Chaos in Dynamical Systems

Author: Edward Ott

Publisher: Cambridge University Press

ISBN: 9780521010849

Category: Mathematics

Page: 478

View: 8128

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New edition of the best-selling graduate textbook on chaos for scientists and engineers.

Chaos in Discrete Dynamical Systems

A Visuell Introduction in 2 Dimensions

Author: Ralph Abraham,Laura Gardini,C. Mira

Publisher: Springer Science & Business Media

ISBN: 9780387943008

Category: Computers

Page: 246

View: 9321

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Chaos Theory is a synonym for dynamical systems theory, a branch of mathematics. Dynamical systems come in three flavors: flows (continuous dynamical systems), cascades (discrete, reversible, dynamical systems), and semi-cascades (discrete, irreversible, dynamical systems). Flows and semi-cascades are the classical systems iuntroduced by Poincare a centry ago, and are the subject of the extensively illustrated book: "Dynamics: The Geometry of Behavior," Addison-Wesley 1992 authored by Ralph Abraham and Shaw. Semi- cascades, also know as iterated function systems, are a recent innovation, and have been well-studied only in one dimension (the simplest case) since about 1950. The two-dimensional case is the current frontier of research. And from the computer graphcis of the leading researcher come astonishing views of the new landscape, such as the Julia and Mandelbrot sets in the beautiful books by Heinz-Otto Peigen and his co-workers. Now, the new theory of critical curves developed by Mira and his students and Toulouse provide a unique opportunity to explain the basic concepts of the theory of chaos and bifurcations for discete dynamical systems in two-dimensions. The materials in the book and on the accompanying disc are not solely developed only with the researcher and professional in mind, but also with consideration for the student. The book is replete with some 100 computer graphics to illustrate the material, and the CD-ROM contains full-color animations that are tied directly into the subject matter of the book, itself. In addition, much of this material has also been class-tested by the authors. The cross-platform CD also contains a software program called ENDO, which enables users to create their own 2-D imagery with X-Windows. Maple scripts are provided which give the reader the option of working directly with the code from which the graphcs in the book were

Bifurcation and Chaos in Simple Dynamical Systems

Author: Jan Awrejcewicz

Publisher: World Scientific

ISBN: 9789810200381

Category: Science

Page: 126

View: 9875

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This book presents a detailed analysis of bifurcation and chaos in simple non-linear systems, based on previous works of the author. Practical examples for mechanical and biomechanical systems are discussed. The use of both numerical and analytical approaches allows for a deeper insight into non-linear dynamical phenomena. The numerical and analytical techniques presented do not require specific mathematical knowledge.

Chaos in nonlinear dynamical systems

proceedings of a workshop held at the U.S. Army Research Office, Research Triangle Park, North Carolina, March 13-15, 1984

Author: J. Chandra,United States. Army Research Office. Mathematical Sciences Division

Publisher: Society for Industrial & Applied

ISBN: N.A

Category: Mathematics

Page: 191

View: 4679

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Discrete Dynamical Systems, Bifurcations and Chaos in Economics

Author: Wei-Bin Zhang

Publisher: Elsevier

ISBN: 9780080462462

Category: Mathematics

Page: 460

View: 8429

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This book is a unique blend of difference equations theory and its exciting applications to economics. It deals with not only theory of linear (and linearized) difference equations, but also nonlinear dynamical systems which have been widely applied to economic analysis in recent years. It studies most important concepts and theorems in difference equations theory in a way that can be understood by anyone who has basic knowledge of calculus and linear algebra. It contains well-known applications and many recent developments in different fields of economics. The book also simulates many models to illustrate paths of economic dynamics. A unique book concentrated on theory of discrete dynamical systems and its traditional as well as advanced applications to economics Mathematical definitions and theorems are introduced in a systematic and easily accessible way Examples are from almost all fields of economics; technically proceeding from basic to advanced topics Lively illustrations with numerous figures Numerous simulation to see paths of economic dynamics Comprehensive treatment of the subject with a comprehensive and easily accessible approach

In the Wake of Chaos

Unpredictable Order in Dynamical Systems

Author: Stephen H. Kellert

Publisher: University of Chicago Press

ISBN: 0226429822

Category: Science

Page: 190

View: 7442

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Chaos theory has captured scientific and popular attention. What began as the discovery of randomness in simple physical systems has become a widespread fascination with "chaotic" models of everything from business cycles to brainwaves to heart attacks. But what exactly does this explosion of new research into chaotic phenomena mean for our understanding of the world? In this timely book, Stephen Kellert takes the first sustained look at the broad intellectual and philosophical questions raised by recent advances in chaos theory—its implications for science as a source of knowledge and for the very meaning of that knowledge itself.

Chaos in Classical and Quantum Mechanics

Author: Martin C. Gutzwiller

Publisher: Springer Science & Business Media

ISBN: 9780387971735

Category: Mathematics

Page: 432

View: 6415

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Describes the chaos apparent in simple mechanical systems with the goal of elucidating the connections between classical and quantum mechanics. It develops the relevant ideas of the last two decades via geometric intuition rather than algebraic manipulation. The historical and cultural background against which these scientific developments have occurred is depicted, and realistic examples are discussed in detail. This book enables entry-level graduate students to tackle fresh problems in this rich field.

Bifurcations and Chaos in Piecewise-smooth Dynamical Systems

Author: Zhanybai T. Zhusubaliyev,Erik Mosekilde

Publisher: World Scientific

ISBN: 9789812564436

Category: Bifurcation theory

Page: 376

View: 3554

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Technical problems often lead to differential equations withpiecewise-smooth right-hand sides. Problems in mechanicalengineering, for instance, violate the requirements of smoothness ifthey involve collisions, finite clearances, or stickOCoslipphenomena."

Toward Analytical Chaos in Nonlinear Systems

Author: Albert C. J. Luo

Publisher: John Wiley & Sons

ISBN: 1118658612

Category: Science

Page: 268

View: 8786

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Analytical Periodic Flows and Chaos in Nonlinear Systems systematically presents a new approach to analytically determine the periodic flows and chaos in nonlinear dynamical systems. The new analytical technique presented in this book is based on Fourier Series theory and the coefficient variation method in differential equations to obtain an autonomous dynamical system of coefficients. To investigate such an autonomous system, the stable and unstable solutions of periodic motions and chaos can be obtained. The book covers the mathematical theory of analytical solutions of periodic flows and chaos in nonlinear dynamical systems, and includes many examples of nonlinear systems frequently encountered in engineering and physics. The analytical techniques in this book can provide a better understanding of regularity and complexity of periodic motions and chaos in nonlinear dynamical systems. The method is straightforward to learn for students and researchers in nonlinear dynamics.

Bifurcation and Chaos in Complex Systems

Author: N.A

Publisher: Elsevier

ISBN: 9780080462660

Category: Science

Page: 400

View: 8453

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The book presents the recent achievements on bifurcation studies of nonlinear dynamical systems. The contributing authors of the book are all distinguished researchers in this interesting subject area. The first two chapters deal with the fundamental theoretical issues of bifurcation analysis in smooth and non-smooth dynamical systems. The cell mapping methods are presented for global bifurcations in stochastic and deterministic, nonlinear dynamical systems in the third chapter. The fourth chapter studies bifurcations and chaos in time-varying, parametrically excited nonlinear dynamical systems. The fifth chapter presents bifurcation analyses of modal interactions in distributed, nonlinear, dynamical systems of circular thin von Karman plates. The theories, methods and results presented in this book are of great interest to scientists and engineers in a wide range of disciplines. This book can be adopted as references for mathematicians, scientists, engineers and graduate students conducting research in nonlinear dynamical systems. · New Views for Difficult Problems · Novel Ideas and Concepts · Hilbert's 16th Problem · Normal Forms in Polynomial Hamiltonian Systems · Grazing Flow in Non-smooth Dynamical Systems · Stochastic and Fuzzy Nonlinear Dynamical Systems · Fuzzy Bifurcation · Parametrical, Nonlinear Systems · Mode Interactions in nonlinear dynamical systems

Introduction to Discrete Dynamical Systems and Chaos

Author: Mario Martelli

Publisher: John Wiley & Sons

ISBN: 1118031121

Category: Mathematics

Page: 344

View: 1816

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A timely, accessible introduction to the mathematics ofchaos. The past three decades have seen dramatic developments in thetheory of dynamical systems, particularly regarding the explorationof chaotic behavior. Complex patterns of even simple processesarising in biology, chemistry, physics, engineering, economics, anda host of other disciplines have been investigated, explained, andutilized. Introduction to Discrete Dynamical Systems and Chaos makes theseexciting and important ideas accessible to students and scientistsby assuming, as a background, only the standard undergraduatetraining in calculus and linear algebra. Chaos is introduced at theoutset and is then incorporated as an integral part of the theoryof discrete dynamical systems in one or more dimensions. Both phasespace and parameter space analysis are developed with ampleexercises, more than 100 figures, and important practical examplessuch as the dynamics of atmospheric changes and neuralnetworks. An appendix provides readers with clear guidelines on how to useMathematica 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 thatcan be assigned for classroom investigation. Based on the author's1993 book, but boasting at least 60% new, revised, and updatedmaterial, the present Introduction to Discrete Dynamical Systemsand Chaos is a unique and extremely useful resource for allscientists interested in this active and intensely studiedfield. An Instructor's Manual presenting detailed solutions to all theproblems in the book is available upon request from the Wileyeditorial department.

Nonlinear Dynamics and Chaos in Agricultural Systems

Author: K. Sakai

Publisher: Gulf Professional Publishing

ISBN: 9780444506467

Category: Mathematics

Page: 204

View: 9468

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An introduction to the analysis of chaos for readers majoring in agricultural science and an introduction to agricultural science for readers majoring in mathematical science and other fields. Hopes some readers will pursue further studies on the chaos of arable land. (Pref.)

Frontiers in the Study of Chaotic Dynamical Systems with Open Problems

Author: Elhadj Zeraoulia,Julien C. Sprott

Publisher: World Scientific

ISBN: 9814340693

Category: Science

Page: 258

View: 6835

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This collection of review articles is devoted to new developments in the study of chaotic dynamical systems with some open problems and challenges. The papers, written by many of the leading experts in the field, cover both the experimental and theoretical aspects of the subject. This edited volume presents a variety of fascinating topics of current interest and problems arising in the study of both discrete and continuous time chaotic dynamical systems. Exciting new techniques stemming from the area of nonlinear dynamical systems theory are currently being developed to meet these challenges. Presenting the state-of-the-art of the more advanced studies of chaotic dynamical systems, Frontiers in the Study of Chaotic Dynamical Systems with Open Problems is devoted to setting an agenda for future research in this exciting and challenging field.

A First Course In Chaotic Dynamical Systems

Theory And Experiment

Author: Robert L. Devaney

Publisher: Westview Press

ISBN: 9780813345475

Category: Science

Page: 320

View: 884

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A First Course in Chaotic Dynamical Systems: Theory and Experiment is the first book to introduce modern topics in dynamical systems at the undergraduate level. Accessible to readers with only a background in calculus, the book integrates both theory and computer experiments into its coverage of contemporary ideas in dynamics. It is designed as a gradual introduction to the basic mathematical ideas behind such topics as chaos, fractals, Newton's method, symbolic dynamics, the Julia set, and the Mandelbrot set, and includes biographies of some of the leading researchers in the field of dynamical systems. Mathematical and computer experiments are integrated throughout the text to help illustrate the meaning of the theorems presented.Chaotic Dynamical Systems Software, Labs 1–6 is a supplementary laboratory software package, available separately, that allows a more intuitive understanding of the mathematics behind dynamical systems theory. Combined with A First Course in Chaotic Dynamical Systems, it leads to a rich understanding of this emerging field.

Chaos

An Introduction to Dynamical Systems

Author: Kathleen Alligood,Tim Sauer,J.A. Yorke

Publisher: Springer

ISBN: 3642592813

Category: Mathematics

Page: 603

View: 7530

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BACKGROUND Sir Isaac Newton hrought to the world the idea of modeling the motion of physical systems with equations. It was necessary to invent calculus along the way, since fundamental equations of motion involve velocities and accelerations, of position. His greatest single success was his discovery that which are derivatives the motion of the planets and moons of the solar system resulted from a single fundamental source: the gravitational attraction of the hodies. He demonstrated that the ohserved motion of the planets could he explained hy assuming that there is a gravitational attraction he tween any two ohjects, a force that is proportional to the product of masses and inversely proportional to the square of the distance between them. The circular, elliptical, and parabolic orhits of astronomy were v INTRODUCTION no longer fundamental determinants of motion, but were approximations of laws specified with differential equations. His methods are now used in modeling motion and change in all areas of science. Subsequent generations of scientists extended the method of using differ ential equations to describe how physical systems evolve. But the method had a limitation. While the differential equations were sufficient to determine the behavior-in the sense that solutions of the equations did exist-it was frequently difficult to figure out what that behavior would be. It was often impossible to write down solutions in relatively simple algebraic expressions using a finite number of terms. Series solutions involving infinite sums often would not converge beyond some finite time.

Chaos in Circuits and Systems

Author: Guanrong Chen,Tetsushi Ueta

Publisher: World Scientific

ISBN: 9789812705303

Category: Chaotic behavior in systems

Page: 641

View: 7096

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In this volume, leading experts present current achievements in the forefront of research in the challenging field of chaos in circuits and systems, with emphasis on engineering perspectives, methodologies, circuitry design techniques, and potential applications of chaos and bifurcation. A combination of overview, tutorial and technical articles, the book describes state-of-the-art research on significant problems in this field. It is suitable for readers ranging from graduate students, university professors, laboratory researchers and industrial practitioners to applied mathematicians and physicists in electrical, electronic, mechanical, physical, chemical and biomedical engineering and science.

Lectures on Chaotic Dynamical Systems

Author: Valentin Senderovich Afraĭmovich,Sze-Bi Hsu

Publisher: American Mathematical Soc.

ISBN: 9780821888315

Category: Mathematics

Page: 353

View: 4383

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This book is devoted to chaotic nonlinear dynamics. It presents a consistent, up-to-date introduction to the field of strange attractors, hyperbolic repellers, and nonlocal bifurcations. The authors keep the highest possible level of ''physical'' intuition while staying mathematically rigorous. In addition, they explain a variety of important nonstandard algorithms and problems involving the computation of chaotic dynamics. The book will help readers who are not familiar withnonlinear dynamics to understand and enjoy sophisticated modern monographs on dynamical systems and chaos. Intended for courses in either mathematics, physics, or engineering, prerequisites are calculus, differential equations, and functional analysis.

Dynamical Systems

Differential Equations, Maps, and Chaotic Behaviour

Author: C.M. Place

Publisher: Routledge

ISBN: 1351454277

Category: Mathematics

Page: 330

View: 4665

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This text discusses the qualitative properties of dynamical systems including both differential equations and maps. The approach taken relies heavily on examples (supported by extensive exercises, hints to solutions and diagrams) to develop the material, including a treatment of chaotic behavior. The unprecedented popular interest shown in recent years in the chaotic behavior of discrete dynamic systems including such topics as chaos and fractals has had its impact on the undergraduate and graduate curriculum. However there has, until now, been no text which sets out this developing area of mathematics within the context of standard teaching of ordinary differential equations. Applications in physics, engineering, and geology are considered and introductions to fractal imaging and cellular automata are given.