This book is about the dynamics of coupled map lattices (CML) and of related spatially extended systems. It will be useful to post-graduate students and researchers seeking an overview of the state-of-the-art and of open problems in this area of nonlinear dynamics. The special feature of this book is that it describes the (mathematical) theory of CML and some related systems and their phenomenology, with some examples of CML modeling of concrete systems (from physics and biology). More precisely, the book deals with statistical properties of (weakly) coupled chaotic maps, geometric aspects of (chaotic) CML, monotonic spatially extended systems, and dynamical models of specific biological systems.

Complexity science has been a source of new insight in physical and social systems and has demonstrated that unpredictability and surprise are fundamental aspects of the world around us. This book is the outcome of a discussion meeting of leading scholars and critical thinkers with expertise in complex systems sciences and leaders from a variety of organizations, sponsored by the Prigogine Center at The University of Texas at Austin and the Plexus Institute, to explore strategies for understanding uncertainty and surprise. Besides contributions to the conference, it includes a key digest by the editors as well as a commentary by the late nobel laureate Ilya Prigogine, "Surprises in half of a century." The book is intended for researchers and scientists in complexity science, as well as for a broad interdisciplinary audience of both practitioners and scholars. It will well serve those interested in the research issues and in the application of complexity science to physical and social systems.

Models form the basis of any decision. They are used in di?erent context and for di?erent purposes: for identi?cation, prediction, classi?cation, or control of complex systems. Modeling is done theory-driven by logical-mathematical methods or data-driven based on observational data of the system and some algorithm or software for analyzing this data. Today, this approach is s- marized as Data Mining. There are many Data Mining algorithms known like Arti?cial Neural N- works, Bayesian Networks, Decision Trees, Support Vector Machines. This book focuses on another method: the Group Method of Data Handling. - thoughthismethodologyhasnotyetbeenwellrecognizedintheinternational science community asa verypowerfulmathematicalmodeling andknowledge extraction technology, it has a long history. Developed in 1968bythe Ukrainianscientist A.G. Ivakhnenko it combines the black-box approach and the connectionism of Arti?cial Neural Networks with well-proven Statistical Learning methods and with more behavior- justi?ed elements of inductive self-organization.Over the past 40 years it has been improving and evolving, ?rst by works in the ?eld of what was known in the U.S.A. as Adaptive Learning Networks in the 1970s and 1980s and later by signi? cantcontributions from scientists from Japan,China, Ukraine, Germany. Many papers and books have been published on this modeling technology, the vast majority of them in Ukrainian and Russian language.

The conference Chaos in Astronomy was held in Athens from 17 to 20 September 2007 and was dedicated to the memory of Nikos Voglis, who was directoroftheResearchCenterforAstronomyoftheAcademyofAthensuntil his death on 9 February 2007. It was attended by 73 registered participants coming from 18 di?erent countries. A total of 40 oral papers were delivered including a conference summary. Furthermore 16 posters were presented. The conference was the main event in a series of talks, public lectures and d- cussion meetings about "Chaos in Astronomy" that have taken place at the Research Center for Astronomy. We underline three special talks that have been given by D. Kazanas (NASA-Goddard Space Flight Center), D. Lynden- Bell (Cambridge, UK) and C. Tsallis (Brazilian Academy of Science). The main sponsor of the conference was the "Alexander S. Onassis" Fo- dation. In addition the conference has been supported by the General S- retariat for Research and Technology, the National Observatory of Athens, and the Hellenic Ministry of Culture. We are grateful to all sponsors for their generosity. We also express our gratitude to the Academy of Athens and to the - rectorship of the Biomedical Research Foundation, where the conference took place.

This volume is published as the proceedings of the third Russian-German - vanced Research Workshop on Computational Science and High Performance Computing in Novosibirsk, Russia, in July 2007. The contributions of these proceedings were provided and edited by the - thors, chosen after a careful selection and reviewing. The workshop was organized by the High Performance Computing Center Stuttgart(Stuttgart,Germany)andtheInstituteofComputationalTechnologies SBRAS(Novosibirsk,Russia)intheframeworkofactivitiesoftheGerman-Russian CenterforComputationalTechnologiesandHighPerformanceComputing. Thee event is held biannually and has already become a good tradition for German and Russian scientists. The ?rst Workshop took place in September 2003 in Novosibirskand the second Workshopwas hosted by Stuttgart in March 2005. Both workshops gave the possibility of sharing and discussing the latest results and developing further scienti?c contacts in the ?eld of computational science and high performance computing. The topics of the current workshop include software and hardware for high performancecomputation,numericalmodellingingeophysicsandcomputational ?uid dynamics, mathematical modelling of tsunami waves, simulation of fuel cellsandmodern? breopticsdevices,numericalmodellingincryptographypr- lems andaeroacoustics,interval analysis,toolsfor Gridapplications,researchon service-oriented architecture (SOA) and telemedicine technologies. Theparticipationofrepresentativesofmajorresearchorganizationsengagedin the solution of the most complex problems of mathematical modelling, devel- ment of new algorithms,programsandkey elementsof informationtechnologies, elaboration and implementation of software and hardware for high performance computing systems,provideda highlevelof competenceofthe workshop. Among the German participants were the heads and leading specialists of the HighPerformanceComputingCenterStuttgart(HLRS)(UniversityofStuttgart), NECHighPerformanceComputingEuropeGmbH,SectionofAppliedMathem- ics(UniversityofFreiburgi.Br.),InstituteofAerodynamics(RWTHAachen),- gionalComputingCenterErlangen(RRZE(UniversityofErlangen-Nuremberg), Center for High Performance Computing (ZHR) (Dresden University of Technology).

This monograph is devoted to construction of novel theoretical approaches of m- eling non-homogeneous structural members as well as to development of new and economically ef?cient (simultaneously keeping the required high engineering ac- racy)computationalalgorithmsofnonlineardynamics(statics)ofstronglynonlinear behavior of either purely continuous mechanical objects (beams, plates, shells) or hybrid continuous/lumped interacting mechanical systems. In general, the results presented in this monograph cannot be found in the - isting literature even with the published papers of the authors and their coauthors. We take a challenging and originally developed approach based on the integrated mathematical-numerical treatment of various continuous and lumped/continuous mechanical structural members, putting emphasis on mathematical and physical modeling as well as on the carefully prepared and applied novel numerical - gorithms used to solve the derived nonlinear partial differential equations (PDEs) mainly via Bubnov-Galerkin type approaches. The presented material draws on the ?elds of bifurcation, chaos, control, and s- bility of the objects governed by strongly nonlinear PDEs and ordinary differential equations (ODEs),and may have a positive impact on interdisciplinary ? elds of n- linear mechanics, physics, and applied mathematics. We show, for the ?rst time in a book, the complexity and fascinating nonlinear behavior of continual mechanical objects, which cannot be found in widely reported bifurcational and chaotic dyn- ics of lumped mechanical systems, i. e. , those governed by nonlinear ODEs.

Nonlinear dynamics has become an important field of research in recent years in many areas of the natural sciences. In particular, it has potential applications in biology and medicine; nonlinear data analysis has helped to detect the progress of cardiac disease, physiological disorders, for example episodes of epilepsy, and others. This book focuses on the current trends of research concerning the prediction of sudden cardiac death and the onset of epileptic seizures, using the nonlinear analysis based on ECG and EEG data. Topics covered include the analysis of cardiac models and neural models. The book is a collection of recent research papers by leading physicists, mathematicians, cardiologists and neurobiologists who are actively involved in using the concepts of nonlinear dynamics to explore the functional behaviours of heart and brain under normal and pathological conditions. This collection is intended for students in physics, mathematics and medical sciences, and researchers in interdisciplinary areas of physics and biology.

This valuable book contributes substantively to the current state-of-the-art of macroeconomics. It provides a method for building models in which business cycles and economic growth emerge from the interactions of a large number of heterogeneous agents. Drawing from recent advances in agent-based computational modeling, the authors show how insights from dispersed fields can be fruitfully combined to improve our understanding of macroeconomic dynamics.

Chaos is a fascinating phenomenon that has been observed in nature, laboratory, and has been applied in various real-world applications. Chaotic systems are deterministic with no random elements involved yet their behavior appears to be random. Obser- tions of chaotic behavior in nature include weather and climate, the dynamics of sat- lites in the solar system, the time evolution of the magnetic field of celestial bodies, population growth in ecology, to mention only a few examples. Chaos has been observed in the laboratory in a number of systems such as electrical circuits, lasers, chemical reactions, fluid dynamics, mechanical systems, and magneto-mechanical devices. Chaotic behavior has also found numerous applications in electrical and communication engineering, information and communication technologies, biology and medicine. To the best of our knowledge, this is the first book edited on chaos applications in intelligent computing. To access the latest research related to chaos applications in intelligent computing, we launched the book project where researchers from all over the world provide the necessary coverage of the mentioned field. The primary obj- tive of this project was to assemble as much research coverage as possible related to the field by defining the latest innovative technologies and providing the most c- prehensive list of research references.

The present volume, published at the occasion of his 100th birthday anniversary, is a collection of articles that reviews the impact of Kolomogorov's work in the physical sciences and provides an introduction to the modern developments that have been triggered in this way to encompass recent applications in biology, chemistry, information sciences and finance.

In recent years statistical physics has made significant progress as a result of advances in numerical techniques. While good textbooks exist on the general aspects of statistical physics, the numerical methods and the new developments based on large-scale computing are not usually adequately presented. In this book 16 experts describe the application of methods of statistical physics to various areas in physics such as disordered materials, quasicrystals, semiconductors, and also to other areas beyond physics, such as financial markets, game theory, evolution, and traffic planning, in which statistical physics has recently become significant. In this way the universality of the underlying concepts and methods such as fractals, random matrix theory, time series, neural networks, evolutionary algorithms, becomes clear. The topics are covered by introductory, tutorial presentations.

TheconferenceChanceinPhysics: FoundationsandPerspectiveswasheldfrom 29thNovemberto3rdDecember1999inIschia, Italy. Itwassponsoredbythe IstitutoItalianoPerGliStudiFiloso?ciinNaples, bytheDeutscheForschun- gemeinschaft(DFG), andbytheSocietaItalianaDiFondamentiDellaFisica. SponsoringbytheInternationalSchoolforAdvancedStudies(ISAS)ofTrieste, Italy, madethecompilationofthisvolumepossible;thefundingbytheIs- tutoItalianoPerGliStudiFiloso?ciwascrucialfortheconferenceandisvery gratefullyacknowledged. TheIstitutomanagedtoprovideauniqueatmosphere foraninterdisciplinarymeeting, andtheseproceedingsre?ectindeedthevery friendlybutneverthelessintenseandneverendingdiscussionsononeofthemost debatedissuesofscience: probability, andinparticularprobabilityinphysics. Wegratefullyacknowlegdetheorganisationalworkaswellastheeditorialwork donebyoursecretaryofthemeetingPhDstudentRoderichTumulka. Themeetingwasintendedtostimulaterenewedre?ectiononthefundam- talandpracticalaspectsofprobabilityinphysics, inparticularthefoundations ofstatisticalsechanics, theprobabilityinthefoundationsofquantummech- ics, thealgebraicviewofprobabilityandthephilosophyofprobabilityinits interrelationwithphysics. Questionslikewhatprobabilityis, orwhatitisabout, orhowprobability entersphysicsareofasubtlekind. Theyaredi?cultinvariousways, often mixedupwiththeenormouscomplexityandtheinescapablelackofmat- maticalrigorinthephysicalapplication, orwiththefoundationalproblemsof quantummechanics, wheretheprobabilisticignoranceconcerningthevaluesof certainphysicalquantitieshasevenbeenelevatedtoamatterofprinciple. At present, theunderstandingofprobabilityinphysicsisalmostaspersonalasthe understandingofquantumtheory. Theaimoftheconferencewasthustofocusonideasaboutprobabilityin physics, itsmeaninganditsphilosophicalimplications, byreviewingthedi?erent facetsofprobabilityinphysicsinitsmodernsettingsandbytakingintoaccount modernquantumtheorieswithoutobservers, wheretheoriginofprobabilityis notmysti?edbydogmatism. Thereviewsweregiveninone-hourtalks, andthediscussionswereheldin theformofroundtables, whereshortercontributionswerealsogiven. Thespeakerswereaskednottodilutethemainthemesoftheconference withtechnicalitiesandtofocussharplyontheissueofprobability. Thiswas VI Preface takentoheartbyallspeakersandthemeetingthusprovedverysuccessful. The contributionsinthisvolumeconsequentlyfocusonconceptualissues, andthey makeworthwhilereadingforspecialistsinthe?eldoffoundationsaswellasfor nonspecialists, becauseextensivetechnicalpriorknowledgeisnotrequired. The contributionshavebeenleftintheordertheywerediscussedinthemeeting, whichprovedtobeaverynaturalone: 1. ClassicalStatisticalMechanics, whereBoltzmann'sunderstandingofstat- ticalmechnanicsasarisingfromkineticgastheoryisreviewedandputinto modernperspectives, withanoutlookonrelativisticstatisticalmechanics. Therelativelackofemphasisonthee?ectofchaoticbehaviouronthefo- dationsofprobabilityisnoteworthy. 2. QuantumMechanics, wherewereviewthoseontologicalquantumtheories, thathavebeenmostseriouslydiscussedintherecentyears. Amongthese areadeterministictheory(Bohmianmechanics)andboththeintrinsically randomtheoriesofwavepacketreductionandtheoperator-basedconsistent (decoherent)histories. Itstartswiththe"orthodox"view, againwith- phasisontheprobabilisticaspectsofthesetheories. 3. Chaoticsystems, wherethedynamicalaspectsforthefoundationsofpro- bilityinphysicsareadressed. 4. PhilosophyofProbability, wheretheissuesoftheearliersectionsarefurther scrutinizedonphilosophicalgrounds. Thesecontributionshavenoabstracts. T

The aim of the book is the presentation of the fundamental mathematical and physical concepts of continuum mechanics of solids in a unified description so as to bring young researchers rapidly close to their research area. Accordingly, emphasis is given to concepts of permanent interest, and details of minor importance are omitted. The formulation is achieved systematically in absolute tensor notation, which is almost exclusively used in modern literature. This mathematical tool is presented such that study of the book is possible without permanent reference to other works.

This book contains the courses given at the Fourth School on Statistical Physics and Cooperative Systems held at Santiago, Chile, from 12th to 16th December 1994. This School brings together scientists working on subjects related to recent trends in complex systems. Some of these subjects deal with dynamical systems, ergodic theory, cellular automata, symbolic and arithmetic dynamics, spatial systems, large deviation theory and neural networks. Scientists working in these subjects come from several aeras: pure and applied mathematics, non linear physics, biology, computer science, electrical engineering and artificial intelligence. Each contribution is devoted to one or more of the previous subjects. In most cases they are structured as surveys, presenting at the same time an original point of view about the topic and showing mostly new results. The expository text of Roberto Livi concerns the study of coupled map lattices (CML) as models of spatially extended dynamical systems. CML is one of the most used tools for the investigation of spatially extended systems. The paper emphasizes rigorous results about the dynamical behavior of one dimensional CML; i.e. a uniform real local function defined in the interval [0,1], interacting with its nearest neighbors in a one dimensional lattice.

By now, most academics have heard something about the new science of complexity. In a manner reminiscent of Einstein and the last hundred years of physics, complexity science has captured the public imagination. (R) One can go to Amazon. com and purchase books on complexification (Casti 1994), emergence (Holland 1998), small worlds (Barabasi 2003), the web of life (Capra 1996), fuzzy thinking (Kosko 1993), global c- plexity (Urry 2003) and the business of long-tails (Anderson 2006). Even television has incorporated the topics of complexity science. Crime shows (R) (R) such as 24 or CSI typically feature investigators using the latest advances in computational modeling to "simulate scenarios" or "data mine" all p- sible suspects-all of which is done before the crime takes place. The (R) World Wide Web is another example. A simple search on Google. Com using the phrase "complexity science" gets close to a million hits! C- plexity science is ubiquitous. What most scholars do not realize, however, is the remarkable role sociologists are playing in this new science. C- sider the following examples. 0. 1 Sociologists in Complexity Science The first example comes from the new science of networks (Barabasi 2003). By now, most readers are familiar with the phenomena known as six-degrees of separation-the idea that, because most large networks are comprised of a significant number of non-random weak-ties, the nodes (e. g. , people, companies, etc.

In the modern world of gigantic datasets, which scientists and practioners of all fields of learning are confronted with, the availability of robust, scalable and easy-to-use methods for pattern recognition and data mining are of paramount importance, so as to be able to cope with the avalanche of data in a meaningful way. This concise and pedagogical research monograph introduces the reader to two specific aspects - clustering techniques and dimensionality reduction - in the context of complex network analysis. The first chapter provides a short introduction into relevant graph theoretical notation; chapter 2 then reviews and compares a number of cluster definitions from different fields of science. In the subsequent chapters, a first-principles approach to graph clustering in complex networks is developed using methods from statistical physics and the reader will learn, that even today, this field significantly contributes to the understanding and resolution of the related statistical inference issues. Finally, an application chapter examines real-world networks from the economic realm to show how the network clustering process can be used to deal with large, sparse datasets where conventional analyses fail.

The theory of stochastic processes originally grew out of efforts to describe Brownian motion quantitatively. Today it provides a huge arsenal of methods suitable for analyzing the influence of noise on a wide range of systems. The credit for acquiring all the deep insights and powerful methods is due ma- ly to a handful of physicists and mathematicians: Einstein, Smoluchowski, Langevin, Wiener, Stratonovich, etc. Hence it is no surprise that until - cently the bulk of basic and applied stochastic research was devoted to purely mathematical and physical questions. However, in the last decade we have witnessed an enormous growth of results achieved in other sciences - especially chemistry and biology - based on applying methods of stochastic processes. One reason for this stochastics boom may be that the realization that noise plays a constructive rather than the expected deteriorating role has spread to communities beyond physics. Besides their aesthetic appeal these noise-induced, noise-supported or noise-enhanced effects sometimes offer an explanation for so far open pr- lems (information transmission in the nervous system and information p- cessing in the brain, processes at the cell level, enzymatic reactions, etc.). They may also pave the way to novel technological applications (noise-- hanced reaction rates, noise-induced transport and separation on the na- scale, etc.). Key words to be mentioned in this context are stochastic r- onance, Brownian motors or ratchets, and noise-supported phenomena in excitable systems.

This volume contains the papers presented at the IUTAM Symposium on Geometry and Statistics of Turbulence, held in November 1999, at the Shonan International Village Center, Hayama (Kanagawa-ken), Japan. The Symposium was proposed in 1996, aiming at organizing concen trated discussions on current understanding of fluid turbulence with empha sis on the statistics and the underlying geometric structures. The decision of the General Assembly of International Union of Theoretical and Applied Mechanics (IUTAM) to accept the proposal was greeted with enthusiasm. Turbulence is often characterized as having the properties of mixing, inter mittency, non-Gaussian statistics, and so on. Interest is growing recently in how these properties are related to formation and evolution of struc tures. Note that the intermittency is meant for passive scalars as well as for turbulence velocity or rate of dissipation. There were eighty-eight participants in the Symposium. They came from thirteen countries, and fifty-seven papers were presented. The presenta tions comprised a wide variety of fundamental subjects of mathematics, statistical analyses, physical models as well as engineering applications. Among the subjects discussed are (a) Degree of self-similarity in cascade, (b) Fine-scale structures and degree of Markovian property in turbulence, (c) Dynamics of vorticity and rates of strain, (d) Statistics associated with vortex structures, (e) Topology, structures and statistics of passive scalar advection, (f) Partial differential equations governing PDFs of velocity in crements, (g) Thermal turbulences, (h) Channel and pipe flow turbulences, and others."

By bringing together various ideas and methods for extracting the slow manifolds, the authors show that it is possible to establish a more macroscopic description in nonequilibrium systems. The book treats slowness as stability. A unifying geometrical viewpoint of the thermodynamics of slow and fast motion enables the development of reduction techniques, both analytical and numerical. Examples considered in the book range from the Boltzmann kinetic equation and hydrodynamics to the Fokker-Planck equations of polymer dynamics and models of chemical kinetics describing oxidation reactions. Special chapters are devoted to model reduction in classical statistical dynamics, natural selection, and exact solutions for slow hydrodynamic manifolds. The book will be a major reference source for both theoretical and applied model reduction. Intended primarily as a postgraduate-level text in nonequilibrium kinetics and model reduction, it will also be valuable to PhD students and researchers in applied mathematics, physics and various fields of engineering.

Intended for self-study, this second volume presents a systematic approach for deriving model equations of planar and spatial mechanisms. The necessary theoretical foundations have been laid in the first volume. The focus is on the application of the modeling methodology to various examples of rigid-body mechanisms, simple planar ones as well as more challenging spatial problems. A rich variety of joint models, active constraints, as well as active and passive force elements is treated. The book is intended for self-study by working engineers and students concerned with the control of mechanical systems, i.e. robotics, mechatronics, vehicles, and machine tools. Its examples can be used as models for university lectures.

This book is drawn from across many active fields of mathematics and physics. It has connections to atmospheric dynamics, spherical codes, graph theory, constrained optimization problems, Markov Chains, and Monte Carlo methods. It addresses how to access interesting, original, and publishable research in statistical modeling of large-scale flows and several related fields. The authors explicitly reach around the major branches of mathematics and physics, showing how the use of a few straightforward approaches can create a cornucopia of intriguing questions and the tools to answer them.

The main theme of this book is the theory of heights as they appear in various guises. This includes a large body of results on Mahlers measure of the height of a polynomial. The authors'approach is very down to earth as they cover the rationals, assuming no prior knowledge of elliptic curves. The chapters include examples and particular computations, with all special calculation included so as to be self-contained. The authors devote space to discussing Mahlers measure and to giving some convincing and original examples to explain this phenomenon. XXXXXXX NEUER TEXT The main theme of this book is the theory of heights as it appears in various guises. To this End.txt.Int.:, it examines the results of Mahlers measure of the height of a polynomial, which have never before appeared in book form. The authors take a down-to-earth approach that includes convincing and original examples. The book uncovers new and interesting connections between number theory and dynamics and will be interesting to researchers in both number theory and nonlinear dynamics."

This book is devoted to one of the most interesting and rapidly developing areas of modern nonlinear physics and mathematics - theoretical, analytical andnumerical, studyofthestructureanddynamicsofone-dimensionalaswell as two- and three-dimensional solitons and nonlinear wave packets described by the Korteweg-de Vries (KdV), Kadomtsev-Petviashvili (KP), nonlinear Schr] odinger (NLS) and derivative nonlinear Schr] odinger (DNLS) classes of equations. Special attention is paid to generalizations (relevant to various complex physical media) of these equations, accounting for higher-order d- persion corrections, in?uence of dissipation, instabilities, and stochastic ?- tuations of the wave ?elds. We present here a coordinated approach to the theory, simulations, and applications of the nonlinear one-, two-, and three-dimensional solitary wave solutions. Overall, the content of the book is a systematic account of results notonlyalreadyknownintheliterature, butalsothoseofneworiginalstudies related to the theory of models allowing soliton solutions, and analyses of the stability and asymptotics of these solutions. We give signi?cant consideration to numerical methods and results of numerical simulations of the structure and dynamics of solitons and nonlinear wave packets. Together with deep insights into the theory, applications to various branches of modern physics are considered, especially to plasma physics (such as space plasmas including ionospheric and magnetospheric processes), hydrodynamics, and atmosphere dynamics. Presently, thetheoryofone-dimensionalnonlinearequationsoftheclasses consideredbytheauthorsiswelldeveloped, andtheprogressinstudiesofthe structure and evolution of one-dimensional solitons and wave packets is ob- ous. This progress was especially fast after the discovery of hidden algebraic symmetries of the KdV, NLS, and other (integrable by the inverse scatt- ing transform (IST) method) classes of one-dimensional evolution equations

Adaptronic structures and systems are engineered to adjust automatically to variable operating and environmental conditions, through the use of feedback control. The authors of this book have taken on the task of comprehensively describing the current state of the art in this highly modern and broadly interdisciplinary field. The book presents selected examples of applications, and goes on to demonstrate current development trends.

Intended for beginners in ergodic theory, this introductory textbook addresses students as well as researchers in mathematical physics. The main novelty is the systematic treatment of characteristic problems in ergodic theory by a unified method in terms of convergent power series and renormalization group methods, in particular. Basic concepts of ergodicity, like Gibbs states, are developed and applied to, e.g., Asonov systems or KAM Theroy. Many examples illustrate the ideas and, in addition, a substantial number of interesting topics are treated in the form of guided problems.