The first systematic presentation of the theory of dynamical systems under the influence of randomness, this book includes products of random mappings as well as random and stochastic differential equations. The basic multiplicative ergodic theorem is presented, providing a random substitute for linear algebra. On its basis, many applications are detailed. Numerous instructive examples are treated analytically or numerically.

The world around us, natural or man-made, is built and held together by solid materials. Understanding their behaviour is the task of solid mechanics, which is in turn applied to many areas, from earthquake mechanics to industry, construction to biomechanics. The variety of materials (metals, rocks, glasses, sand, flesh and bone) and their properties (porosity, viscosity, elasticity, plasticity) is reflected by the concepts and techniques needed to understand them: a rich mixture of mathematics, physics and experiment. These are all combined in this unique book, based on years of experience in research and teaching. Starting from the simplest situations, models of increasing sophistication are derived and applied. The emphasis is on problem-solving and building intuition, rather than a technical presentation of theory. The text is complemented by over 100 carefully-chosen exercises, making this an ideal companion for students taking advanced courses, or those undertaking research in this or related disciplines.

Statistical physics concepts such as stochastic dynamics, short- and long-range correlations, self-similarity and scaling, permit an understanding of the global behavior of economic systems without first having to work out a detailed microscopic description of the system. This pioneering text explores the use of these concepts in the description of financial systems, the dynamic new specialty of econophysics. The authors illustrate the scaling concepts used in probability theory, critical phenomena, and fully-developed turbulent fluids and apply them to financial time series. They also present a new stochastic model that displays several of the statistical properties observed in empirical data. Physicists will find the application of statistical physics concepts to economic systems fascinating. Economists and other financial professionals will benefit from the book's empirical analysis methods and well-formulated theoretical tools that will allow them to describe systems composed of a huge number of interacting subsystems.

This book offers a fundamental explanation of nonlinear oscillations in physical systems. Originally intended for electrical engineers, it remains an important reference for the increasing numbers of researchers studying nonlinear phenomena in physics, chemical engineering, biology, medicine, and other fields.

Originally published in 1986.

The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These paperback editions preserve the original texts of these important books while presenting them in durable paperback editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Les oscillations complexes mises en evidence dans les systemes physiologiques s'analysent par des modeles. Cet ouvrage se propose de presenter et de developper les mathematiques necessaires a leur comprehension. On presente en particulier les notions d'excitabilite, de bistabilite, de synchronisation et d'oscillations en salves dans le cadre de l'analyse qualitative."

Solitons are waves with exceptional stability properties which appear in many areas of physics. The basic properties of solitons are introduced here using examples from macroscopic physics (e.g. blood pressure pulses and fibre optical communications). The book then presents the main theoretical methods before discussing applications from solid state or atomic physics such as dislocations, excitations in spin chains, conducting polymers, ferroelectrics and Bose-Einstein condensates. Examples are also taken from biological physics and include energy transfer in proteins and DNA fluctuations. Throughout the book the authors emphasise a fresh approach to modelling nonlinearities in physics. Instead of a perturbative approach, nonlinearities are treated intrinsically and the analysis based on the soliton equations introduced in this book. Based on the authors' graduate course, this textbook gives an instructive view of the physics of solitons for students with a basic knowledge of general physics, and classical and quantum mechanics.

The book provides a general introduction to the theory of large deviations and a wide overview of the metastable behaviour of stochastic dynamics. With only minimal prerequisites, the book covers all the main results and brings the reader to the most recent developments. Particular emphasis is given to the fundamental Freidlin-Wentzell results on small random perturbations of dynamical systems. Metastability is first described on physical grounds, following which more rigorous approaches to its description are developed. Many relevant examples are considered from the point of view of the so-called pathwise approach. The first part of the book develops the relevant tools including the theory of large deviations which are then used to provide a physically relevant dynamical description of metastability. Written to be accessible to graduate students, this book provides an excellent route into contemporary research.

The techniques that can be used to solve non-linear problems are far different than those that are used to solve linear problems. Many courses in analysis and applied mathematics attack linear cases simply because they are easier to solve and do not require a large theoretical background in order to approach them. Professor Schechter's 2005 book is devoted to non-linear methods using the least background material possible and the simplest linear techniques. An understanding of the tools for solving non-linear problems is developed whilst demonstrating their application to problems in one dimension and then leading to higher dimensions. The reader is guided using simple exposition and proof, assuming a minimal set of pre-requisites. For completion, a set of appendices covering essential basics in functional analysis and metric spaces is included, making this ideal as an accompanying text on an upper-undergraduate or graduate course, or even for self-study.

The time variability of many natural and social phenomena is not well described by standard methods of data analysis. However, nonlinear time series analysis uses chaos theory and nonlinear dynamics to understand seemingly unpredictable behavior. The results are applied to real data from physics, biology, medicine, and engineering in this volume. Researchers from all experimental disciplines, including physics, the life sciences, and the economy, will find the work helpful in the analysis of real world systems. First Edition Hb (1997): 0-521-55144-7 First Edition Pb (1997): 0-521-65387-8

This book presents a unified approach for obtaining the limiting distributions of minimum distance, M and R estimators corresponding to non-smooth underlying scores in a large class of dynamic non-linear models including ARCH models. It also discusses classes of goodness-of-t tests for fitting an error distribution in some of these models and/or fitting a regression-autoregressive function without assuming the knowledge of the error distribution. The main tool is the asymptotic equicontinuity of certain basic weighted residual empirical processes in the uniform and L2 metrics. The contents of this monograph should be useful to graduate students and research scholars in statistics, econometrics, and finance. This book is a an updated edition of the author's monograph Weighted Empirical Processes and Liner Models (IMS Lecture Notes-Monograph 21, 1992). The new edition differs from the previous one in many ways. To mention just a few: It includes asymptotically distribution free tests for fitting a regression and/or an autoregressive models; the asymptotic distributions of auto-regression quantiles and rank scores; and above all the weak convergence of the residual empirical processes useful in nonlinear ARCH models. Hira L. Koul is a professor of statistics at Michigan State University. He is a Fellow of the IMS and an Elected Member of the International Statistical Institute. He was awarded the prestigious Humboldt Research Award for Senior Researchers in 1995. He has been on the editorial boards of the Annals of Statistics, Sankhya, and J. Indian Statistical Association. Currently he is a Coordinating Editor of the Journal of Statistical Planning and Inference, and an Associate Editor of Statistics and Probability Letters.

The theory of dynamical systems has given rise to the vast new area variously called applied dynamics, nonlinear science, or chaos theory. This introductory text covers the central topological and probabilistic notions in dynamics ranging from Newtonian mechanics to coding theory. The only prerequisite is a basic undergraduate analysis course. The authors use a progression of examples to present the concepts and tools for describing asymptotic behavior in dynamical systems, gradually increasing the level of complexity. Subjects include contractions, logistic maps, equidistribution, symbolic dynamics, mechanics, hyperbolic dynamics, strange attractors, twist maps, and KAM-theory.

This book provides a broad introduction to the subject of dynamical systems, suitable for a one or two-semester graduate course. In the first chapter, the authors introduce over a dozen examples, and then use these examples throughout the book to motivate and clarify the development of the theory. Topics include topological dynamics, symbolic dynamics, ergodic theory, hyperbolic dynamics, one-dimensional dynamics, complex dynamics, and measure-theoretic entropy. The authors top off the presentation with some beautiful and remarkable applications of dynamical systems to areas such as number theory, data storage, and internet search engines.

This book is a text/monograph designed to provide an overview of optimal computational methods for the solution of nonlinear equations, fixed points of contractive and noncontractive mapping, and for the computation of the topological degree. The worst-case settings are analysed here. Several classes of functions are studied with special empahsis on tight complexity bounds and methods which are close to or achieve these bounds. Each chapter ends with exercises, including companies and open-ended research based exercises.

This is a comprehensive introduction to the exciting scientific field of nonlinear dynamics for students, scientists, and engineers, and requires only minimal prerequisites in physics and mathematics. The book treats all the important areas in the field and provides an extensive and up-to-date bibliography of applications in all fields of science, social science, economics, and even the arts.

Statistical physics concepts such as stochastic dynamics, short- and long-range correlations, self-similarity and scaling, permit an understanding of the global behavior of economic systems without first having to work out a detailed microscopic description of the system. This pioneering text explores the use of these concepts in the description of financial systems, the dynamic new specialty of econophysics. The authors illustrate the scaling concepts used in probability theory, critical phenomena, and fully-developed turbulent fluids and apply them to financial time series. They also present a new stochastic model that displays several of the statistical properties observed in empirical data. Physicists will find the application of statistical physics concepts to economic systems fascinating. Economists and other financial professionals will benefit from the book's empirical analysis methods and well-formulated theoretical tools that will allow them to describe systems composed of a huge number of interacting subsystems.

Complex behavior can occur in any system made up of large numbers of interacting constituents, be they atoms in a solid, cells in a living organism, or consumers in a national economy. Analysis of this behavior often involves making important assumptions and approximations, the exact nature of which vary from subject to subject. Foundations of Complex-system Theories begins with a description of the general features of complexity and then examines a range of important concepts, such as theories of composite systems, collective phenomena, emergent properties, and stochastic processes. Each topic is discussed with reference to the fields of statistical physics, evolutionary biology, and economics, thereby highlighting recurrent themes in the study of complex systems. This detailed yet nontechnical book will appeal to anyone who wants to know more about complex systems and their behavior. It will also be of great interest to specialists studying complexity in the physical, biological, and social sciences.

This monograph grew out of the authors' efforts to provide a natural geometric description for the class of maps invariant under parabolic renormalization and for the Inou-Shishikura fixed point itself as well as to carry out a computer-assisted study of the parabolic renormalization operator. It introduces a renormalization-invariant class of analytic maps with a maximal domain of analyticity and rigid covering properties and presents a numerical scheme for computing parabolic renormalization of a germ, which is used to compute the Inou-Shishikura renormalization fixed point. Inside, readers will find a detailed introduction into the theory of parabolic bifurcation, Fatou coordinates, Ecalle-Voronin conjugacy invariants of parabolic germs, and the definition and basic properties of parabolic renormalization. The systematic view of parabolic renormalization developed in the book and the numerical approach to its study will be interesting to both experts in the field as well as graduate students wishing to explore one of the frontiers of modern complex dynamics.

Un systeme dynamique discret est un ensemble fini d'elements, prenant chacun un nombre fini d'etats, et evoluant, dans un temps discret, par interactions mutuelles. Ce livre est consacre a l'analyse de la dynamique temporelle de tels systemes. Grace a des outils de metrique discrete, on etablit des resultats de convergence globale (contraction booleenne) convergence locale vers un point fixe ou vers un cycle, et ceci pour differents modes operatoires.

Dynamics Reported reports on recent developments in dynamical systems theory. Dynamical systems theory of course originated from ordinary differential equations. Today, dynamical systems theory covers a much larger area, including dynamical processes described by functional and integral equations, by partial and stochastic differential equations, etc. Dynamical systems theory has evolved remarkably rapidly in the recent years. A wealth of new phenomena, new ideas and new techniques proved to be of considerable interest to scientists in rather different fields. It is not surprising that thousands of publications on the theory itself and on its various applications have appeared and still will appear. Dynamics Reported presents carefully written articles on major subjects in dynamical systems and their applications, addressed not only to specialists but also to a broader range of readers. Topics are advanced while detailed exposition of ideas, restriction to typical results, rather than to the most general ones, and last but not least lucid proofs help to gain an utmost degree of clarity. It is hoped that Dynamics Reported will stimulate exchange of ideas among those working in dynamical systems and moreover will be useful for those entering the field.

A modern introduction to synchronization phenomena, this text presents recent discoveries and the current state of research in the field, from low-dimensional systems to complex networks. The book describes some of the main mechanisms of collective behaviour in dynamical systems, including simple coupled systems, chaotic systems, and systems of infinite-dimension. After introducing the reader to the basic concepts of nonlinear dynamics, the book explores the main synchronized states of coupled systems and describes the influence of noise and the occurrence of synchronous motion in multistable and spatially-extended systems. Finally, the authors discuss the underlying principles of collective dynamics on complex networks, providing an understanding of how networked systems are able to function as a whole in order to process information, perform coordinated tasks, and respond collectively to external perturbations. The demonstrations, numerous illustrations and application examples will help advanced graduate students and researchers gain an organic and complete understanding of the subject.

Mathematical Modeling for Epidemiology and Ecology provides readers with the mathematical tools needed to understand and use mathematical models and read advanced mathematical biology books. It presents mathematics in biological contexts, focusing on the central mathematical ideas and the biological implications, with detailed explanations. The author assumes no mathematics background beyond elementary differential calculus. An introductory chapter on basic principles of mathematical modeling is followed by chapters on empirical modeling and mechanistic modeling. These chapters contain a thorough treatment of key ideas and techniques that are often neglected in mathematics books, such as the Akaike Information Criterion. The second half of the book focuses on analysis of dynamical systems, emphasizing tools to simplify analysis, such as the Routh-Hurwitz conditions and asymptotic analysis. Courses can be focused on either half of the book or thematically chosen material from both halves, such as a course on mathematical epidemiology. The biological content is self-contained and includes many topics in epidemiology and ecology. Some of this material appears in case studies that focus on a single detailed example, and some is based on recent research by the author on vaccination modeling and scenarios from the COVID-19 pandemic. The problem sets feature linked problems where one biological setting appears in multi-step problems that are sorted into the appropriate section, allowing readers to gradually develop complete investigations of topics such as HIV immunology and harvesting of natural resources. Some problems use programs written by the author for Matlab or Octave; these combine with more traditional mathematical exercises to give students a full set of tools for model analysis. Each chapter contains additional case studies in the form of projects with detailed directions. New appendices contain mathematical details on optimization, numerical solution of differential equations, scaling, linearization, and sophisticated use of elementary algebra to simplify problems.

This introductory textbook/reference addresses the fundamental and mostly applied kinds of models. The focus is on models of dynamic systems that move and change over time. However, the work also proposes new methods of uncertainty treatment, offering supporting examples. Topics and features: Chapters suitable for textbook use in teaching modeling and simulation Includes sections of questions and answers, helpful in didactic work Proposes new methodology in addition to examining conventional approaches Offers some cognitive, more abstract models to give a wider insight on model building The book's readership may consist of researchers working on multidisciplinary problems, as well educators and students. It may be used while teaching computer simulation, applied mathematics, system analysis and system dynamics.

At a first glance the reader of this book might be puzzled by the variety of its topics which range from phase-transition-like phenomena of chemical reactions, lasers and electrical currents to biological systems, like neuron networks and membranes, to population dynamics and socio logy. When looking more closely at the different subjects the reader will recognize, however, that this book deals with one main problem: the behaviour of systems which are composed of many elements of one or a few kinds. We are sure the reader will be surprised in the same way as the participants of a recent symposium on synergetics, who recognized that such systems have amazingly common features. Though the subsystems (e. g. electrons, cells, human beings) are quite different in nature, their joint action is governed by only a few principles which lead to strikingly similar phenomena. It hardly needs to be mentioned that once such common principles are established, they are of an enormous stimulus and help for future research. Though the articles of this book are based on invited papers given at the first International Symposium on Synergetics at Schlof. l. Elmau from April 30 to May 6, 1972, it differs from usual conference proceedings in a distinct way. The authors and subjects were chosen from the very beginning so that fmally a well organized total book arises. We hope that the reader will feel the same pleasure and enthusiasm the participants at the symposium had."

Nonlinear Structures & Systems, Volume 1: Proceedings of the 37th IMAC, A Conference and Exposition on Structural Dynamics, 2019, the first volume of eight from the Conference brings together contributions to this important area of research and engineering. The collection presents early findings and case studies on fundamental and applied aspects of Nonlinear Dynamics, including papers on: Nonlinear Reduced-order Modeling Jointed Structures: Identification, Mechanics, Dynamics Experimental Nonlinear Dynamics Nonlinear Model & Modal Interactions Nonlinear Damping Nonlinear Modeling & Simulation Nonlinearity & System Identification

Divorce rates are at an all-time high. But without a theoretical understanding of the processes related to marital stability and dissolution, it is difficult to design and evaluate new marriage interventions. The Mathematics of Marriage provides the foundation for a scientific theory of marital relations. The book does not rely on metaphors, but develops and applies a mathematical model using difference equations. The work is the fulfillment of the goal to build a mathematical framework for the general system theory of families first suggested by Ludwig Von Bertalanffy in the 1960s.The book also presents a complete introduction to the mathematics involved in theory building and testing, and details the development of experiments and models. In one "marriage experiment," for example, the authors explored the effects of lowering or raising a couple's heart rates. Armed with their mathematical model, they were able to do real experiments to determine which processes were affected by their interventions.Applying ideas such as phase space, null clines, influence functions, inertia, and uninfluenced and influenced stable steady states (attractors), the authors show how other researchers can use the methods to weigh their own data with positive and negative weights. While the focus is on modeling marriage, the techniques can be applied to other types of psychological phenomena as well.