1. 1 Preface Many phenomena from physics, biology, chemistry and economics are modeled by di?erential equations with parameters. When a nonlinear equation is est- lished, its behavior/dynamics should be understood. In general, it is impossible to ?nd a complete dynamics of a nonlinear di?erential equation. Hence at least, either periodic or irregular/chaotic solutions are tried to be shown. So a pr- erty of a desired solution of a nonlinear equation is given as a parameterized boundary value problem. Consequently, the task is transformed to a solvability of an abstract nonlinear equation with parameters on a certain functional space. When a family of solutions of the abstract equation is known for some para- ters, the persistence or bifurcations of solutions from that family is studied as parameters are changing. There are several approaches to handle such nonl- ear bifurcation problems. One of them is a topological degree method, which is rather powerful in cases when nonlinearities are not enough smooth. The aim of this book is to present several original bifurcation results achieved by the author using the topological degree theory. The scope of the results is rather broad from showing periodic and chaotic behavior of non-smooth mechanical systems through the existence of traveling waves for ordinary di?erential eq- tions on in?nite lattices up to study periodic oscillations of undamped abstract waveequationsonHilbertspaceswithapplicationstononlinearbeamandstring partial di?erential equations. 1.

The goal of this book is to explore some of the connections between control theory and geometric mechanics; that is, control theory is linked with a geometric view of classical mechanics in both its Lagrangian and Hamiltonian formulations and in particular with the theory of mechanical systems subject to motion constraints. The synthesis of topics is appropriate as there is a particularly rich connection between mechanics and nonlinear control theory. The aim is to provide a unified treatment of nonlinear control theory and constrained mechanical systems that incorporates material that has not yet made its way into texts and monographs.This book is intended for graduate students who wish to learn this subject and researchers in the area who want to enhance their techniques.

The present volume contains expanded and substantially reworked records of invitedlecturesdeliveredduringthe38thKarpaczWinterSchoolofTheoretical Physics on "Dynamical Semigroups: Dissipation, Chaos, Quanta", which took placeinLadek , Zdr' oj,(Poland)intheperiod6-15February2002. Themainpurposeoftheschoolwastocreateaplatformfortheconfrontation ofviewpointsandresearchmethodologiesrepresentedbytwogroupsofexperts actually working in the very same area of theoretical physics. This situation is quite distinct in non-equilibrium statistical physics of open systems, where classicalandquantumaspectsareaddressedseparatelybymeansofverydi?erent andevenincompatibleformaltools. TheschooltopicsselectionbytheLecturersreads:dissipativedynamicsand chaoticbehaviour,modelsofenvironment-systemcouplingandmodelsofth- mostats;non-equilibriumstatisticalmechanicsandfarfromequilibriumphen- ena;quantumopensystems,decoherenceandlinkstoquantumchaos;quantum andclassicalapplicationsofMarkovsemigroupsandthevalidityofMarkovian approximations. Theorganizingprincipleforthewholeendeavourwastheissueofthedyn- ics of open systems and more speci?cally -15February2002. Themainpurposeoftheschoolwastocreateaplatformfortheconfrontation ofviewpointsandresearchmethodologiesrepresentedbytwogroupsofexperts actually working in the very same area of theoretical physics. This situation is quite distinct in non-equilibrium statistical physics of open systems, where classicalandquantumaspectsareaddressedseparatelybymeansofverydi?erent andevenincompatibleformaltools. TheschooltopicsselectionbytheLecturersreads:dissipativedynamicsand chaoticbehaviour,modelsofenvironment-systemcouplingandmodelsofth- mostats;non-equilibriumstatisticalmechanicsandfarfromequilibriumphen- ena;quantumopensystems,decoherenceandlinkstoquantumchaos;quantum andclassicalapplicationsofMarkovsemigroupsandthevalidityofMarkovian approximations. Theorganizingprincipleforthewholeendeavourwastheissueofthedyn- ics of open systems and more speci?cally - thedynamics of dissipation. Since this research area is extremely broad and varied, no single book can cover all importantdevelopments. Therefore,linkswithdynamicalchaoswerechosento representasupplementaryconstraint. Theprogrammeoftheschoolandits?naloutcomeintheformofthepresent volumehasbeenshapedwiththehelpofthescienti?ccommitteecomprising:R. Alicki,Ph. Blanchard,J. R. Dorfman,G. Gallavotti,P. Gaspard,I. Guarneri, ? F. Haake, M. Ku's, A. Lasota, B. Zegarlinski ' and K. Zyczkowski. Some of the committeememberstookchargeoflecturingtoo. Weconveyourthankstoall ofthem. Wewouldliketoexpresswordsofgratitudetomembersofthelocalorgan- ingcommittee,W. Ceg laandP. Lugiewicz, fortheirhelp. Specialthanksmust beextendedtoMrsAnnaJadczykforherhelpatvariousstagesoftheschool organizationandthecompetenteditorialassistance. Theschoolwas?nanciallysupportedbytheUniversityofWroc law,Univ- sityofZielonaG' ora,PolishMinistryofEducation,PolishAcademyofSciences, FoundationfortheKarpaczWinterSchoolofTheoreticalPhysicsandthe- nationfromtheDrWilhelmHeinrichHeraeusundElseHeraeusStiftung. Wrocla wandZielonaG' ora,Poland PiotrGarbaczewski June2002 RobertOlkiewicz TableofContents Introduction...1 ChapterI NonequilibriumDynamics SomeRecentAdvancesinClassicalStatisticalMechanics E. G. D. Cohen...7 DeterministicThermostatsandFluctuationRelations L. Rondoni...35 WhatIstheMicroscopicResponseofaSystem DrivenFarFromEquilibrium? C. Jarzynski...63 Non-equilibriumStatisticalMechanics ofClassicalandQuantumSystems D. Kusnezov,E. Lutz,K. Aoki...8 3 ChapterII DynamicsofRelaxationandChaoticBehaviour DynamicalTheoryofRelaxation inClassicalandQuantumSystems P. Gaspard...111 RelaxationandNoiseinChaoticSystems S. Fishman,S. Rahav...165 FractalStructuresinthePhaseSpace ofSimpleChaoticSystemswithTransport J. R. Dorfman...193 ChapterIII DynamicalSemigroups MarkovSemigroupsandTheirApplications R. Rudnicki,K. Pich'or,M. Tyran-Kaminska ' ...215 VIII TableofContents InvitationtoQuantumDynamicalSemigroups R. Alicki...239 FiniteDissipativeQuantumSystems M. Fannes...265 CompletePositivityinDissipativeQuantumDynamics F. Benatti,R. Floreanini,R. Romano...283 QuantumStochasticDynamicalSemigroup W. A. Majewski ...305 ChapterIV Driving,DissipationandControlinQuantumSystems DrivenChaoticMesoscopicSystems, DissipationandDecoherence D. Cohen...317 QuantumStateControlinCavityQED T. WellensandA. Buchleitner...351 SolvingSchrodinger'sEquationforanOpenSystem andItsEnvironment W. T. Strunz...377 ChapterV DynamicsofLargeSystems ThermodynamicBehaviorofLargeDynamicalSystems -Quantum1dConductorandClassicalMultibakerMap- S. Tasaki...395 CoherentandDissipativeTransport inAperiodicSolids:AnOverview J. Bellissard...

This book provides an introduction to the relatively new discipline of arithmetic dynamics. Whereas classical discrete dynamics is the study of iteration of self-maps of the complex plane or real line, arithmetic dynamics is the study of the number-theoretic properties of rational and algebraic points under repeated application of a polynomial or rational function. A principal theme of arithmetic dynamics is that many of the fundamental problems in the theory of Diophantine equations have dynamical analogs.This graduate-level text provides an entry for students into an active field of research and serves as a standard reference for researchers.

This work is devoted to an intensive study in contact mechanics, treating the nonsmooth dynamics of contacting bodies. Mathematical modeling is illustrated and discussed in numerous examples of engineering objects working in different kinematic and dynamic environments.

Topics covered in five self-contained chapters examine non-steady dynamic phenomena which are determined by key factors: i.e., heat conduction, thermal stresses, and the amount of wearing. New to this monograph is the importance of the inertia factor, which is considered on par with thermal stresses.

Nonsmooth Dynamics of Contacting Thermoelastic Bodies is an engaging accessible practical reference for engineers (civil, mechanical, industrial) and researchers in theoretical and applied mechanics, applied mathematics, physicists, and graduate students.

Based on a one-year course taught by the author to graduates at the University of Missouri, this book provides a student-friendly account of some of the standard topics encountered in an introductory course of ordinary differential equations. In a second semester, these ideas can be expanded by introducing more advanced concepts and applications. A central theme in the book is the use of Implicit Function Theorem, while the latter sections of the book introduce the basic ideas of perturbation theory as applications of this Theorem. The book also contains material differing from standard treatments, for example, the Fiber Contraction Principle is used to prove the smoothness of functions that are obtained as fixed points of contractions. The ideas introduced in this section can be extended to infinite dimensions.

This book is written by leading scholars in Network Science, Nonlinear Science and Infrastructure Systems, expressly to develop common theoretical underpinnings for better solutions to modern infrastructural problems. The book is dedicated to the formulation of infrastructural tools that will better solve problems from transportation networks to telecommunications, Internet, supply chains and more.

The study of hyperbolic systems is a core theme of modern dynamics. On surfaces the theory of the ?ne scale structure of hyperbolic invariant sets and their measures can be described in a very complete and elegant way, and is the subject of this book, largely self-contained, rigorously and clearly written. It covers the most important aspects of the subject and is based on several scienti?c works of the leading research workers in this ?eld. This book ?lls a gap in the literature of dynamics. We highly recommend it for any Ph.D student interested in this area. The authors are well-known experts in smooth dynamical systems and ergodic theory. Now we give a more detailed description of the contents: Chapter1.TheIntroductionisadescriptionofthemainconceptsinhyp- bolic dynamics that are used throughout the book. These are due to Bowen, Hirsch, Man' "e, Palis, Pugh, Ruelle, Shub, Sinai, Smale and others. Stable and r unstable manifolds are shown to beC foliated. This result is very useful in a number of contexts. The existence of smooth orthogonal charts is also proved. This chapter includes proofs of extensions to hyperbolic di?eomorphisms of some results of Man' "e for Anosov maps. Chapter 2. All the smooth conjugacy classes of a given topological model are classi?ed using Pinto's and Rand's HR structures. The a?ne structures of Ghys and Sullivan on stable and unstable leaves of Anosov di?eomorphisms are generalized.

The book addresses mathematicians and physicists, including graduate students, who are interested in quantum dynamical systems and applications of operator algebras and ergodic theory. It is the only monograph on this topic. Although the authors assume a basic knowledge of operator algebras, they give precise definitions of the notions and in most cases complete proofs of the results which are used.

In this book, the major ideas behind Organic Computing are delineated, together with a sparse sample of computational projects undertaken in this new field. Biological metaphors include evolution, neural networks, gene-regulatory networks, networks of brain modules, hormone system, insect swarms, and ant colonies. Applications are as diverse as system design, optimization, artificial growth, task allocation, clustering, routing, face recognition, and sign language understanding.

Unlike the conventional research for the general theory of stability, this mono graph deals with problems on stability and stabilization of dynamic systems with respect not to all but just to a given part of the variables characterizing these systems. Such problems are often referred to as the problems of partial stability (stabilization). They naturally arise in applications either from the requirement of proper performance of a system or in assessing system capa bility. In addition, a lot of actual (or desired) phenomena can be formulated in terms of these problems and be analyzed with these problems taken as the basis. The following multiaspect phenomena and problems can be indicated: * "Lotka-Volterra ecological principle of extinction;" * focusing and acceleration of particles in electromagnetic fields; * "drift" of the gyroscope axis; * stabilization of a spacecraft by specially arranged relative motion of rotors connected to it. Also very effective is the approach to the problem of stability (stabilization) with respect to all the variables based on preliminary analysis of partial sta bility (stabilization). A. M. Lyapunov, the founder of the modern theory of stability, was the first to formulate the problem of partial stability. Later, works by V. V. Rumyan tsev drew the attention of many mathematicians and mechanicians around the world to this problem, which resulted in its being intensively worked out. The method of Lyapunov functions became the key investigative method which turned out to be very effective in analyzing both theoretic and applied problems.

This monograph combines the knowledge of both the field of nonlinear dynamics and non-smooth mechanics, presenting a framework for a class of non-smooth mechanical systems using techniques from both fields. The book reviews recent developments, and opens the field to the nonlinear dynamics community. This book addresses researchers and graduate students in engineering and mathematics interested in the modelling, simulation and dynamics of non-smooth systems and nonlinear dynamics.

The ?eld of applied nonlinear dynamics has attracted scientists and engineers across many different disciplines to develop innovative ideas and methods to study c- plex behavior exhibited by relatively simple systems. Examples include: population dynamics, ?uidization processes, applied optics, stochastic resonance, ?ocking and ?ightformations, lasers, andmechanicalandelectricaloscillators. Acommontheme among these and many other examples is the underlying universal laws of nonl- ear science that govern the behavior, in space and time, of a given system. These laws are universal in the sense that they transcend the model-speci?c features of a system and so they can be readily applied to explain and predict the behavior of a wide ranging phenomena, natural and arti?cial ones. Thus the emphasis in the past decades has been in explaining nonlinear phenomena with signi?cantly less att- tion paid to exploiting the rich behavior of nonlinear systems to design and fabricate new devices that can operate more ef?ciently. Recently, there has been a series of meetings on topics such as Experimental Chaos, Neural Coding, and Stochastic Resonance, which have brought together many researchers in the ?eld of nonlinear dynamics to discuss, mainly, theoretical ideas that may have the potential for further implementation. In contrast, the goal of the 2007 ICAND (International Conference on Applied Nonlinear Dynamics) was focused more sharply on the implementation of theoretical ideas into actual - vices and system

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.

This IMA Volume in Mathematics and its Applications FRACTALS IN MULTIMEDIA is a result of a very successful three-day minisymposium on the same title. The event was an integral part of the IMA annual program on Mathemat ics in Multimedia, 2000-2001. We would like to thank Michael F. Barnsley (Department of Mathematics and Statistics, University of Melbourne), Di etmar Saupe (Institut fUr Informatik, UniversiUit Leipzig), and Edward R. Vrscay (Department of Applied Mathematics, University of Waterloo) for their excellent work as organizers of the meeting and for editing the proceedings. We take this opportunity to thank the National Science Foundation for their support of the IMA. Series Editors Douglas N. Arnold, Director of the IMA Fadil Santosa, Deputy Director of the IMA v PREFACE This volume grew out of a meeting on Fractals in Multimedia held at the IMA in January 2001. The meeting was an exciting and intense one, focused on fractal image compression, analysis, and synthesis, iterated function systems and fractals in education. The central concerns of the meeting were to establish within these areas where we are now and to develop a vision for the future."

In this text the authors consider the Korteweg-de Vries (KdV) equation (ut = - uxxx ] 6uux) with periodic boundary conditions. Derived to describe long surface waves in a narrow and shallow channel, this equation in fact models waves in homogeneous, weakly nonlinear and weakly dispersive media in general.

Viewing the KdV equation as an infinite dimensional, and in fact integrable Hamiltonian system, we first construct action-angle coordinates which turn out to be globally defined. They make evident that all solutions of the periodic KdV equation are periodic, quasi-periodic or almost-periodic in time. Also, their construction leads to some new results along the way.

Subsequently, these coordinates allow us to apply a general KAM theorem for a class of integrable Hamiltonian pde's, proving that large families of periodic and quasi-periodic solutions persist under sufficiently small Hamiltonian perturbations.

The pertinent nondegeneracy conditions are verified by calculating the first few Birkhoff normal form terms -- an essentially elementary calculation.

Ergodic theory is hard to study because it is based on measure theory, which is a technically difficult subject to master for ordinary students, especially for physics majors. Many of the examples are introduced from a different perspective than in other books and theoretical ideas can be gradually absorbed while doing computer experiments. Theoretically less prepared students can appreciate the deep theorems by doing various simulations. The computer experiments are simple but they have close ties with theoretical implications. Even the researchers in the field can benefit by checking their conjectures, which might have been regarded as unrealistic to be programmed easily, against numerical output using some of the ideas in the book. One last remark: The last chapter explains the relation between entropy and data compression, which belongs to information theory and not to ergodic theory. It will help students to gain an understanding of the digital technology that has shaped the modern information society.

The seminal 1970 Moscow thesis of Grigoriy A. Margulis, published for the first time. Entitled "On Some Aspects of the Theory of Anosov Systems," it uses ergodic theoretic techniques to study the distribution of periodic orbits of Anosov flows. The thesis introduces the "Margulis measure" and uses it to obtain a precise asymptotic formula for counting periodic orbits. This has an immediate application to counting closed geodesics on negatively curved manifolds. The thesis also contains asymptotic formulas for the number of lattice points on universal coverings of compact manifolds of negative curvature.

The thesis is complemented by a survey by Richard Sharp, discussing more recent developments in the theory of periodic orbits for hyperbolic flows, including the results obtained in the light of Dolgopyat's breakthroughs on bounding transfer operators and rates of mixing.

The present volume is an introduction to nonlinear waves and soliton theory in the special environment of compact spaces such as closed curves and surfaces and other domain contours.

The first part of the book introduces the mathematical concept required for treating the manifolds considered. An introduction to the theory of motion of curves and surfaces is given. The second and third parts discuss the modeling of various physical solitons on compact systems.

These proceedings are the fifth in the series Traffic and Granular Flow, and we hope they will be as useful a reference as their predecessors. Both the realistic modelling of granular media and traffic flow present important challenges at the borderline between physics and engineering, and enormous progress has been made since 1995, when this series started. Still the research on these topics is thriving, so that this book again contains many new results. Some highlights addressed at this conference were the influence of long range electric and magnetic forces and ambient fluids on granular media, new precise traffic measurements, and experiments on the complex decision making of drivers. No doubt the "hot topics" addressed in granular matter research have diverged from those in traffic since the days when the obvious analogies between traffic jams on highways and dissipative clustering in granular flow intrigued both c- munities alike. However, now just this diversity became a stimulating feature of the conference. Many of us feel that our joint interest in complex systems, where many simple agents, be it vehicles or particles, give rise to surprising and fascin- ing phenomena, is ample justification for bringing these communities together: Traffic and Granular Flow has fostered cooperation and friendship across the scientific disciplines.

This book explains why complex systems research is important in understanding the structure, function and dynamics of complex natural and social phenomena. It illuminates how complex collective behavior emerges from the parts of a system, due to the interaction between the system and its environment. Readers will learn the basic concepts and methods of complex system research. The book is not highly technical mathematically, but teaches and uses the basic mathematical notions of dynamical system theory, making the book useful for students of science majors and graduate courses.

Multibody Mechanics and Visualization is designed to appeal to computer-savvy students who will acquire significant skills in mathematical and physical modelling of mechanical systems in the process of producing attractive computer simulations and animations. The emphasis here is on general skills with all-round applicability rather than the ability to solve "cooked-up problems. The approachable style and clear presentation of this text will help you grasp the essentials of: modeling the kinematics and dynamics of arbitrary multibody mechanisms; formulating a mathematical description of general motions of such mechanisms; implementing the description in a computer-graphics application for the animation/visualization of the movement. Multibody Mechanics and Visualization plays down the prediction of dynamics by formal analysis of differential equations while preparing its students to perform such analyses with greater understanding later. The text relies on the following principles for effective tuition: an inductive approach to learning - discerning general patterns from particular observations; repetition and review of important principles to reinforce your learning through numerous examples; obvious visual guidance that shows you at a glance which information you need for different levels of understanding; computer tools, visual representations and elements of active learning integrated into the text to suit the way you want to learn. Supported in the text in parallel with the theoretical presentation is the simulation and animation application Mambo. In contrast with existing commercially available educational software tools, Mambo requires detailed input from you in order to define the specific geometry of a mechanism as well as the differential equations governing its behavior while allowing you to visualize the results of your efforts. The Mambo toolbox enables you to provide these specifications for mechanisms that would pose insurmountable algebraic challenges to manual calculation. With these tools, you will be able to see the implications of decisions made throughout the modeling process, to check your mathematical analyses, and to enjoy the fruit of your labor Mambo can be freely downloaded from the author's website and runs under any version of MS Windows(r). The toolbox is compatible with the Maple software environment and the Matlab(r) extended symbolic toolbox."

This book covers a new explanation of the origin of Hamiltonian chaos and its quantitative characterization. The author focuses on two main areas: Riemannian formulation of Hamiltonian dynamics, providing an original viewpoint about the relationship between geodesic instability and curvature properties of the mechanical manifolds; and a topological theory of thermodynamic phase transitions, relating topology changes of microscopic configuration space with the generation of singularities of thermodynamic observables. The book contains numerous illustrations throughout and it will interest both mathematicians and physicists.

The notion of a ?xed point plays a crucial role in numerous branches of mat- maticsand its applications. Informationabout the existence of such pointsis often the crucial argument in solving a problem. In particular, topological methods of ?xed point theory have been an increasing focus of interest over the last century. These topological methods of ?xed point theory are divided, roughly speaking, into two types. The ?rst type includes such as the Banach Contraction Principle where the assumptions on the space can be very mild but a small change of the map can remove the ?xed point. The second type, on the other hand, such as the Brouwer and Lefschetz Fixed Point Theorems, give the existence of a ?xed point not only for a given map but also for any its deformations. This book is an exposition of a part of the topological ?xed and periodic point theory, of this second type, based on the notions of Lefschetz and Nielsen numbers. Since both notions are homotopyinvariants, the deformationis used as an essential method, and the assertions of theorems typically state the existence of ?xed or periodic points for every map of the whole homotopy class, we refer to them as homotopy methods of the topological ?xed and periodic point theory.

This book is devoted to applications of complex nonlinear dynamic phenomena to real systems and device applications. In recent decades there has been significant progress in the theory of nonlinear phenomena, but there are comparatively few devices that actually take this rich behavior into account. The text applies and exploits this knowledge to propose devices which operate more efficiently and cheaply, while affording the promise of much better performance.