An Introduction to Dynamical Systems and Chaos

Author: G.C. Layek

Publisher: Springer

ISBN: 8132225562

Category: Mathematics

Page: 622

View: 7207

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The book discusses continuous and discrete systems in systematic and sequential approaches for all aspects of nonlinear dynamics. The unique feature of the book is its mathematical theories on flow bifurcations, oscillatory solutions, symmetry analysis of nonlinear systems and chaos theory. The logically structured content and sequential orientation provide readers with a global overview of the topic. A systematic mathematical approach has been adopted, and a number of examples worked out in detail and exercises have been included. Chapters 1–8 are devoted to continuous systems, beginning with one-dimensional flows. Symmetry is an inherent character of nonlinear systems, and the Lie invariance principle and its algorithm for finding symmetries of a system are discussed in Chap. 8. Chapters 9–13 focus on discrete systems, chaos and fractals. Conjugacy relationship among maps and its properties are described with proofs. Chaos theory and its connection with fractals, Hamiltonian flows and symmetries of nonlinear systems are among the main focuses of this book. Over the past few decades, there has been an unprecedented interest and advances in nonlinear systems, chaos theory and fractals, which is reflected in undergraduate and postgraduate curricula around the world. The book is useful for courses in dynamical systems and chaos, nonlinear dynamics, etc., for advanced undergraduate and postgraduate students in mathematics, physics and engineering.

Dynamical Systems and Chaos

Author: Henk Broer,Floris Takens

Publisher: Springer Science & Business Media

ISBN: 9781441968708

Category: Mathematics

Page: 313

View: 3195

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Over the last four decades there has been extensive development in the theory of dynamical systems. This book aims at a wide audience where the first four chapters have been used for an undergraduate course in Dynamical Systems. Material from the last two chapters and from the appendices has been used quite a lot for master and PhD courses. All chapters are concluded by an exercise section. The book is also directed towards researchers, where one of the challenges is to help applied researchers acquire background for a better understanding of the data that computer simulation or experiment may provide them with the development of the theory.

Introduction to Applied Nonlinear Dynamical Systems and Chaos

Author: Stephen Wiggins

Publisher: Springer Science & Business Media

ISBN: 1475740670

Category: Mathematics

Page: 672

View: 4932

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This volume is an introduction to applied nonlinear dynamics and chaos. The emphasis is on teaching the techniques and ideas that will enable students to take specific dynamical systems and obtain some quantitative information about their behavior. The new edition has been updated and extended throughout, and contains an extensive bibliography and a detailed glossary of terms.

Chaos

An Introduction to Dynamical Systems

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

Publisher: Springer

ISBN: 3642592813

Category: Mathematics

Page: 603

View: 7417

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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.

Introduction to Discrete Dynamical Systems and Chaos

Author: Mario Martelli

Publisher: John Wiley & Sons

ISBN: 1118031121

Category: Mathematics

Page: 344

View: 4192

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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. An Instructor's Manual presenting detailed solutions to all the problems in the book is available upon request from the Wiley editorial department.

Chaos in Dynamical Systems

Author: Edward Ott

Publisher: Cambridge University Press

ISBN: 9780521010849

Category: Mathematics

Page: 478

View: 9763

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

Discrete Dynamical Systems, Bifurcations and Chaos in Economics

Author: Wei-Bin Zhang

Publisher: Elsevier

ISBN: 9780080462462

Category: Mathematics

Page: 460

View: 6509

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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

Nonlinear Dynamical Systems and Chaos

Author: Henk W Broer,I. Hoveijn,F. Takens,S.A. van Gils

Publisher: Birkhäuser

ISBN: 3034875185

Category: Mathematics

Page: 463

View: 9010

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Symmetries in dynamical systems, "KAM theory and other perturbation theories", "Infinite dimensional systems", "Time series analysis" and "Numerical continuation and bifurcation analysis" were the main topics of the December 1995 Dynamical Systems Conference held in Groningen in honour of Johann Bernoulli. They now form the core of this work which seeks to present the state of the art in various branches of the theory of dynamical systems. A number of articles have a survey character whereas others deal with recent results in current research. It contains interesting material for all members of the dynamical systems community, ranging from geometric and analytic aspects from a mathematical point of view to applications in various sciences.

An Exploration of Dynamical Systems and Chaos

Completely Revised and Enlarged Second Edition

Author: John Argyris,Gunter Faust,Maria Haase,Rudolf Friedrich

Publisher: Springer

ISBN: 3662460424

Category: Technology & Engineering

Page: 865

View: 9735

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This book is conceived as a comprehensive and detailed text-book on non-linear dynamical systems with particular emphasis on the exploration of chaotic phenomena. The self-contained introductory presentation is addressed both to those who wish to study the physics of chaotic systems and non-linear dynamics intensively as well as those who are curious to learn more about the fascinating world of chaotic phenomena. Basic concepts like Poincaré section, iterated mappings, Hamiltonian chaos and KAM theory, strange attractors, fractal dimensions, Lyapunov exponents, bifurcation theory, self-similarity and renormalisation and transitions to chaos are thoroughly explained. To facilitate comprehension, mathematical concepts and tools are introduced in short sub-sections. The text is supported by numerous computer experiments and a multitude of graphical illustrations and colour plates emphasising the geometrical and topological characteristics of the underlying dynamics. This volume is a completely revised and enlarged second edition which comprises recently obtained research results of topical interest, and has been extended to include a new section on the basic concepts of probability theory. A completely new chapter on fully developed turbulence presents the successes of chaos theory, its limitations as well as future trends in the development of complex spatio-temporal structures. "This book will be of valuable help for my lectures" Hermann Haken, Stuttgart "This text-book should not be missing in any introductory lecture on non-linear systems and deterministic chaos" Wolfgang Kinzel, Würzburg “This well written book represents a comprehensive treatise on dynamical systems. It may serve as reference book for the whole field of nonlinear and chaotic systems and reports in a unique way on scientific developments of recent decades as well as important applications.” Joachim Peinke, Institute of Physics, Carl-von-Ossietzky University Oldenburg, Germany

Dynamical Systems

Stability, Symbolic Dynamics, and Chaos

Author: Clark Robinson

Publisher: CRC Press

ISBN: 1482227878

Category: Mathematics

Page: 520

View: 3137

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

Discrete dynamical systems and chaos

Author: Mario Martelli

Publisher: Chapman & Hall/CRC

ISBN: N.A

Category: Mathematics

Page: 282

View: 2368

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In a balanced presentation, this Monograph presents definitions and results on dynamical systems and chaos in a manner accessible to undergraduates in a range of disciplines with of theoretical results and practical applications.

An Introduction To Chaotic Dynamical Systems

Author: Robert Devaney

Publisher: CRC Press

ISBN: 0429981937

Category: Science

Page: 360

View: 7649

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The study of nonlinear dynamical systems has exploded in the past 25 years, and Robert L. Devaney has made these advanced research developments accessible to undergraduate and graduate mathematics students as well as researchers in other disciplines with the introduction of this widely praised book. In this second edition of his best-selling text, Devaney includes new material on the orbit diagram fro maps of the interval and the Mandelbrot set, as well as striking color photos illustrating both Julia and Mandelbrot sets. This book assumes no prior acquaintance with advanced mathematical topics such as measure theory, topology, and differential geometry. Assuming only a knowledge of calculus, Devaney introduces many of the basic concepts of modern dynamical systems theory and leads the reader to the point of current research in several areas.

Bifurcation and Chaos in Simple Dynamical Systems

Author: Jan Awrejcewicz

Publisher: World Scientific

ISBN: 9789810200381

Category: Science

Page: 126

View: 9553

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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.

Laws of Chaos

Invariant Measures and Dynamical Systems in One Dimension

Author: Abraham Boyarsky,Pawel Gora

Publisher: Springer Science & Business Media

ISBN: 1461220246

Category: Mathematics

Page: 400

View: 4790

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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.

Differential Equations, Dynamical Systems, and an Introduction to Chaos

Author: Morris W. Hirsch,Stephen Smale,Robert L. Devaney

Publisher: Academic Press

ISBN: 0123820103

Category: Mathematics

Page: 418

View: 9925

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Hirsch, Devaney, and Smale's classic Differential Equations, Dynamical Systems, and an Introduction to Chaos has been used by professors as the primary text for undergraduate and graduate level courses covering differential equations. It provides a theoretical approach to dynamical systems and chaos written for a diverse student population among the fields of mathematics, science, and engineering. Prominent experts provide everything students need to know about dynamical systems as students seek to develop sufficient mathematical skills to analyze the types of differential equations that arise in their area of study. The authors provide rigorous exercises and examples clearly and easily by slowly introducing linear systems of differential equations. Calculus is required as specialized advanced topics not usually found in elementary differential equations courses are included, such as exploring the world of discrete dynamical systems and describing chaotic systems. Classic text by three of the world's most prominent mathematicians Continues the tradition of expository excellence Contains updated material and expanded applications for use in applied studies

Nonlinear Differential Equations and Dynamical Systems

Author: Ferdinand Verhulst

Publisher: Springer Science & Business Media

ISBN: 3642614531

Category: Mathematics

Page: 306

View: 4643

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For lecture courses that cover the classical theory of nonlinear differential equations associated with Poincare and Lyapunov and introduce the student to the ideas of bifurcation theory and chaos, this text is ideal. Its excellent pedagogical style typically consists of an insightful overview followed by theorems, illustrative examples, and exercises.

Bifurcation and Chaos in Complex Systems

Author: N.A

Publisher: Elsevier

ISBN: 9780080462660

Category: Science

Page: 400

View: 2407

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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

Nonlinear Dynamics and Chaos in Agricultural Systems

Author: Kenshi Sakai

Publisher: Gulf Professional Publishing

ISBN: 9780444506467

Category: Mathematics

Page: 204

View: 1754

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This book provides an introduction to the analysis of chaos and chaos theory as it relates to agricultural science. With clear explanations of chaos theory and principles, the first part of the book offers some basic facts, the fundamental terminology, and the concepts of deterministic chaos. The second part of this volume contains rich applications of the theory as applied to real agricultural systems. Applications include a wide area such as alternate bearing in tree crops, weed control and tillage, nonlinear vibrations in agricultural tractors, and piglet pricing analysis. Readers will find useful tools for calculating the order, rules and theory behind complex phenomena observed in arable land.

Chaos and Complexity in Psychology

The Theory of Nonlinear Dynamical Systems

Author: Stephen J. Guastello,Matthijs Koopmans,David Pincus

Publisher: Cambridge University Press

ISBN: 1139867261

Category: Psychology

Page: N.A

View: 1006

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While many books have discussed methodological advances in nonlinear dynamical systems theory (NDS), this volume is unique in its focus on NDS's role in the development of psychological theory. After an introductory chapter covering the fundamentals of chaos, complexity and other nonlinear dynamics, subsequent chapters provide in-depth coverage of each of the specific topic areas in psychology. A concluding chapter takes stock of the field as a whole, evaluating important challenges for the immediate future. The chapters are written by experts in the use of NDS in each of their respective areas, including biological, cognitive, developmental, social, organizational and clinical psychology. Each chapter provides an in-depth examination of theoretical foundations and specific applications and a review of relevant methods. This edited collection represents the state of the art in NDS science across the disciplines of psychology.