What do economic chaos and uncertainties mean in rational or irrational economic theories? How do simple deterministic interactions among a few variables lead to unpredictable complex phenomena? Why is complexity of economies causing so many conflicts and confusions worldwide?This book provides a comprehensive introduction to recent developments of complexity theory in economics. It presents different models based on well-accepted economic mechanisms such as the Solow model, Ramsey model, and Lucas model. It is focused on presenting complex behaviors, such as business cycles, aperiodic motion, bifurcations, catastrophes, chaos, and hidden attractors, in basic economic models with nonlinear behavior. It shows how complex nonlinear phenomena are identified from various economic mechanisms and theories. These models demonstrate that the traditional or dominant economic views on evolution of, for instance, capitalism market, free competition, or Keynesian economics, are not generally valid. Markets are unpredictable and nobody knows with certainty the consequences of policies or other external factors in economic systems with simple interactions.

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.

'This book offers one of the few places where a collection of results from the literature can be found ... The book has an extensive bibliography ... It is very nice to have the compendium of results that is presented here.'zbMATHA mathematical billiard is a mechanical system consisting of a billiard ball on a table of any form (which can be planar or even a multidimensional domain) but without billiard pockets. The ball moves and its trajectory is defined by the ball's initial position and its initial speed vector. The ball's reflections from the boundary of the table are assumed to have the property that the reflection and incidence angles are the same. This book comprehensively presents known results on the behavior of a trajectory of a billiard ball on a planar table (having one of the following forms: circle, ellipse, triangle, rectangle, polygon and some general convex domains). It provides a systematic review of the theory of dynamical systems, with a concise presentation of billiards in elementary mathematics and simple billiards related to geometry and physics.The description of these trajectories leads to the solution of various questions in mathematics and mechanics: problems related to liquid transfusion, lighting of mirror rooms, crushing of stones in a kidney, collisions of gas particles, etc. The analysis of billiard trajectories can involve methods of geometry, dynamical systems, and ergodic theory, as well as methods of theoretical physics and mechanics, which has applications in the fields of biology, mathematics, medicine, and physics.

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.

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.

Covering one of the fastest growing areas of applied mathematics, Nonlinear Dynamics and Chaos: Second Edition, is a fully updated edition of this highly respected text. Covering a breadth of topics, ranging from the basic concepts to applications in the physical sciences, the book is highly illustrated and written in a clear and comprehensible style.

• Provides a self-contained introduction to the theory and applications of nonlinear dynamics and chaos.

• Introduces the concepts of instabilities, bifurcations, and catastrophes.

• Each idea is carefully explained and supported by examples.

• Features many applications to a wide variety of scientific fields.

• Includes an illustrated glossary of geometrical dynamics.

• Features a supplementary bibliography of further reading.

• Assumes minimal background knowledge.

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.

In the past hundred years investigators have learned the significance of complex behavior in deterministic systems. The potential applications of this discovery are as numerous as they are encouraging.This text clearly presents the mathematical foundations of chaotic dynamics, including methods and results at the forefront of current research. The book begins with a thorough introduction to dynamical systems and their applications. It goes on to develop the theory of regular and stochastic behavior in higher-degree-of-freedom Hamiltonian systems, covering topics such as homoclinic chaos, KAM theory, the Melnikov method, and Arnold diffusion. Theoretical discussions are illustrated by a study of the dynamics of small circumasteroidal grains perturbed by solar radiation pressure. With alternative derivations and proofs of established results substituted for those in the standard literature, this work serves as an important source for researchers, students and teachers.Skillfully combining in-depth mathematics and actual physical applications, this book will be of interest to the applied mathematician, the theoretical mechanical engineer and the dynamical astronomer alike.

A combinatorial method is developed in this book to explore the mysteries of chaos, which has became a topic of science since 1975. Using tools from theoretical computer science, formal languages and automata, the complexity of symbolic behaviors of dynamical systems is classified and analysed thoroughly. This book is mainly devoted to explanation of this method and apply it to one-dimensional dynamical systems, including the circle and interval maps, which are typical in exhibiting complex behavior through simple iterated calculations. The knowledge for reading it is self-contained in the book.

Chaos surrounds us. Seemingly random events -- the flapping of a flag, a storm-driven wave striking the shore, a pinball's path -- often appear to have no order, no rational pattern. Explicating the theory of chaos and the consequences of its principal findings -- that actual, precise rules may govern such apparently random behavior -- has been a major part of the work of Edward N. Lorenz. In "The Essence of Chaos," Lorenz presents to the general reader the features of this "new science," with its far-reaching implications for much of modern life, from weather prediction to philosophy, and he describes its considerable impact on emerging scientific fields.

Unlike the phenomena dealt with in relativity theory and quantum mechanics, systems that are now described as "chaotic" can be observed without telescopes or microscopes. They range from the simplest happenings, such as the falling of a leaf, to the most complex processes, like the fluctuations of climate. Each process that qualifies, however, has certain quantifiable characteristics: how it unfolds depends very sensitively upon its present state, so that, even though it is not random, it seems to be. Lorenz uses examples from everyday life, and simple calculations, to show how the essential nature of chaotic systems can be understood. In order to expedite this task, he has constructed a mathematical model of a board sliding down a ski slope as his primary illustrative example. With this model as his base, he explains various chaotic phenomena, including some associated concepts such as strange attractors and bifurcations.

As a meteorologist, Lorenz initially became interested in the field of chaos because of its implications for weather forecasting. In a chapter ranging through the history of weather prediction and meteorology to a brief picture of our current understanding of climate, he introduces many of the researchers who conceived the experiments and theories, and he describes his own initial encounter with chaos.

A further discussion invites readers to make their own chaos. Still others debate the nature of randomness and its relationship to chaotic systems, and describe three related fields of scientific thought: nonlinearity, complexity, and fractality. Appendixes present the first publication of Lorenz's seminal paper "Does the Flap of a Butterfly's Wing in Brazil Set Off a Tornado in Texas?"; the mathematical equations from which the copious illustrations were derived; and a glossary.

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 monograph on a new powerful method discovered by the author for the study of nonlinear dynamical systems relying on reduction of nonlinear differential equations to the linear abstract Schroedinger-like equation in Hilbert space. Besides the possibility of unification of many apparently completely different techniques, the "quantal" Hilbert space formalism introduced enables new original methods to be discovered for solving nonlinear problems arising in investigation of ordinary and partial differential equations as well as difference equations. Applications covered in the book include symmetries and first integrals, linearization transformations, Backlund transformations, stroboscopic maps, functional equations involving the case of Feigenbaum-Cvitanovic renormalization equations and chaos.

This book provides a good coverage of the recent developments and future directions in the study of dissipative systems. The primary thrust here is in exposing the reader to the frontiers of chaos, pointing out clues for further work in nonlinear science. With the aid of various types of mappings, the collapse of tori is investigated. The book contains much valuable introductory material and copious reference lists. Some notes on the historical development of the subject are interspersed in this volume.