This book offers an informal, easy-to-understand account of topics in modern physics and mathematics. The focus is, in particular, on statistical mechanics, soft matter, probability, chaos, complexity, and models, as well as their interplay. The book features 28 key entries and it is carefully structured so as to allow readers to pursue different paths that reflect their interests and priorities, thereby avoiding an excessively systematic presentation that might stifle interest. While the majority of the entries concern specific topics and arguments, some relate to important protagonists of science, highlighting and explaining their contributions. Advanced mathematics is avoided, and formulas are introduced in only a few cases. The book is a user-friendly tool that nevertheless avoids scientific compromise. It is of interest to all who seek a better grasp of the world that surrounds us and of the ideas that have changed our perceptions.

A timely, accessible introduction to the mathematics of chaos

The past three decades have seen dramatic developments in the theory of dynamical systems, particularly regarding the exploration of chaotic behavior. Complex patterns of even simple processes arising in biology, chemistry, physics, engineering, economics, and a host of other disciplines have been investigated, explained, and utilized.

Introduction to Discrete Dynamical Systems and Chaos makes these exciting and important ideas accessible to students and scientists by assuming, as a background, only the standard undergraduate training in calculus and linear algebra. Chaos is introduced at the outset and is then incorporated as an integral part of the theory of discrete dynamical systems in one or more dimensions. Both phase space and parameter space analysis are developed with ample exercises, more than 100 figures, and important practical examples such as the dynamics of atmospheric changes and neural networks.

An appendix provides readers with clear guidelines on how to use Mathematica to explore discrete dynamical systems numerically. Selected programs can also be downloaded from a Wiley ftp site (address in preface). Another appendix lists possible projects that can be assigned for classroom investigation. Based on the author’s 1993 book, but boasting at least 60% new, revised, and updated material, the present Introduction to Discrete Dynamical Systems and Chaos is a unique and extremely useful resource for all scientists interested in this active and intensely studied field.

Focuses on the latest research in the field of differential equations in engineering applications Discusses the most recent research findings that are occurring across different institutions Identifies the gaps in the knowledge of differential equations Presents the most fruitful areas for further research in advanced processes Offers the most forthcoming studies in modeling and simulation along with real-world case studies

A hundred years ago it became known that deterministic systems can exhibit very complex behavior. By proving that ordinary differential equations can exhibit strange behavior, Poincare undermined the founda tions of Newtonian physics and opened a window to the modern theory of nonlinear dynamics and chaos. Although in the 1930s and 1940s strange behavior was observed in many physical systems, the notion that this phenomenon was inherent in deterministic systems was never suggested. Even with the powerful results of S. Smale in the 1960s, complicated be havior of deterministic systems remained no more than a mathematical curiosity. Not until the late 1970s, with the advent of fast and cheap comput ers, was it recognized that chaotic behavior was prevalent in almost all domains of science and technology. Smale horseshoes began appearing in many scientific fields. In 1971, the phrase 'strange attractor' was coined to describe complicated long-term behavior of deterministic systems, and the term quickly became a paradigm of nonlinear dynamics. The tools needed to study chaotic phenomena are entirely different from those used to study periodic or quasi-periodic systems; these tools are analytic and measure-theoretic rather than geometric. For example, in throwing a die, we can study the limiting behavior of the system by viewing the long-term behavior of individual orbits. This would reveal incomprehensibly complex behavior. Or we can shift our perspective: Instead of viewing the long-term outcomes themselves, we can view the probabilities of these outcomes. This is the measure-theoretic approach taken in this book."

Chaos in science has always been a fascinating realm since it challenges the usual scientific approach of reductionism. While carefully distinguishing between complexity, holism, randomness, incompleteness, nondeterminism and stochastic behaviour the authors show that, although many aspects of chaos have been phenomenologically understood, most of its defining principles are still difficult to grasp and formulate. Demonstrating that chaos escapes all traditional methods of description, the authors set out to find new methods to deal with this phenomenon and illustrate their constructive approach with many examples from physics, biology and information technology. While maintaining a high level of rigour, an overly complicated mathematical apparatus is avoided in order to make this book accessible, beyond the specialist level, to a wider interdisciplinary readership.

From Hamiltonian Chaos to Complex Systems: A Nonlinear Physics Approach collects contributions on recent developments in non-linear dynamics and statistical physics with an emphasis on complex systems. This book provides a wide range of state-of-the-art research in these fields. The unifying aspect of this book is demonstration of how similar tools coming from dynamical systems, nonlinear physics, and statistical dynamics can lead to a large panorama of research in various fields of physics and beyond, most notably with the perspective of application in complex systems.

Chaos is the idea that a system will produce very different long-term behaviors when the initial conditions are perturbed only slightly. Chaos is used for novel, time- or energy-critical interdisciplinary applications. Examples include high-performance circuits and devices, liquid mixing, chemical reactions, biological systems, crisis management, secure information processing, and critical decision-making in politics, economics, as well as military applications, etc. This book presents the latest investigations in the theory of chaotic systems and their dynamics. The book covers some theoretical aspects of the subject arising in the study of both discrete and continuous-time chaotic dynamical systems. This book presents the state-of-the-art of the more advanced studies of chaotic dynamical systems.

The latest developments in chaos theory — from an industry expert

Chaos and Order in the Capital Markets was the first book to introduce and popularize chaos as it applies to finance. It has since become the classic source on the topic. This new edition is completely updated to include the latest ripples in chaos theory with new chapters that tie in today's hot innovations, such as fuzzy logic, neural nets, and artificial intelligence.

Critical praise for Peters and the first edition of Chaos and Order in the Capital Markets

"The bible of market chaologists." — BusinessWeek

"Ed Peters has written a first-class summary suitable for any investment professional or skilled investor." — Technical Analysis of Stocks & Commodities

"It ranks among the most provocative financial books of the past few years. Reading this book will provide a generous payback for the time and mental energy expended." — Financial Analysts Journal

This second edition of Chaos and Order in the Capital Markets brings the topic completely up to date with timely examples from today's markets and descriptions of the latest wave of technology, including genetic algorithms, wavelets, and complexity theory.

Chaos and Order in the Capital Markets was the very first book to explore and popularize chaos theory as it applies to finance. It has since become the industry standard, and is regarded as the definitive source to which analysts, investors, and traders turn for a comprehensive overview of chaos theory. Now, this invaluable reference — touted by BusinessWeek as "the bible of market chaologists" — has been updated and revised to bring you the latest developments in the field.

Mainstream capital market theory is based on efficient market assumptions, even though the markets themselves exhibit characteristics that are symptomatic of nonlinear dynamic systems. As it explores — and validates — this nonlinear nature, Chaos and Order repudiates the "random walk" theory and econometrics. It shifts the focus away from the concept of efficient markets toward a more general view of the forces underlying the capital market system.

Presenting new analytical techniques, as well as reexamining methods that have been in use for the past forty years, Chaos and Order offers a thorough examination of chaos theory and fractals as applied to investments and economics. This new edition includes timely examples from today's markets and descriptions of cutting-edge technologies-genetic algorithms, wavelets, complexity theory-and hot innovations, such as fuzzy logic and artificial intelligence.

Beyond the history of current capital market theory, Chaos and Order covers the crucial characteristics of fractals, the analysis of fractal time series through rescaled range analysis (R/S), the specifics of fractal statistics, and the definition and analysis of chaotic systems. It offers an in-depth exploration of:

• Random walks and efficient markets — the development of the efficient market hypothesis (EMH) and modern portfolio theory
• The linear paradigm — why it has failed
• Nonlinear dynamic systems — phase space, the Henon Map, Lyapunov exponents
• Applying chaos and nonlinear methods — neural networks, genetic algorithms
• Dynamical analysis of time series — reconstructing a phase space, the fractal dimension

Tonis Vaga's Coherent Market Hypothesis — the theory of social imitation, control parameters, Vaga's implementations

Plus, Chaos and Order now contains a Windows-compatible disk including data sets for running analyses described in the appendices.

Written by a leading expert in the field, Chaos and Order in the Capital Markets has all the information you need for a complete, up-to-date look at chaos theory. This latest edition will undoubtedly prove to be as invaluable as the first.

The classical three-body problem is of great importance for its applications to astronomy and space navigation, and also as a simple model of a non-integrable Hamiltonian dynamical system. A central role is played by periodic orbits, of which a large number have been computed numerically. Here the author explains and organizes this material through a systematic study of generating families, which are the limits of families of periodic orbits when the mass ratio of the two main bodies becomes vanishingly small. The most critical part is the study of bifurcations. Many cases are distinguished and studied separately and detailed recipies are given. Their use is illustrated by determining generating families, and comparing them with numerical computations for the Earth+Moon and Sun-Jupiter systems.

The study of nonlinear dynamical systems has been gathering momentum since the late 1950s. It now constitutes one of the major research areas of modern theoretical physics. The twin themes of fractals and chaos, which are linked by attracting sets in chaotic systems that are fractal in structure, are currently generating a great deal of excitement. The degree of structure robustness in the presence of stochastic and quantum noise is thus a topic of interest. Chaos, Noise and Fractals discusses the role of fractals in quantum mechanics, the influence of phase noise in chaos and driven optical systems, and the arithmetic of chaos. The book represents a balanced overview of the field and is a worthy addition to the reading lists of researchers and students interested in any of the varied, and sometimes bizarre, aspects of this intriguing subject.

From its original meaning as a gaping void, or the emptiness that precedes the whole of creation, chaos has taken on the exclusive meaning of confusion, pandemonium and mayhem. This definition has become the overarching word to describe any challenge to the established order; be it railway strikes or political dissent, any unexpected event is routinely described in the media and popular parlance as 'chaos'. In his incisive new study, Stuart Walton argues that this is a pitifully one-dimensional view of the world, as he looks to many of the great social, political, artistic and philosophical advances that have emerged from periods of disorder and from the refusal to think within the standard paradigms. Exploring this worldview, Walton contends that we are superstitious about states of affairs in which anything could happen because we have been taught to prefer the imposition of rules in every aspect of our lives, from our diets to our romances. Indeed, in An Excursion through Chaos he demonstrates how it is these very restrictions that are responsible for the alienation that has characterised postwar society, a state of disengagement that could have been avoided if we had taken a less fearful attitude towards the unravelling of order.

This new edition strives yet again to provide readers with a working knowledge of chaos theory and dynamical systems through parallel introductory explanations in the book and interaction with carefully-selected programs supplied on the accompanying diskette. The programs enable readers, especially advanced-undergraduate students in physics, engineering, and math, to tackle relevant physical systems quickly on their PCs, without distraction from algorithmic details. For the third edition of Chaos: A Program Collection for the PC, each of the previous twelve programs is polished and rewritten in C++ (both Windows and Linux versions are included). A new program treats kicked systems, an important class of two-dimensional problems, which is introduced in Chapter 13. Each chapter follows the structure: theoretical background; numerical techniques; interaction with the program; computer experiments; real experiments and empirical evidence; reference. Interacting with the many numerical experiments have proven to help readers to become familiar with this fascinating topic and even to enjoy the experience.

'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.

Several distinctive aspects make Dynamical Systems unique, including:
o treating the subject from a mathematical perspective with the proofs of most of the results included
o providing a careful review of background materials
o introducing ideas through examples and at a level accessible to a beginning graduate student
o focusing on multidimensional systems of real variables

The book treats the dynamics of both iteration of functions and solutions of ordinary differential equations. Many concepts are first introduced for iteration of functions where the geometry is simpler, but results are interpreted for differential equations. The dynamical systems approach of the book concentrates on properties of the whole system or subsets of the system rather than individual solutions. The more local theory discussed deals with characterizing types of solutions under various hypothesis, and later chapters address more global aspects.

A periodically forced mathematical pendulum is one of the typical and popular nonlinear oscillators that possess complex and rich dynamical behaviors. Although the pendulum is one of the simplest nonlinear oscillators, yet, until now, we are still not able to undertake a systematical study of periodic motions to chaos in such a simplest system due to lack of suitable mathematical methods and computational tools. To understand periodic motions and chaos in the periodically forced pendulum, the perturbation method has been adopted. One could use the Taylor series to expend the sinusoidal function to the polynomial nonlinear terms, followed by traditional perturbation methods to obtain the periodic motions of the approximated differential system.This book discusses Hamiltonian chaos and periodic motions to chaos in pendulums. This book first detects and discovers chaos in resonant layers and bifurcation trees of periodic motions to chaos in pendulum in the comprehensive fashion, which is a base to understand the behaviors of nonlinear dynamical systems, as a results of Hamiltonian chaos in the resonant layers and bifurcation trees of periodic motions to chaos. The bifurcation trees of travelable and non-travelable periodic motions to chaos will be presented through the periodically forced pendulum.

This textbook on the theory of nonlinear dynamical systems for nonmathematical advanced undergraduate or graduate students is also a reference book for researchers in the physical and social sciences. It provides a comprehensive introduction including linear systems, stability theory of nonlinear systems, bifurcation theory, chaotic dynamics. Discussion of the measure--theoretic approach to dynamical systems and the relation between deterministic systems and stochastic processes is featured. There are a hundred exercises and an associated website provides a software program, computer exercises and answers to selected book exercises.

A powerful new way to navigate today's unprecedented market conditions

"Bill Williams' pioneering application of chaos theory to the financial markets is leading technical analysis into the twenty-first century and beyond. New Trading Dimensions presents a complete, highly original, and intriguing trading method with clear, detailed illustrations, and challenging practice pages. Bill's wisdom, technical expertise, and skillful teaching style make this a revolutionary must-have new book for stock and commodity traders." —Tom Bierovic, Product Manager for User Education, Omega Research, Inc.

"Bill hits the nail on the head. The essence of successful trading is a combination of knowing who you are and allowing the market to reveal its secrets. Bill Williams has the gift of explaining these concepts better than anyone I know. This is a compelling work that belongs in every trader's library." —George Angell, author, Profitable Day-Trading with Precision

"Bill Williams is one of the great educators of our time. He freely shares his knowledge and experience in this inexpensive book. This book is required reading for all market technicians. The principles are sound as we have tested them with our software." —John Hill, President, Futures Truth, Co.

"Bill Williams has always been an excellent teacher, taking complex terms and concepts and translating them into a clear, commonsense approach to trading. This book provides a complete trading program that reflects Bill's years of wisdom and experience in the marketplace." —Darrell Jobman, Editorial Consultant and former Editor-in-Chief of Futures magazine

"Bill uses the hidden structure of chaos theory to skillfully guide the reader to the correct psychological profile for success in trading. Bill then goes on to provide a comprehensive structure, a trading paradigm, directly derived from his research and considerable trading experience. You will be challenged, interested, and have a mind-opening experience that's even fun." —Joe DiNapoli, author, Trading with DiNapoli Levels

As today's market environment continues to change dramatically, more and more traders are discovering that traditional forecasting methods—pure technical analysis and fundamental analysis—just do not work. Sending out contradictory messages, these opposing schools of thought leave investors baffled about the future direction of the market, and consequently, at a loss as to how to tailor their trading systems. As a result, many practitioners have now turned to a new forecasting "cocktail" that combines traditional charting methodologies with chaos theory and human psychology. In this groundbreaking book, Bill Williams, a seasoned trader at the forefront of this dynamic new approach, explains exactly what it is, how it works in current stock and commodity markets, and how to use it to your advantage.

Based on human nature rather than the vagaries of the market, the new trading dimension works on the premise that we trade not the market, but our own belief system. By assessing what your personal biases are, you can determine how they influence your ultimate success—or failure—and then adjust your trading strategies accordingly.

Written by an expert in the field who has been featured in Futures, Worth, Success, and other prominent publications, New Trading Dimensions takes the latest in scientific knowledge about human behavior and applies it directly to the fields of stock and commodity investing and trading. With straightforward guidelines, it shows you how to adopt the right attitude toward the behavior of the market and use the right tools (ATTITOOLS) for profitable trading. Packed with practice exercises, specific applications to different types of investments, and a detailed review of important market signals, here's where you'll learn how to:

• Discover what the market wants and align your own beliefs with the direction of the market
• Apply chaos theory to trading and investing
• Use Williams' "Market Alligator" for analyzing and profiting from the markets
• Employ a multidimensional trading program that includes such tools and techniques as fractals, oscillators, AC signals, psychological zones, and balance lines
• Exit trades in a timely fashion to reap high returns

Drawing on the author's more than forty years of experience as both a successful trader and seasoned trainer, this invaluable guide offers a breakthrough method that has proven its ability to turn investors into consistent winners.

Self-organized criticality, the spontaneous development of systems to a critical state, is the first general theory of complex systems with a firm mathematical basis. This theory describes how many seemingly desperate aspects of the world, from stock market crashes to mass extinctions, avalanches to solar flares, all share a set of simple, easily described properties.
..".a'must read'...Bak writes with such ease and lucidity, and his ideas are so intriguing...essential reading for those interested in complex systems...it will reward a sufficiently skeptical reader." -NATURE
..".presents the theory (self-organized criticality) in a form easily absorbed by the non-mathematically inclined reader." -BOSTON BOOK REVIEW
"I picture Bak as a kind of scientific musketeer; flamboyant, touchy, full of swagger and ready to join every fray... His book is written with panache. The style is brisk, the content stimulating. I recommend it as a bracing experience." -NEW SCIENTIST

The main goal is to offer to readers a panorama of recent progress in nonlinear physics, complexity and transport with attractive chapters readable by a broad audience. It allows to gain an insight into these active fields of research and notably promotes the interdisciplinary studies from mathematics to experimental physics. To reach this aim, the book collects a selection of contributions to the third edition of the CCT conference (Marseilles, 1-5 June 2015).

This book aims to familiarize the reader with the essential properties of the chaotic dynamics of Hamiltonian systems by avoiding specialized mathematical tools, thus making it easily accessible to a broader audience of researchers and students. Unique material on the most intriguing and fascinating topics of unsolved and current problems in contemporary chaos theory is presented. The coverage includes: separatrix chaos; properties and a description of systems with non-ergodic dynamics; the distribution of Poincare recurrences and their role in transport theory; dynamical models of the Maxwell's Demon, the occurrence of persistent fluctuations, and a detailed discussion of their role in the problem underlying the foundation of statistical physics; the emergence of stochastic webs in phase space and their link to space tiling with periodic (crystal type) and aperiodic (quasi-crystal type) symmetries.This second edition expands on pseudochaotic dynamics with weak mixing and the new phenomenon of fractional kinetics, which is crucial to the transport properties of chaotic motion.The book is ideally suited to all those who are actively working on the problems of dynamical chaos as well as to those looking for new inspiration in this area. It introduces the physicist to the world of Hamiltonian chaos and the mathematician to actual physical problems.The material can also be used by graduate students.

This book focuses on the computational analysis of nonlinear vibrations of structural members (beams, plates, panels, shells), where the studied dynamical problems can be reduced to the consideration of one spatial variable and time. The reduction is carried out based on a formal mathematical approach aimed at reducing the problems with infinite dimension to finite ones. The process also includes a transition from governing nonlinear partial differential equations to a set of finite number of ordinary differential equations.Beginning with an overview of the recent results devoted to the analysis and control of nonlinear dynamics of structural members, placing emphasis on stability, buckling, bifurcation and deterministic chaos, simple chaotic systems are briefly discussed. Next, bifurcation and chaotic dynamics of the Euler-Bernoulli and Timoshenko beams including the geometric and physical nonlinearity as well as the elastic-plastic deformations are illustrated. Despite the employed classical numerical analysis of nonlinear phenomena, the various wavelet transforms and the four Lyapunov exponents are used to detect, monitor and possibly control chaos, hyper-chaos, hyper-hyper-chaos and deep chaos exhibited by rectangular plate-strips and cylindrical panels.The book is intended for post-graduate and doctoral students, applied mathematicians, physicists, teachers and lecturers of universities and companies dealing with a nonlinear dynamical system, as well as theoretically inclined engineers of mechanical and civil engineering.

A leading pioneer in the field offers practical applications of this innovative science. Peters describes complex concepts in an easy-to-follow manner for the non-mathematician. He uses fractals, rescaled range analysis and nonlinear dynamical models to explain behavior and understand price movements. These are specific tools employed by chaos scientists to map and measure physical and now, economic phenomena.

This book aims to provide a lively working knowledge of the thermodynamic control of microscopic simulations, while summarizing the historical development of the subject, along with some personal reminiscences. Many computational examples are described so that they are well-suited to learning by doing. The contents enhance the current understanding of the reversibility paradox and are accessible to advanced undergraduates and researchers in physics, computation, and irreversible thermodynamics.

This volume provides a self-contained survey of the mechanisms presiding information processing and communication. The main thesis is that chaos and complexity are the basic ingredients allowing systems composed of interesting subunits to generate and process information and communicate in a meaningful way. Emphasis is placed on communication in the form of games and on the related issue of decision making under conditions of uncertainty. Biological, cognitive, physical, engineering and societal systems are approached from a unifying point of view, both analytically and by numerical simulation, using the methods of nonlinear dynamics and probability theory. Epistemological issues in connection with incompleteness and self-reference are also addressed.

The book Complexity and Control: Towards a Rigorous Behavioral Theory of Complex Dynamical Systems is a graduate-level monographic textbook, intended to be a novel and rigorous contribution to modern Complexity Theory.This book contains 11 chapters and is designed as a one-semester course for engineers, applied and pure mathematicians, theoretical and experimental physicists, computer and economic scientists, theoretical chemists and biologists, as well as all mathematically educated scientists and students, both in industry and academia, interested in predicting and controlling complex dynamical systems of arbitrary nature.