Nonautonomous dynamical systems provide a mathematical framework for temporally changing phenomena, where the law of evolution varies in time due to seasonal, modulation, controlling or even random effects. Our goal is to provide an approach to the corresponding geometric theory of nonautonomous discrete dynamical systems in infinite-dimensional spaces by virtue of 2-parameter semigroups (processes). These dynamical systems are generated by implicit difference equations, which explicitly depend on time. Compactness and dissipativity conditions are provided for such problems in order to have attractors using the natural concept of pullback convergence. Concerning a necessary linear theory, our hyperbolicity concept is based on exponential dichotomies and splittings. This concept is in turn used to construct nonautonomous invariant manifolds, so-called fiber bundles, and deduce linearization theorems. The results are illustrated using temporal and full discretizations of evolutionary differential equations.

Fully revised to match the more traditional sequence of course materials, this full-color second edition presents the basic principles and methods of thermodynamics using a clear and engaging style and a wealth of end-of-chapter problems. It includes five new chapters on topics such as mixtures, psychometry, chemical equilibrium, and combustion, and discussion of the Second Law of Thermodynamics has been expanded and divided into two chapters, allowing instructors to introduce the topic using either the cycle analysis in Chapter 6 or the definition of entropy in Chapter 7. Online ancillaries including new LMS testbanks, a password-protected solutions manual, prepared PowerPoint lecture slides, instructional videos, and figures in electronic format are available at www.cambridge.org/thermo

Lectures: J. Chazarain, A. Piriou: Probl mes mixtes hyperboliques: Premi re partie: Les probl mes mixtes hyperboliques v rifiant 1a condition de Lopatinski uniforme; Deuxi me partie: Propagation et r flexion des singularit s.- L. G rding: Introduction to hyperbolicity.- T. Kato: Linear and quasi-linear equations of evolution of hyperbolic type.- K.W. Morton: Numerical methods for non-linear hyperbolic equations of mathematical physics.- Seminars: H. Brezis: First-order quasilinear equation on a torus.

Smooth ergodic theory of deterministic dynamical systems deals with the study of dynamical behaviors relevant to certain invariant measures under differentiable mappingsor ows. The relevance of invariantmeasures is that they describe the f- quencies of visits for an orbit and hence they give a probabilistic description of the evolution of a dynamical system. The fact that the system is differentiable allows one to use techniques from analysis and geometry. The study of transformationsand their long-termbehavior is ubiquitousin ma- ematics and the sciences. They arise not only in applications to the real world but also to diverse mathematical disciplines, including number theory, Lie groups, - gorithms, Riemannian geometry, etc. Hence smooth ergodic theory is the meeting ground of many different ideas in pure and applied mathematics. It has witnessed a great progress since the pioneering works of Sinai, Ruelle and Bowen on Axiom A diffeomorphisms and of Pesin on non-uniformly hyperbolic systems, and now it becomes a well-developed eld.

In recent years, scientists have applied the principles of complex systems science to increasingly diverse fields. The results have been nothing short of remarkable: their novel approaches have provided answers to long-standing questions in biology, ecology, physics, engineering, computer science, economics, psychology and sociology. "Unifying Themes in Complex Systems" is a well established series of carefully edited conference proceedings that serve the purpose of documenting and archiving the progress of cross-fertilization in this field.

About NECSI: For over 10 years, The New England Complex Systems Institute (NECSI) has been instrumental in the development of complex systems science and its applications. NECSI conducts research, education, knowledge dissemination, and community development around the world for the promotion of the study of complex systems and its application for the betterment of society. NECSI hosts the International Conference on Complex Systems and publishes the NECSI Book Series in conjunction with Springer Publishers.

Vector?eldsonmanifoldsplaymajorrolesinmathematicsandothersciences. In particular, the Poincar' e-Hopf index theorem and its geometric count- part,the Gauss-Bonnettheorem, giveriseto the theoryof Chernclasses,key invariants of manifolds in geometry and topology. One has often to face problems where the underlying space is no more a manifold but a singular variety. Thus it is natural to ask what is the "good" notionofindexofavector?eld,andofChernclasses,ifthespaceacquiress- gularities.Thequestionwasexploredbyseveralauthorswithvariousanswers, starting with the pioneering work of M.-H. Schwartz and R. MacPherson. We present these notions in the framework of the obstruction theory and the Chern-Weil theory. The interplay between these two methods is one of the main features of the monograph. Marseille Jean-Paul Brasselet Cuernavaca Jos' e Seade Tokyo Tatsuo Suwa September 2009 v Acknowledgements Parts of this monograph were written while the authors were staying at various institutions, such as Hokkaido University and Niigata University in Japan, CIRM, Universit' e de la Mediterran' ee and IML at Marseille, France, the Instituto de Matem' aticas of UNAM at Cuernavaca, Mexico, ICTP at Trieste, Italia, IMPA at Rio de Janeiro, and USP at S" ao Carlos in Brasil, to name a few, and we would like to thank them for their generous hospitality and support. Thanks are also due to people who helped us in many ways, in particular our co-authors of results quoted in the book: Marcelo Aguilar, Wolfgang Ebeling, Xavier G' omez-Mont, Sabir Gusein-Zade, LeDung " Tran ' g, Daniel Lehmann, David Massey, A.J. Parameswaran, Marcio Soares, Mihai Tibar, Alberto Verjovsky,andmanyother colleagueswho helped usin variousways.

How can our societies be stabilized in a crisis? Why can we enjoy and understand Shakespeare? Why are fruitflies uniform? How do omnivorous eating habits aid our survival? What makes the Mona Lisa s smile beautiful? How do women keep our social structures intact? Could there possibly be a single answer to all these questions? This book shows that the statement: "weak links stabilize complex systems" provides the key to understanding each of these intriguing puzzles, and many others too. The author (recipient of several distinguished science communication prizes) uses weak (low affinity, low probability) interactions as a thread to introduce a vast variety of networks from proteins to economics and ecosystems. Many people, from Nobel Laureates to high-school students have helped to make the book understandable to all interested readers. This unique book and the ideas it develops will have a significant impact on many, seemingly diverse, fields of study."

Evolution is a critical challenge for many areas of science, technology and development of society. The book reviews general evolutionary facts such as origin of life and evolution of the genome and clues to evolution through simple systems. Emerging areas of science such as "systems biology" and "bio-complexity" are founded on the idea that phenomena need to be understood in the context of highly interactive processes operating at different levels and on different scales. This is where physics meets complexity in nature, and where we must begin to learn about complexity if we are to understand it. Similarly, there is an increasingly urgent need to understand and predict the evolutionary behavior of highly interacting man-made systems, in areas such as communications and transport, which permeate the modern world. The same applies to the evolution of human networks such as social, political and financial systems, where technology has tended to vastly increase both the complexity and speed of interaction, which is sometimes effectively instantaneous. The book contains reviews on such diverse areas as evolution experiments with microorganisms, the origin and evolution of viruses, evolutionary dynamics of genes and environment in cancer development, aging as an evolution-facilitating program, evolution of vision and evolution of financial markets.

In the study of mathematical models that arise in the context of concrete - plications, the following two questions are of fundamental importance: (i) we- posedness of the model, including existence and uniqueness of solutions; and (ii) qualitative properties of solutions. A positive answer to the ?rst question, - ing of prime interest on purely mathematical grounds, also provides an important test of the viability of the model as a description of a given physical phenomenon. An answer or insight to the second question provides a wealth of information about the model, hence about the process it describes. Of particular interest are questions related to long-time behavior of solutions. Such an evolution property cannot be v- i?ed empirically, thus any in a-priori information about the long-time asymptotics can be used in predicting an ultimate long-time response and dynamical behavior of solutions. In recent years, this set of investigations has attracted a great deal of attention. Consequent efforts have then resulted in the creation and infusion of new methods and new tools that have been responsible for carrying out a successful an- ysis of long-time behavior of several classes of nonlinear PDEs.

Based on lecture notes of two summer schools with a mixed audience from mathematical sciences, epidemiology and public health, this volume offers a comprehensive introduction to basic ideas and techniques in modeling infectious diseases, for the comparison of strategies to plan for an anticipated epidemic or pandemic, and to deal with a disease outbreak in real time. It covers detailed case studies for diseases including pandemic influenza, West Nile virus, and childhood diseases. Models for other diseases including Severe Acute Respiratory Syndrome, fox rabies, and sexually transmitted infections are included as applications. Its chapters are coherent and complementary independent units. In order to accustom students to look at the current literature and to experience different perspectives, no attempt has been made to achieve united writing style or unified notation.

Notes on some mathematical background (calculus, matrix algebra, differential equations, and probability) have been prepared and may be downloaded at the web site of the Centre for Disease Modeling (www.cdm.yorku.ca).

The papers collected in this volume are contributions to the 43rd session of the Seminaire de mathematiques superieures (SMS) on "Morse Theoretic Methods in Nonlinear Analysis and Symplectic Topology." This session took place at the Universite de Montreal in July 2004 and was a NATO Advanced Study Institute (ASI). The aim of the ASI was to bring together young researchers from various parts of the world and to present to them some of the most signi cant recent advances in these areas. More than 77 mathematicians from 17 countries followed the 12 series of lectures and participated in the lively exchange of ideas. The lectures covered an ample spectrum of subjects which are re ected in the present volume: Morse theory and related techniques in in nite dim- sional spaces, Floer theory and its recent extensions and generalizations, Morse and Floer theory in relation to string topology, generating functions, structure of the group of Hamiltonian di?eomorphisms and related dynamical problems, applications to robotics and many others. We thank all our main speakers for their stimulating lectures and all p- ticipants for creating a friendly atmosphere during the meeting. We also thank Ms. Diane Belanger, our administrative assistant, for her help with the organi- tion and Mr. Andre Montpetit, our technical editor, for his help in the preparation of the volume."

This book provides a challenging and stimulating introduction to the contemporary topics of complexity and criticality, and explores their common basis of scale invariance, a central unifying theme of the book.Criticality refers to the behaviour of extended systems at a phase transition where scale invariance prevails. The many constituent microscopic parts bring about macroscopic phenomena that cannot be understood by considering a single part alone. The phenomenology of phase transitions is introduced by considering percolation, a simple model with a purely geometrical phase transition, thus enabling the reader to become intuitively familiar with concepts such as scale invariance and renormalisation. The Ising model is then introduced, which captures a thermodynamic phase transition from a disordered to an ordered system as the temperature is lowered in zero external field. By emphasising analogies between percolation and the Ising model, the reader's intuition of phase transitions is developed so that the underlying theoretical formalism may be appreciated fully. These equilibrium systems undergo a phase transition only if an external agent finely tunes certain external parameters to particular values.Besides fractals and phase transitions, there are many examples in Nature of the emergence of such complex behaviour in slowly driven non-equilibrium systems: earthquakes in seismic systems, avalanches in granular media and rainfall in the atmosphere. A class of non-equilibrium systems, not constrained by having to tune external parameters to obtain critical behaviour, is addressed in the framework of simple models, revealing that the repeated application of simple rules may spontaneously give rise to emergent complex behaviour not encoded in the rules themselves. The common basis of complexity and criticality is identified and applied to a range of non-equilibrium systems. Finally, the reader is invited to speculate whether self-organisation in non-equilibrium systems might be a unifying concept for disparate fields such as statistical mechanics, geophysics and atmospheric physics.Visit for animations for the models in the book (available for Windows and Linux), solutions to exercises, as well as a list with corrections.

Almost all process systems are nonlinear in nature. Nonlinear control is traditionally an area of interest in process systems engineering which is of great practical importance. These facts notwithstanding, many process engineers have difficulty with the paradigms and results of modern nonlinear control theory because they lack the mathematical background usually associated with such methods or because of their computational difficulty and small-scale applicability in the general case. Analysis and Control of Nonlinear Process Systems overcomes these barriers.

Features:

a [ The necessary mathematical preliminaries for readers from a process engineering background.

a [ Constant reference to the widely-known finite-dimensional linear time-invariant continuous case as a basis for extension to the nonlinear situation.

a [ The most promising theories and analytical methods for nonlinear process control laid out clearly and straightforwardly with exercises to reaffirm the techniques as they are taught.

a [ Emphasis on the importance of process knowledge and first-principles-based models in obtaining feasible and effective solutions in particular circumstances from general cases.

a [ Illustration of applications with simple examples and case studies.

Analysis and Control of Nonlinear Process Systems will interest graduate process engineers wishing to study advanced control methods either with a view to further research or application in industry as well as to academics seeking to move process control courses into more complicated but up-to-date territory. It will also be a great assistance to those in their senior undergraduate years who will form the next generation ofindustrial process engineers and need unfussy access to the most modern nonlinear control ideas.

This book contains the proceedings of the conference "Fractals in Graz 2001 - Analysis, Dynamics, Geometry, Stochastics" that was held in the second week of June 2001 at Graz University of Technology, in the capital of Styria, southeastern province of Austria. The scientific committee of the meeting consisted of M. Barlow (Vancouver), R. Strichartz (Ithaca), P. Grabner and W. Woess (both Graz), the latter two being the local organizers and editors of this volume. We made an effort to unite in the conference as well as in the present pro ceedings a multitude of different directions of active current work, and to bring together researchers from various countries as well as research fields that all are linked in some way with the modern theory of fractal structures. Although (or because) in Graz there is only a very small group working on fractal structures, consisting of "non-insiders," we hope to have been successful with this program of wide horizons. All papers were written upon explicit invitation by the editors, and we are happy to be able to present this representative panorama of recent work on poten tial theory, random walks, spectral theory, fractal groups, dynamic systems, fractal geometry, and more. The papers presented here underwent a refereeing process."

Consisting of 16 refereed original contributions, this volume presents a diversified collection of recent results in control of distributed parameter systems. Topics addressed include - optimal control in fluid mechanics - numerical methods for optimal control of partial differential equations - modeling and control of shells - level set methods - mesh adaptation for parameter estimation problems - shape optimization Advanced graduate students and researchers will find the book an excellent guide to the forefront of control and estimation of distributed parameter systems.

This volume is based on the course notes of the 2nd NCN Pedagogical School, the second in the series of Pedagogical Schools in the frame work of the European TMR project, "Breakthrough in the control of nonlinear systems (Nonlinear Control Network)". The school consists of four courses that have been chosen to give a broad range of techniques for the analysis and synthesis of nonlinear control systems, and have been developed by leading experts in the field. The topics covered are: Differential Algebraic Methods in Nonlinear Systems; Nonlinear QFT; Hybrid Systems; Physics in Control.The book has a pedagogical character, and is specially directed to postgraduates in most areas of engineering and applied sciences like mathematics and physics. It will also be of interest to researchers and practitioners needing a solid introduction to the above topics.

Control of nonlinear systems, one of the most active research areas in control theory, has always been a domain of natural convergence of research interests in applied mathematics and control engineering. The theory has developed from the early phase of its history, when the basic tool was essentially only the Lyapunov second method, to the present day, where the mathematics ranges from differential geometry, calculus of variations, ordinary and partial differential equations, functional analysis, abstract algebra and stochastic processes, while the applications to advanced engineering design span a wide variety of topics, which include nonlinear controllability and observability, optimal control, state estimation, stability and stabilization, feedback equivalence, motion planning, noninteracting control, disturbance attenuation, asymptotic tracking. The reader will find in the book methods and results which cover a wide variety of problems: starting from pure mathematics (like recent fundamental results on (non)analycity of small balls and the distance function), through its applications to all just mentioned topics of nonlinear control, up to industrial applications of nonlinear control algorithms.

This book combines real problems of practical interest with an application of profound theory. The mathematical model is derived step by step on the basis of physical principles, and the physics behind the control problems serves as a basis for the controller design. The book demonstrates how the physics behind the mathematical models can help to successfully apply a certain control strategy. The book aims to show the practical relevance of the presented methods, methods which are often criticised as only of theoretical interest, through an examination of their industrial applications. Throughout, the book gives the unique mathematical formulation of the different disciplines involved, namely electrical, hydraulic and mechanical engineering. Yet it also points out the common mathematical structure of the different physical models. This makes it possible to transfer reliable control strategies between the disciplines.

Control of nonlinear systems, one of the most active research areas in control theory, has always been a domain of natural convergence of research interests in applied mathematics and control engineering. The theory has developed from the early phase of its history, when the basic tool was essentially only the Lyapunov second method, to the present day, where the mathematics ranges from differential geometry, calculus of variations, ordinary and partial differential equations, functional analysis, abstract algebra and stochastic processes, while the applications to advanced engineering design span a wide variety of topics, which include nonlinear controllability and observability, optimal control, state estimation, stability and stabilization, feedback equivalence, motion planning, noninteracting control, disturbance attenuation, asymptotic tracking. The reader will find in the book methods and results which cover a wide variety of problems: starting from pure mathematics (like recent fundamental results on (non)analycity of small balls and the distance function), through its applications to all just mentioned topics of nonlinear control, up to industrial applications of nonlinear control algorithms.

This book examines the control problem for wheeled mobile robots. Several novel control strategies are developed and the stability of each controller is examined utilizing Lyapunov techniques. The performance of each controller is either illustrated through simulation results or experimental results. The final chapter describes how the control techniques developed for wheeled mobile robots can be applied to solve other problems with similar governing differential equations (e.g., twin rotor helicopters, surface vessels). Several appendices are included to provide the reader with the mathematical background utilized in the control development and stability analysis. Two appendices are also included that provide specific details with regard to the modifications that were done to commercially available mobile robots (e.g., a K2A manufactured by Cybermotion Inc. and a Pioneer II manufactured by Activemedia) to experimentally demonstrate the performance of the torque input controllers.

Chaos occurs widely in both natural and man-made systems. Recently, examples of the potential usefulness of chaotic behavior have caused growing interest among engineers and applied scientists. In this book the new mathematical ideas in nonlinear dynamics are described in such a way that engineers can apply them to real physical systems.
From a review of the first edition by Prof. El Naschie, University of Cambridge: "Small is beautiful and not only that, it is comprehensive as well. These are the spontaneous thoughts which came to my mind after browsing in this latest book by Prof. Thomas Kapitaniak, probably one of the most outstanding scientists working on engineering applications of Nonlinear Dynamics and Chaos today. A more careful reading reinforced this first impression....The presentation is lucid and user friendly with theory, examples, and exercises."

Generalized method of moments (GMM) estimation of nonlinear systems has two important advantages over conventional maximum likelihood (ML) estimation: GMM estimation usually requires less restrictive distributional assumptions and remains computationally attractive when ML estimation becomes burdensome or even impossible. This book presents an in-depth treatment of the conditional moment approach to GMM estimation of models frequently encountered in applied microeconometrics. It covers both large sample and small sample properties of conditional moment estimators and provides an application to empirical industrial organization. With its comprehensive and up-to-date coverage of the subject which includes topics like bootstrapping and empirical likelihood techniques, the book addresses scientists, graduate students and professionals in applied econometrics.

Flow line design is one of the major tasks in production management. The decision to install a set of machines and buffers is often highly irreversible. It determines both cost and revenue to a large extent. In order to assess the economic impact of any possible flow line design, production rates and inventory levels have to be estimated. These performance measures depend on the allocation of buffers whenever the flow of material is occasionally disrupted, for example due to machine failures or quality problems. The book describes analytical methods that can be used to evaluate flow lines much faster than with simulation techniques. Based on these fast analytical techniques, it is possible to determine a flow line design that maximizes the net present value of the flow line investment. The flow of material through the line may be non-linear, for example due to assembly operations or quality inspections.

The past decade has witnessed an increasing interest in observers for nonlinear systems. This subject is relevant in different contexts such as synchronization of complex dynamical systems, fault detection and isolation, and output feedback control. This book contains the contributions that are to be presented at the workshop "New Directions in Nonlinear Observer Design", to be held from June 24-26, 1999, in Geiranger Fjord, Norway. The workshop has been organised by Olav Egeland, Thor I. Fossen and Henk Nijmeijer; it will include participants from Africa, Asia, Europe and USA and it will focus on recent developments in the above mentioned areas. The contributions form a good review of present achievements and challenges in nonlinear observer design. The workshop is supported by the Strategic University Program on Marine Cybernetics at the Norwegian University of Science and Technology and ABB.

This book contains several contemporary topics in the areas of mathematical modelling and computation for complex systems. The readers find several new mathematical methods, mathematical models and computational techniques having significant relevance in studying various complex systems. The chapters aim to enrich the understanding of topics presented by carefully discussing the associated problems and issues, possible solutions and their applications or relevance in other scientific areas of study and research. The book is a valuable resource for graduate students, researchers and educators in understanding and studying various new aspects associated with complex systems. Key Feature * The chapters include theory and application in a mix and balanced way. * Readers find reasonable details of developments concerning a topic included in this book. * The text is emphasized to present in self-contained manner with inclusion of new research problems and questions.