The theory of Lyapunov exponents originated over a century ago in the study of the stability of solutions of differential equations. Written by one of the subject's leading authorities, this book is both an account of the classical theory, from a modern view, and an introduction to the significant developments relating the subject to dynamical systems, ergodic theory, mathematical physics and probability. It is based on the author's own graduate course and is reasonably self-contained with an extensive set of exercises provided at the end of each chapter. This book makes a welcome addition to the literature, serving as a graduate text and a valuable reference for researchers in the field.

Covering a broad range of topics, this text provides a comprehensive survey of the modeling of chaotic dynamics and complexity in the natural and social sciences. Its attention to models in both the physical and social sciences and the detailed philosophical approach make this a unique text in the midst of many current books on chaos and complexity. Including an extensive index and bibliography along with numerous examples and simplified models, this is an ideal course text.

The new discipline of chaotics will alter our thinking about the real forces of change in our society. As presented here, chaotics emphasizes that the real world cannot be understood in terms of conventional deterministic philosophies or standard chaos theory, but that complexity in itself has a powerful but subtle role to play. How does this apply to business and society? To what degree are our lives governed by misguided notions--or do our businesses succeed by chance--because real societal and business forces and their effects are not really understood? Beginning with the foundations of the discipline, this book applies chaotics to business and wealth creation and to society. On the social side, it examines a sea-change in the philosophy of everyday living, be it the concept of employment or our relationship to the environment. The book examines personal identity and its loss in modern society, as well as the search for new contacts and gratification through technology. The authors look at the stunted growth of philosophy against science but emphasize what philosophy has to tell us in a chaotic world. A major new text which will be of interest to professionals and scholars in business, government, and society.

Using phase-plane analysis, findings from the theory of topological horseshoes and linked-twist maps, this book presents a novel method to prove the existence of chaotic dynamics. In dynamical systems, complex behavior in a map can be indicated by showing the existence of a Smale-horseshoe-like structure, either for the map itself or its iterates. This usually requires some assumptions about the map, such as a diffeomorphism and some hyperbolicity conditions. In this text, less stringent definitions of a horseshoe have been suggested so as to reproduce some geometrical features typical of the Smale horseshoe, while leaving out the hyperbolicity conditions associated with it. This leads to the study of the so-called topological horseshoes. The presence of chaos-like dynamics in a vertically driven planar pendulum, a pendulum of variable length, and in other more general related equations is also proved.

This book starts with a discussion of nonlinear ordinary differential equations, bifurcation theory and Hamiltonian dynamics. It then embarks on a systematic discussion of the traditional topics of modern nonlinear dynamics -- integrable systems, Poincare maps, chaos, fractals and strange attractors. The Baker s transformation, the logistic map and Lorenz system are discussed in detail in view of their central place in the subject. There is a detailed discussion of solitons centered around the Korteweg-deVries equation in view of its central place in integrable systems. Then, there is a discussion of the Painleve property of nonlinear differential equations which seems to provide a test of integrability. Finally, there is a detailed discussion of the application of fractals and multi-fractals to fully-developed turbulence -- a problem whose understanding has been considerably enriched by the application of the concepts and methods of modern nonlinear dynamics. On the application side, there is a special emphasis on some aspects of fluid dynamics and plasma physics reflecting the author s involvement in these areas of physics. A few exercises have been provided that range from simple applications to occasional considerable extension of the theory. Finally, the list of references given at the end of the book contains primarily books and papers used in developing the lecture material this volume is based on.

This book has grown out of the author s lecture notes for an interdisciplinary graduate-level course on nonlinear dynamics. The basic concepts, language and results of nonlinear dynamical systems are described in a clear and coherent way. In order to allow for an interdisciplinary readership, an informal style has been adopted and the mathematical formalism has been kept to a minimum.

This book is addressed to first-year graduate students in applied mathematics, physics, and engineering, and is useful also to any theoretically inclined researcher in the physical sciences and engineering.

This second edition constitutes an extensive rewrite of the text involving refinement and enhancement of the clarity and precision, updating and amplification of several sections, addition of new material like theory of nonlinear differential equations, solitons, Lagrangian chaos in fluids, and critical phenomena perspectives on the fluid turbulence problem and many new exercises."

A billiard is a dynamical system in which a point particle alternates between free motion and specular reflections fromthe boundaryof a domain."Exterior Billiards" presents billiards in the complement of domains and their applications in aerodynamics and geometrical optics.

This book distinguishes itself from existing literature by presenting billiard dynamics "outside" bounded domains, including scattering, resistance, invisibility and retro-reflection. It begins with an overview of the mathematical notations used throughout the book and a brief review of the main results. Chapters 2 and 3 are focused on problems of minimal resistance and Newton s problem in media with positive temperature. In chapters 4 and 5, scattering of billiards bynonconvex and rough domains is characterized and some related special problems of optimal mass transportation are studied. Applications in aerodynamics are addressed next and problems of invisibility and retro-reflection within the framework of geometric optics conclude the text.

The book will appeal to mathematicians working in dynamical systems and calculus of variations. Specialists working in the areas of applications discussed will also find it useful."

This monograph presents key method to successfully manage the growing complexity of systems where conventional engineering and scientific methodologies and technologies based on learning and adaptability come to their limits and new ways are nowadays required. The transition from adaptable to evolvable and finally to self-evolvable systems is highlighted, self-properties such as self-organization, self-configuration, and self-repairing are introduced and challenges and limitations of the self-evolvable engineering systems are evaluated."

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 Mathematical Elasticity set contains three self-contained volumes that together provide the only modern treatise on elasticity. They introduce contemporary research on three-dimensional elasticity, the theory of plates, and the theory of shells. Each volume contains proofs, detailed surveys of all mathematical prerequisites, and many problems for teaching and self-study. An extended preface and extensive bibliography have been added to each volume to highlight the progress that has been made since the original publication. The first book, Three-Dimensional Elasticity, covers the modeling and mathematical analysis of nonlinear three-dimensional elasticity. In volume two, Theory of Plates, asymptotic methods provide a rigorous mathematical justification of the classical two-dimensional linear plate and shallow shell theories. The objective of Theory of Shells, the final volume, is to show how asymptotic methods provide a rigorous mathematical justification of the classical two-dimensional linear shell theories: membrane, generalized membrane, and flexural. These classic textbooks are for advanced undergraduates, first-year graduate students, and researchers in pure or applied mathematics or continuum mechanics. They are appropriate for courses in mathematical elasticity, theory of plates and shells, continuum mechanics, computational mechanics, and applied mathematics in general.

An electrifying introduction to complexity theory, the science of how complex systems behave, that explains the interconnectedness of all things and that Deepak Chopra says, “will change the way you understand yourself and the universe.� Nothing in the universe is more complex than life. Throughout the skies, in oceans, and across lands, life is endlessly on the move. In its myriad forms—from cells to human beings, social structures, and ecosystems--life is open-ended, evolving, unpredictable, yet adaptive and self-sustaining. Complexity theory addresses the mysteries that animate science, philosophy, and metaphysics: how this teeming array of existence, from the infinitesimal to the infinite, is in fact a seamless living whole and what our place, as conscious beings, is within it. Physician, scientist, and philosopher Neil Theise makes accessible this “theory of being,� one of the pillars of modern science, and its holistic view of human existence. He notes the surprising underlying connections within a universe that is itself one vast complex system—between ant colonies and the growth of forests, cancer and economic bubbles, murmurations of starlings and crowds walking down the street. The implications of complexity theory are profound, providing insight into everything from the permeable boundaries of our bodies to the nature of consciousness. Notes on Complexity is an invitation to trade our limited, individualistic view for the expansive perspective of a universe that is dynamic, cohesive, and alive—a whole greater than the sum of its parts. Theise takes us to the exhilarating frontiers of human knowledge and in the process restores wonder and meaning to our experience of the everyday.

Dissipative Quantum Chaos and Decoherence provides an overview of the state of the art of research in this exciting field. The main emphasis is on the development of a semiclassical formalism that allows one to incorporate the effect of dissipation and decoherence in a precise, yet tractable way into the quantum mechanics of classically chaotic systems. The formalism is employed to reveal how the spectrum of the quantum mechanical propagator of a density matrix is determined by the spectrum of the corresponding classical propagator of phase space density. Simple quantum--classical hybrid formulae for experimentally relevant correlation functions and time-dependent expectation values of observables are derived. The problem of decoherence is treated in detail, and highly unexpected cases of very slow decoherence are revealed, with important consequences for the long-debated realizability of Schrödinger cat states as well as for the construction of quantum computers.

The Mathematical Elasticity set contains three self-contained volumes that together provide the only modern treatise on elasticity. They introduce contemporary research on three-dimensional elasticity, the theory of plates, and the theory of shells. Each volume contains proofs, detailed surveys of all mathematical prerequisites, and many problems for teaching and self-study. An extended preface and extensive bibliography have been added to each volume to highlight the progress that has been made since the original publication. The first book, Three-Dimensional Elasticity, covers the modeling and mathematical analysis of nonlinear three-dimensional elasticity. In volume two, Theory of Plates, asymptotic methods provide a rigorous mathematical justification of the classical two-dimensional linear plate and shallow shell theories. The objective of Theory of Shells, the final volume, is to show how asymptotic methods provide a rigorous mathematical justification of the classical two-dimensional linear shell theories: membrane, generalized membrane, and flexural. These classic textbooks are for advanced undergraduates, first-year graduate students, and researchers in pure or applied mathematics or continuum mechanics. They are appropriate for courses in mathematical elasticity, theory of plates and shells, continuum mechanics, computational mechanics, and applied mathematics in general.

This book discusses dynamical systems that are typically driven by stochastic dynamic noise. It is written by two statisticians essentially for the statistically inclined readers, although readers whose primary interests are in determinate systems will find some of the methodology explained in this book of interest. The statistical approach adopted in this book differs in many ways from the deterministic approach to dynamical systems. Even the very basic notion of initial-value sensitivity requires careful development in the new setting provided. This book covers, in varying depth, many of the contributions made by the statisticians in the past twenty years or so towards our understanding of estimation, the Lyapunov-like index, the nonparametric regression, and many others, many of which are motivated by their dynamical system counterparts but have now acquired a distinct statistical flavour. Kung-Sik Chan is a professor at the University of Iowa, Department of Statistics and Actuarial Science. He is an elected member of the International Statistical Institute. He has served on the editorial boards of the Journal of Business and Economic Statistics and Statistica Sinica. He received a Faculty Scholar Award from the University of Iowa in 1996. Howell Tong holds the Chair of Statistics at the London School of Economics and the University of Hong Kong. He is a foreign member of the Norwegian Academy of Science and Letters, an elected member of the International Statistical Institute and a Council member of its Bernoulli Society, an elected fellow of the Institute of Mathematical Statistics, and an honorary fellow of the Institute of Actuaries (London). He was the Founding Dean of the Graduate School and sometimes the Acting Pro-Vice Chancellor (Research) at the University of Hong Kong. He has served on the editorial boards of several international journals, including Biometrika, Journal of Royal Statistical Society (Series B), Statistica Sinica, and others. He is a guest professor of the Academy of Mathematical and System Sciences of the Chinese Academy of Sciences and received a National Natural Science Prize (China) in the category of Mathematics and Mechanics (Class II) in 2001. He has also held visiting professorships at various universities, including the Imperial College in London, the ETH in Zurich, the Fourier University in Grenoble, the Wall Institute at the University of British Columbia, Vancouver, and the Chinese University of Hong Kong.

This book is the first systematic presentation of the theory of dynamical systems under the influence of randomness. It includes products of random mappings as well as random and stochastic differential equations. The basic mulitplicative ergodic theorem is presented and provides a random substitute for linear algebra. On its basis random invariant manifolds are constructed, systems are simplified by smooth random coordinate transformations (random normal forms), and qualitative changes in families of random systems (random bifurcation theory) are studied. Numerous instructive examples are treated analytically or numerically. The main intention, however, is to present a reliable and rather complete source of reference which lays the foundation for future work and applications.

This volume contains a selection of the most important papers in the theory of chaotic attractors over the past 40 years. It is dedicated to James Yorke - a pioneer in the field and a recipient of the 2003 Japan prize - on the occasion of his 60th birthday. The volume includes an introduction to Yorke's work and an overview of key developments in the theory of chaotic attractors.

This book presents and extend different known methods to solve different types of strong nonlinearities encountered by engineering systems. A better knowledge of the classical methods presented in the first part lead to a better choice of the so-called base functions . These are absolutely necessary to obtain the auxiliary functions involved in the optimal approaches which are presented in the second part.
Every chapter introduces a distinct approximate method applicable to nonlinear dynamical systems. Each approximate analytical approach is accompanied by representative examples related to nonlinear dynamical systems from to various fields of engineering."

This book gives a unified treatment of a variety of mathematical systems generating densities, ranging from one-dimensional discrete time transformations through continuous time systems described by integro-partial-differential equations. Examples have been drawn from a variety of the sciences to illustrate the utility of the techniques presented. This material was organized and written to be accessible to scientists with knowledge of advanced calculus and differential equations. In various concepts from measure theory, ergodic theory, the geometry of manifolds, partial differential equations, probability theory and Markov processes, and chastic integrals and differential equations are introduced. The past few years have witnessed an explosive growth in interest in physical, biological, and economic systems that could be profitably studied using densities. Due to the general inaccessibility of the mathematical literature to the non-mathematician, there has been little diffusion of the concepts and techniques from ergodic theory into the study of these "chaotic" systems. This book intends to bridge that gap.

The Mathematical Elasticity set contains three self-contained volumes that together provide the only modern treatise on elasticity. They introduce contemporary research on three-dimensional elasticity, the theory of plates, and the theory of shells. Each volume contains proofs, detailed surveys of all mathematical prerequisites, and many problems for teaching and self-study. An extended preface and extensive bibliography have been added to each volume to highlight the progress that has been made since the original publication. The first book, Three-Dimensional Elasticity, covers the modeling and mathematical analysis of nonlinear three-dimensional elasticity. In volume two, Theory of Plates, asymptotic methods provide a rigorous mathematical justification of the classical two-dimensional linear plate and shallow shell theories. The objective of Theory of Shells, the final volume, is to show how asymptotic methods provide a rigorous mathematical justification of the classical two-dimensional linear shell theories: membrane, generalized membrane, and flexural. These classic textbooks are for advanced undergraduates, first-year graduate students, and researchers in pure or applied mathematics or continuum mechanics. They are appropriate for courses in mathematical elasticity, theory of plates and shells, continuum mechanics, computational mechanics, and applied mathematics in general.