only if the number of degrees of freedom exceeds 2, diffusion in the Zaslavsky webs is possible at one and half degrees of freedom. This diffusion is rather universal phenomenon and its speed is much greater than that of Arnold diffusion. Beautiful symmetries of the Zaslavsky webs and their properties in different branches of physics have been described in the book "Weak chaos and Quasi-Regular Structures" （Nauka,Moscow, 1991 and Cambridge University Press, Cambridge, 1991） coauthored with R. Sagdeev, D. Usikov, and A. Chernikov. In 1991, George emigrated to the USA and became a Professor of Physics and Mathematics at Physical Department of the New York University and at the Courant Institute of Mathematical Sciences. The last 17 years of his life he de-voted to principal problems of Hamiltonian chaos connected with anomalous kinetics and fractional dynamics, foundations of statistical mechanics, chaotic advection, quantum chaos, and long-range propagation of acoustic waves in the ocean. In his New York period George published two important books on the Hamiltonian chaos: "Physics of Chaos in Hamiltonian Systems" （Imperial College Press, London, 1998）and "Hamiltonian chaos and Fractional Dynamics" （Oxford University Press, NY,2005）. His last book "Ray and wave chaos in ocean acoustics: chaos in waveguides"（World Scientific Press, Singapore, 2010）, written with D. Makarov, S. Prants, and A. Virovlynsky, reviews original results on chaos with acoustic waves in the under-water sound channel.George was a very creative scientist and a very good teacher whose former stu-dents and collaborators are working now in America, Europe and Asia. He authored and coauthored 9 books and more than 300 papers in journals. Many of his works are widely cited. George worked hard all his life. He loved music, theater, literature and was an expert in good vines and food. Only a few people knew that he loved to paint. In the last years he has spent every summer in Provence, France, working ,writing books and papers and painting in water colors. The album with his watercolors was issued in 2009 in Moscow.

内容概要

In memory of Dr. George Zaslavsky, Long-range Interactions, Stochasticity and Fractional Dynamics covers'the recent developments of long-range interaction, fractional dynamics, brain dynamics and stochastic theory of turbulence, each chapter was written by established scientists in the field. The book is dedicated to Dr. George Zaslavsky, who was one of three founders of the theory of Hamiltonian chaos. The book discusses self-similarity and stochasticity and fractionality for discrete and continuous dynamical systems, as well as long-range interactions and diluted networks. A comprehensive theory for brain dynamics is also presented. In addition, the complexity and stochasticity for soliton chains and turbulence are addressed.     The book is intended for researchers in the field of nonlinear dynamics in mathematics, physics and engineering.

作者简介

编者：罗朝俊 （墨西哥）阿弗莱诺维奇（Valentin Afraimovich） 丛书主编：（瑞典）伊布拉基莫夫Dr. Albert C.J. Luo is a Professor at Southern Illinois University Edwardsville,USA.Dr. Valentin Afraimovich is a Proiessor at San Luis Potosi University, Mexico.

书籍目录

1  Fractional Zaslavsky and Henon Discrete Maps  Vasily E. Tarasov  1.1  Introduction  1.2  Fractional derivatives    1.2.1  Fractional Riemann-Liouville derivatives    1.2.2  Fractional Caputo derivatives    1.2.3  Fractional Liouville derivatives    1.2.4  Interpretation of equations with fractional derivatives.    1.2.5  Discrete maps with memory  1.3  Fractional Zaslavsky maps    1.3.1  Discrete Chirikov and Zaslavsky maps    1.3.2  Fractional universal and Zaslavsky map    1.3.3  Kicked damped rotator map    1.3.4  Fractional Zaslavsky map from fractional differential equations  1.4  Fractional H6non map    1.4.1  Henon map    1.4.2  Fractional Henon map  1.5  Fractional derivative in the kicked term and Zaslavsky map    1.5.1  Fractional equation and discrete map    1.5.2  Examples  1.6  Fractional derivative in the kicked damped term and generalizations of Zaslavsky and Henon maps    1.6.1  Fractional equation and discrete map    1.6.2  Fractional Zaslavsky and Henon maps  1.7  Conclusion  References2  Self-similarity, Stochasticity and Fractionality  Vladimir V Uchaikin  2.1  Introduction    2.1.1  Ten years ago    2.1.2  Two kinds of motion    2.1.3  Dynamic self-similarity    2.1.4  Stochastic self-similarity    2.1.5  Self-similarity and stationarity  2.2  From Brownian motion to Levy motion    2.2.1  Brownian motion    2.2.2  Self-similar Brownian motion in nonstationary nonhomogeneous environment    2.2.3  Stable laws    2.2.4  Discrete time Levy motion    2.2.5  Continuous time Levy motion    2.2.6  Fractional equations for continuous time Levy motion  2.3  Fractional Brownian motion    2.3.1  Differential Brownian motion process    2.3.2  Integral Brownian motion process    2.3.3  Fractional Brownian motion    2.3.4  Fractional Gaussian noises    2.3.5  Barnes and Allan model    2.3.6  Fractional Levy motion  2.4  Fractional Poisson motion    2.4.1  Renewal processes    2.4.2  Self-similar renewal processes    2.4.3  Three forms of fractal dust generator    2.4.4  nth arrival time distribution    2.4.5  Fractional Poisson distribution  2.5  Fractional compound Poisson process    2.5.1  Compound Poisson process    2.5.2  Levy-Poisson motion    2.5.3  Fractional compound Poisson motion    2.5.4  A link between solutions    2.5.5  Fractional generalization of the Levy motion    Acknowledgments  Appendix. Fractional operators  References3  Long-range Interactions and Diluted Networks  Antonia Ciani, Duccio Fanelli and Stefano Ruffo  3.1  Long-range interactions    3.1.1  Lack of additivity    3.1.2  Equilibrium anomalies: Ensemble inequivalence, negative specific heat and temperature jumps    3.1.3  Non-equilibrium dynamical properties    3.1.4  Quasi Stationary States    3.1.5  Physical examples    3.1.6  General remarks and outlook  3.2  Hamiltonian Mean Field model: equilibrium and out-of- equilibrium features    3.2.1  The model    3.2.2  Equilibrium statistical mechanics    3.2.3  On the emergence of Quasi Stationary States: Non-    equilibrium dynamics  3.3  Introducing dilution in the Hamiltonian Mean Field model    3.3.1  Hamiltonian Mean Field model on a diluted network    3.3.2  On equilibrium solution of diluted Hamiltonian Mean Field    3.3.3  On Quasi Stationary States in presence of dilution    3.3.4  Phase transition  3.4  Conclusions    Acknowledgments  References4  Metastability and Transients in Brain Dynamics: Problems and Rigorous Results  Valentin S. Afraimovich, Mehmet K. Muezzinoglu and  Mikhail I. Rabinovich  4.1  Introduction: what we discuss and why now    4.1.1  Dynamical modeling of cognition    4.1.2  Brain imaging    4.1.3  Dynamics of emotions  4.2  Mental modes    4.2.1  State space    4.2.2  Functional networks    4.2.3  Emotion-cognition tandem    4.2.4  Dynamical model of consciousness  4.3  Competition--robustness and sensitivity    4.3.1  Transients versus attractors in brain    4.3.2  Cognitive variables    4.3.3  Emotional variables    4.3.4  Metastability and dynamical principles    4.3.5  Winnerless competition--structural stability of transients    4.3.6  Examples: competitive dynamics in sensory systems    4.3.7  Stable heteroclinic channels  4.4  Basic ecological model    4.4.1  The Lotka-Volterra system    4.4.2  Stress and hysteresis    4.4.3  Mood and cognition    4.4.4  Intermittent heteroclinic channel  4.5  Conclusion    Acknowledgments  Appendix 1  Appendix 2  References5  Dynamics of Soliton Chains: From Simple to Complex and Chaotic Motions  Konstantin A. Gorshkov, Lev A. Ostrovsky and Yury A. Stepanyants  5.1  Introduction  5.2  Stable soliton lattices and a hierarchy of envelope solitons  5.3  Chains of solitons within the framework of the Gardner model  5.4  Unstable soliton lattices and stochastisation  5.5  Soliton stochastisation and strong wave turbulence in a resonator with external sinusoidal pumping  5.6  Chains of two-dimensional solitons in positive-dispersion media  5.7  Conclusion  Few words in memory of George M. Zaslavsky  References6  What is Control of Turbulence in Crossed Fields?-Don't Even Think of Eliminating All Vortexes!  Dimitri Volchenkov  6.1  Introduction  6.2  Stochastic theory of turbulence in crossed fields: vortexes of all sizes die out, but one    6.2.1  The method of renormalization group    6.2.2  Phenomenology of fully developed isotropic turbulence    6.2.3  Quantum field theory formulation of stochastic Navier-Stokes turbulence    6.2.4  Analytical properties of Feynman diagrams    6.2.5  Ultraviolet renormalization and RG-equations    6.2.6  What do the RG representations sum?    6.2.7  Stochastic magnetic hydrodynamics    6.2.8  Renormalization group in magnetic hydrodynamics    6.2.9  Critical dimensions in magnetic hydrodynamics    6.2.10  Critical dimensions of composite operators in magnetic hydrodynamics    6.2.11  Operators of the canonical dimension d = 2    6.2.12  Vector operators of the canonical dimension d = 3    6.2.13  Instability in magnetic hydrodynamics    6.2.14  Long life to eddies of a preferable size  6.3  In search of lost stability    6.3.1  Phenomenology of long-range turbulent transport in the scrape-off layer (SOL) of thermonuclear reactors    6.3.2  Stochastic models of turbulent transport in cross-field systems    6.3.3  Iterative solutions in crossed fields    6.3.4  Functional integral formulation of cross-field turbulent transport    6.3.5  Large-scale instability of iterative solutions    6.3.6  Turbulence stabilization by the poloidal electric drift    6.3.7  Qualitative discrete time model of anomalous transport in the SOL  6.4  Conclusion  References7  Entropy and Transport in Billiards  M. Courbage and S.M. Saberi Fathi  7.1  Introduction  7.2  Entropy    7.2.1  Entropy in the Lorentz gas    7.2.2  Some dynamical properties of the barrier billiard model  7.3  Transport    7.3.1  Transport in Lorentz gas    7.3.2  Transport in the barrier billiard  7.4  Concluding remarks  ReferencesIndex

章节摘录

插图：Note that the continuous limit of discrete systems with power-law long-range interactions gives differential equations with derivatives of non-integer orders with respect to coordinates （Tarasov and Zaslavsky, 2006; Tarasov, 2006）. Fractional differentiation with respect to time is characterized by long-term memory effects that correspond to intrinsic dissipative processes in the physical systems. The memory effects to discrete maps mean that their present state evolution depends on all past states. The discrete maps with memory are considered in the papers （Fulinski and Kleczkowski, 1987;Fick et al., 1991; Giona, 1991; Hartwich and Fick, 1993; Gallas, 1993; Stanislavsky,2006; Tarasov and Zaslavsky, 2008; Tarasov, 2009; Edelman and Tarasov, 2009）.The interesting question is a connection of fractional equations of motion and thediscrete maps with memory. This derivation is realized for universal and standard maps in （Tarasov and Zaslavsky, 2008; Tarasov, 2009）.   It is important to derive discrete maps with memory from equations of motion with fractional derivatives. It was shown （Zaslavsky et al., 2006） that perturbed by aperiodic force, the nonlinear system with fractional derivative exhibits a new type of chaotic motion called the fractional chaotic attractor.

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《长距离相互作用、随机及分数维动力学》编辑推荐：Nonlinear Physical Science focuses on the recent advances of fundamental theories and principles, analytical and symbolic approaches, as well as computational techniques in nonlinear physical science and nonlinear mathematics with engineering applications.

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