An intriguing and illuminating look at how randomness, chance, and probability affect our daily lives.
Successes and failures in life are often attributed to clear causes, when actually they are profoundly influenced by randomness and chance. Here, with the sense of narrative and imaginative approach of a storyteller, Leonard Mlodinow vividly demonstrates how wine ratings, corporate success, school grades, and political polls are less reliable than we believe. Showing us the true nature of chance and revealing the psychological illusions that cause us to misjudge the world around us, Mlodinow provides the tools we need for more informed decision making. From the classroom to the courtroom and from financial markets to supermarkets, Mlodinow's insights will intrigue, awe, and inspire.

This book demonstrates that while elliptic and hyperbolic tori determine the distribution of maximal invariant tori, they themselves form n-parameter families. Therefore, torus bifurcations of high co-dimension may be found in a single given Hamiltonian system, absent untypical conditions or external parameters. The text moves logically from the integrable case, in which symmetries allow for reduction to bifurcating equilibria, to non-integrability, where smooth parametrisations must be replaced by Cantor sets.

This book describes a family of algorithms for studying the global structure of systems. By a finite covering of the phase space we construct a directed graph with vertices corresponding to cells of the covering and edges corresponding to admissible transitions. The method is used, among other things, to locate the periodic orbits and the chain recurrent set, to construct the attractors and their basins, to estimate the entropy, and more.

This is an advanced textbook on the subject of turbulence, and is suitable for engineers, physical scientists and applied mathematicians. The aim of the book is to bridge the gap between the elementary accounts of turbulence found in undergraduate texts, and the more rigorous monographs on the subject. Throughout, the book combines the maximum of physical insight with the minimum of mathematical detail. Chapters 1 to 5 may be appropriate as background material for an advanced undergraduate or introductory postgraduate course on turbulence, while chapters 6 to 10 may be suitable as background material for an advanced postgraduate course on turbulence, or act as a reference source for professional researchers. This second edition covers a decade of advancement in the field, streamlining the original content while updating the sections where the subject has moved on. The expanded content includes large-scale dynamics, stratified & rotating turbulence, the increased power of direct numerical simulation, two-dimensional turbulence, Magnetohydrodynamics, and turbulence in the core of the Earth

This book is devoted to the phenomenon of quasi-periodic motion in dynamical systems. Such a motion in the phase space densely fills up an invariant torus. This phenomenon is most familiar from Hamiltonian dynamics. Hamiltonian systems are well known for their use in modelling the dynamics related to frictionless mechanics, including the planetary and lunar motions. In this context the general picture appears to be as follows. On the one hand, Hamiltonian systems occur that are in complete order: these are the integrable systems where all motion is confined to invariant tori. On the other hand, systems exist that are entirely chaotic on each energy level. In between we know systems that, being sufficiently small perturbations of integrable ones, exhibit coexistence of order (invariant tori carrying quasi-periodic dynamics) and chaos (the so called stochastic layers). The Kolmogorov-Arnol'd-Moser (KAM) theory on quasi-periodic motions tells us that the occurrence of such motions is open within the class of all Hamiltonian systems: in other words, it is a phenomenon persistent under small Hamiltonian perturbations. Moreover, generally, for any such system the union of quasi-periodic tori in the phase space is a nowhere dense set of positive Lebesgue measure, a so called Cantor family. This fact implies that open classes of Hamiltonian systems exist that are not ergodic. The main aim of the book is to study the changes in this picture when other classes of systems - or contexts - are considered.

Given the ease with which computers can do iteration it is now possible for almost anyone to generate beautiful images whose roots lie in discrete dynamical systems. Images of Mandelbrot and Julia sets abound in publications both mathematical and not. The mathematics behind the pictures are beautiful in their own right and are the subject of this text. Mathematica programs that illustrate the dynamics are included in an appendix.

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

Developed and class-tested by a distinguished team of authors at two universities, this text is intended for courses in nonlinear dynamics in either mathematics or physics. The only prerequisites are calculus, differential equations, and linear algebra. Along with discussions of the major topics, including discrete dynamical systems, chaos, fractals, nonlinear differential equations and bifurcations, the text also includes Lab Visits -- short reports that illustrate relevant concepts from the physical, chemical and biological sciences. There are Computer Experiments throughout the text that present opportunities to explore dynamics through computer simulations, designed for use with any software package. And each chapter ends with a Challenge, guiding students through an advanced topic in the form of an extended exercise.

This book focuses on complex analytic dynamics, which dates from 1916 and is currently attracting considerable interest. The text provides a comprehensive, well-organized treatment of the foundations of the theory of iteration of rational functions of a complex variable. The coverage extends from early memoirs of Fatou and Julia to important recent results and methods of Sullivan and Shishikura. Many details of the proofs have not appeared in print before.

This book is about the explicit elimination of fast oscillatory scales in dynamical systems, which is important for efficient computer-simulations and our understanding of model hierarchies. The author presents his new direct method, homogenization in time, based on energy principles and weak convergence techniques. How to use this method is shown in several general cases taken from classical and quantum mechanics. The results are applied to special problems from plasma physics, molecular dynamics and quantum chemistry. Background material from functional analysis is provided and explained to make this book accessible for a general audience of graduate students and researchers.

UNDERSTANDING NONLINEAR DYNAMICS is based on an undergraduate course taught for many years to students in the biological sciences. The text provides a clear and accessible development of many concepts from contemporary dynamics, including stability and multistability, cellular automata and excitable media, fractals, cycles, and chaos. A chapter on time-series analysis builds on this foundation to provide an introduction to techniques for extracting information about dynamics from data. The text will be useful for courses offered in the life sciences or other applied science programs, or as a supplement to emphasize the application of subjects presented in mathematics or physics courses. Extensive examples are derived from the experimental literature, and numerous exercise sets can be used in teaching basic mathematical concepts and their applications. Concrete applications of the mathematics are illustrated in such areas as biochemistry, neurophysiology, cardiology, and ecology. The text also provides an entry point for researchers not familiar with mathematics but interested in applications of nonlinear dynamics to the life sciences.

Considerable work has been done on chaotic dynamics in the field of economic growth and dynamic macroeconomic models during the last two decades. This book considers numerous new developments: introduction of infrastructure in growth models, heterogeneity of agents, hysteresis systems, overlapping models with "pay-as-you-go" systems, keynesian approaches with finance considerations, interactions between relaxation cycles and chaotic dynamics, methodological issues, long memory processes and fractals... A volume of contributions which shows the relevance and fruitfulness of non-linear analysis for the explanation of complex dynamics in economic systems.

Our DMV Seminar on 'Classical Nonintegrability, Quantum Chaos' intended to introduce students and beginning researchers to the techniques applied in nonin tegrable classical and quantum dynamics. Several of these lectures are collected in this volume. The basic phenomenon of nonlinear dynamics is mixing in phase space, lead ing to a positive dynamical entropy and a loss of information about the initial state. The nonlinear motion in phase space gives rise to a linear action on phase space functions which in the case of iterated maps is given by a so-called transfer operator. Good mixing rates lead to a spectral gap for this operator. Similar to the use made of the Riemann zeta function in the investigation of the prime numbers, dynamical zeta functions are now being applied in nonlinear dynamics. In Chapter 2 V. Baladi first introduces dynamical zeta functions and transfer operators, illustrating and motivating these notions with a simple one-dimensional dynamical system. Then she presents a commented list of useful references, helping the newcomer to enter smoothly into this fast-developing field of research. Chapter 3 on irregular scattering and Chapter 4 on quantum chaos by A. Knauf deal with solutions of the Hamilton and the Schr6dinger equation. Scatter ing by a potential force tends to be irregular if three or more scattering centres are present, and a typical phenomenon is the occurrence of a Cantor set of bounded orbits. The presence of this set influences those scattering orbits which come close."

Although chaotic behaviour had often been observed numerically earlier, the first mathematical proof of the existence, with positive probability (persistence) of strange attractors was given by Benedicks and Carleson for the Henon family, at the beginning of 1990's. Later, Mora and Viana demonstrated that a strange attractor is also persistent in generic one-parameter families of diffeomorphims on a surface which unfolds homoclinic tangency. This book is about the persistence of any number of strange attractors in saddle-focus connections. The coexistence and persistence of any number of strange attractors in a simple three-dimensional scenario are proved, as well as the fact that infinitely many of them exist simultaneously.

The Session was intended to give a broad survey of the mathematical problems arising in the chaotic transition of deterministic dynamical systems, both in classical and quantum mechanics.
The lectures of Mather and Forni thoroughly cover the area- preserving twist maps, and include an up-to-date version of the Aubry-Mather theory.
The lectures of Bellissard describe the quantum aspects: classical limit, localization, spectral properties of the relevant SchrAdinger operators and thus represent an exhaus- tive introduction to the mathematics of "quantum chaos."
The lectures of Degli Esposti et al. reviews equidistribu- tion of unstable periodic orbits and classical limit of quantized toral symplectomorphisms.

The chapters of this book were written by structural engineers. The approach, therefore, is not aiming toward a scientific modelling of the response but to the definition of engineering procedures for detecting and avoiding undesired phenomena. In this sense chaotic and stochastic behaviour can be tackled in a similar manner. This aspect is illustrated in Chapter 1. Chapters 2 and 3 are entirely devoted to Stochastic Dynamics and cover single-degree-of-freedom systems and impact problems, respectively. Chapter 4 provides details on the numerical tools necessary for evaluating the main indexes useful for the classification of the motion and for estimating the response probability density function. Chapter 5 gives an overview of random vibration methods for linear and nonlinear multi-degree-of-freedom systems. The randomness of the material characteristics and the relevant stochastic models ar considered in Chapter 6. Chapter 7, eventually, deals with large engineering sytems under stochastic excitation and allows for the stochastic nature of the mechanical and geometrical properties.

The purpose of this volume is to give a detailed account of a series of re sults concerning some ergodic questions of quantum mechanics which have the past six years following the formulation of a generalized been addressed in Kolmogorov-Sinai entropy by A.Connes, H.Narnhofer and W.Thirring. Classical ergodicity and mixing are fully developed topics of mathematical physics dealing with the lowest levels in a hierarchy of increasingly random behaviours with the so-called Bernoulli systems at its apex showing a structure that characterizes them as Kolmogorov (K-) systems. It seems not only reasonable, but also inevitable to use classical ergodic theory as a guide in the study of ergodic behaviours of quantum systems. The question is which kind of random behaviours quantum systems can exhibit and whether there is any way of classifying them. Asymptotic statistical independence and, correspondingly, complete lack of control over the distant future are typical features of classical K-systems. These properties are fully characterized by the dynamical entropy of Kolmogorov and Sinai, so that the introduction of a similar concept for quantum systems has provided the opportunity of raising meaningful questions and of proposing some non-trivial answers to them. Since in the following we shall be mainly concerned with infinite quantum systems, the algebraic approach to quantum theory will provide us with the necessary analytical tools which can be used in the commutative context, too."

Our life is a highly nonlinear process. It starts with birth and ends with death; in between there are a lot of ups and downs. Quite often, we believe that stable and steady situations, probably easy to capture by linearization, are paradisiacal, but already after a short period of everyday routine we usually become bored and seek change, that is, nonlinearities. If we reflect for a while, we notice that our life and our perceptions are mainly determined by nonlinear phenomena, for example, events occurring suddenly and unexpectedly. One may be surprised by how long scientists tried to explain our world by models based on a linear ansatz. Due to the lack of typical nonlinear patterns, although everybody experienced nonlinearities, nobody could classify them and, thus, . study them further. The discoveries of the last few decades have finally provided access to the world of nonlinear phenomena and have initiated a unique inter disciplinary field of research: nonlinear science. In contrast to the general tendency of science to become more branched out and specialized as the result of any progress, nonlinear science has brought together many different disciplines. This has been motivated not only by the immense importance of nonlinearities for science, but also by the wonderful simplicity ohhe concepts. Models like the logistic map can be easily understood by high school students and have brought revolutionary new insights into our scientific under standing."

The treatment of chaotic dynamics in mathematics and physics during last two decades has led to a number of new concepts for the investigation of complex behavior in nonlinear dynamical processes. The aim the CISM course Engineering Applications of Dynamics of Chaos of which this is the proceedings volume was to make these concepts available to engineers and applied scientists possessing only such modest knowledges in mathematics which are usual for engineers, for example graduating from a Technical University. The contents of the articles contributed by leading experts in this field cover not only theoretical foundations and algorithmic and computational aspects but also applications to engineering problems. In the first article an introduction into the basic concepts for the investigation of chaotic behavior of dynamical systems is given which is followed in the second article by an extensive treatment of approximative analytical methods to determine the critical parameter values describing the onset of chaos. The important relation between chaotic dynamics and the phenomenon of turbulence is treated in the third article by studying instabilities various fluid flows. In this contribution also an introduction into interesting phenomenon of pattern formation is given. The fourth and fifth articles present various applications to nonlinear oscillations including roll motions of ships, rattling oscillations in gear boxes, tumbling oscillations of satellites, flutter motions of fluid carrying pipes and vibrations of robot arms. In the final article a short treatment of hyperchaos is given.

This book is devoted to the history of chaos theory, from celestial mechanics (three-body problem) to electronics and meteorology. Many illustrative examples of chaotic behaviors exist in various contexts found in nature (chemistry, astrophysics, biomedicine). This book includes the most popular systems from chaos theory (Lorenz, Roessler, van der Pol, Duffing, logistic map, Lozi map, Henon map etc.) and introduces many other systems, some of them very rarely discussed in textbooks as well as in scientific papers. The contents are formulated with an original approach as compared to other books on chaos theory.

Why do traffic jams seem to happen for no apparent reason? Can major earthquakes be predicted? Why does the stock market have its ups and downs? How do species evolve? Where do galaxies come from? What is the origin of life on Earth? "What if all these questions had a single answer?
Over the past two decades, no field of scientific inquiry has had a more striking impact across a wide array of disciplines-from biology to physics, computing to meteorology-than that known as chaos and complexity, the study of complex systems. Now astrophysicist John Gribbin draws on his expertise to explore, in sparkling prose that communicates not only the wonder but the substance of cutting-edge science, the principles behind chaos and complexity. He reveals the remarkable ways these two revolutionary theories have been applied over the last twenty years to explain all sorts of phenomena-from weather patterns to mass extinctions.
Grounding these paradigm-shifting ideas in their historical context, Gribbin also traces their development from Newton to Darwin to Lorenz, Prigogine, and Lovelock, demonstrating how-far from overturning all that has gone before-chaos and complexity are the triumphant extensions of simple scientific laws. More astonishing, he shows how chaos and complexity permeate the universe on every scale, governing the evolution of life and galaxies alike. As profound as it is provocative, "Deep Simplicity takes us to the brink of understanding life itself.

A unique and accessible book providing a unified framework for studying quantum and classical dynamical systems, both finite and infinite, conservative and dissipative. Many examples and references are included throughout, making it an ideal text for graduate students in physics and mathematics.

This book brings together different work in the new field of physics called the chaos theory, an extension of classical mechanics, in which simple and complex causes are seen to interact. Mathematics may only be able to solve simple linear equations which experiment has pushed nature into obeying in a limited way, but now that computers can map the whole plane of solutions of non-linear equations a new vision of nature is revealed. The implications are staggeringly universal in all areas of scientific work and philosophical thought.

A bold, visionary, and mind-bending exploration of how the geometry of chaos can explain our uncertain world - from weather and pandemics to quantum physics and free will Covering a breathtaking range of topics - from climate change to the foundations of quantum physics, from economic modelling to conflict prediction, from free will to consciousness and spirituality - The Primacy of Doubt takes us on a unique journey through the science of uncertainty. A key theme that unifies these seemingly unconnected topics is the geometry of chaos: the beautiful and profound fractal structures that lie at the heart of much of modern mathematics. Royal Society Research Professor Tim Palmer shows us how the geometry of chaos not only provides the means to predict the world around us, it suggests new insights into some of the most astonishing aspects of our universe and ourselves. This important and timely book helps the reader makes sense of uncertainty in a rapidly changing world.

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.