Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields (Applied Mathematical Sciences Vol. 42)

Book Description

From the reviews: "This book is concerned with the application of methods from dynamical systems and bifurcation theories to the study of nonlinear oscillations. Chapter 1 provides a review of basic results in the theory of dynamical systems, covering both ordinary differential equations and discrete mappings. Chapter 2 presents 4 examples from nonlinear oscillations. Chapter 3 contains a discussion of the methods of local bifurcation theory for flows and maps, including center manifolds and normal forms. Chapter 4 develops analytical methods of averaging and perturbation theory. Close analysis of geometrically defined two-dimensional maps with complicated invariant sets is discussed in chapter 5. Chapter 6 covers global homoclinic and heteroclinic bifurcations. The final chapter shows how the global bifurcations reappear in degenerate local bifurcations and ends with several more models of physical problems which display these behaviors." #Book Review - Engineering Societies Library, New York#1 "An attempt to make research tools concerning `strange attractors' developed in the last 20 years available to applied scientists and to make clear to research mathematicians the needs in applied works. Emphasis on geometric and topological solutions of differential equations. Applications mainly drawn from nonlinear oscillations." #American Mathematical Monthly#2

Customer Reviews:

Will never collect dust...........2001-06-03

This book has been a continuing source of information and guidance for 18 years now. Students and researchers in many different fields have used this book due to its breadth and detail of coverage. The book does require a fairly advanced mathematical background, but the authors do include a glossary for the reader lacking this.

Chapter one is an overview of differential equations and dynamical systems. All the concepts needed for a study of such systems are discussed in great detail and also very informally, stressing instead the understanding of the concepts, and not merely their definition. Some of the proofs of the main results, such as the Hartman-Grobman and the stable manifold theorems, are omitted however.

This is followed in Chapter 2 by a very intuitive discussion of the van der Pols equation, Duffings equation, the Lorenz equations, and the bouncing ball. Numerical calculations are effectively employed to illustrate some of the main properties of the systems modeled by these equations.

A taste of bifurcation theory follows in Chapter 3. Center manifolds are defined and many examples are given, but the proof of the center manifold theorem is omitted unfortunately. Normal forms and Hopf bifurcations are treated in detail.

Averaging methods are discussed in Chapter 4, with part of the averaging theorem proved using a version of Gronwall's lemma. Several interesting examples of averaging are given, along with a discussion of to what extent the bifurcation properties of the averaged equations carry over to the original equations. Most importantly, this chapter discusses the Melnikov function, so very important in the study of small perturbations of dynamical systems with a hyperbolic fixed point. A full proof that simple zeros of the Melnikov function imply the transversal intersection of the stable and unstable manifolds is given.

Chapter 5 moves on to results of a more purely mathematical nature, where symbolic dynamics and the Smale horseshoe map are discussed. The proofs of the stable manifold theorem and the Palis lambda lemma are, however, omitted. Markov partitions and the shadowing lemma are discussed also but the latter is not proven. The authors do however give a proof of the Smale-Birkhoff homoclinic theorem. A purely mathematical overview of attractors is given along with measure-theoretic (ergodic) properties of dynamical systems.

The (local) bifurcation theory of Chapter 3 is extended to global bifurcations in the next chapter. A very detailed discussion of rotation numbers is given but the KAM theory is only briefly mentioned. The main emphasis is on 1-dimensional maps, the Lorentz system, and Silnikov theory. The authors give a very detailed treatment of wild hyperbolic sets.

The book ends with a discussion of bifurcations from equilibrium points that have multiple degeneracies. The discussion is more motivated from a physical standpont than the last few chapters. But some interesting mathematical constructions are employed, namely the role of k-jets, which have fascinating connections with algebraic goemetry, via the "blowing-up" techniques.

The concepts in the book have proven to have enduring value in the study of dynamical systems, and this book will no doubt continue to serve students and researchers in the years to come.

Background.......2001-01-11

Guckenheimer is one of my favourite book in nonlinear science. Another absolute reference. This books deserved to be milestone in nonlinear dynamics.

Changed the Nature of Science As We Know It........2000-01-26

This book has clearly withstood the test of time in over 15 years of continuous publication. On my bookcase, it stands among my most treasured and well-worn classics of fluid mechanics and differential equations--Hirsch and Smale, Birkhoff and Rota, Chandrasekhar, Bachelor, Lamb, Landau and Lifschitz... It changed many of the unquestioned assumptions of many fields besides my own. It redefined the terms of many scientific debates. And, it changed my life.

I obtained Guckenheimer and Holmes' classic when it first came out in 1983. It was so clear, concise and intellectually engaging that it inspired me to wonder whether the system of equations I was studying for my Ph.D. research at the time--the governing equations of thermal convection at infinite Prandtl number (which govern plate tectonics in the earth's mantle)--might have a chaotic solution. Guckenheimer and Holmes outlined a clear methodology to find out the answer.

My advisor at the University of Chicago thought not. Only steady solutions could be admitted in the absence of external forcing due to the lack of momentum transfer--this belief was widely held at the time, despite certain oscillatory solutions found by Fritz Busse (then at UCLA) and chaotic solutions found in certain limiting cases by Andrew Fowler at Oxford.

In despair, I left my studies at Chicago to work as a Unix sysadmin at my undergraduate alma mater --Cornell, where (unbeknownst to me when I took the job) John Guckenheimer had just relocated from UCSC. Delighted to find him there, I sat in on his courses. Later, with his help, I wrote a proposal to NASA to support the completion of my thesis--with him and Donald Turcotte serving as my advisors.

The 3-year fellowship was approved, and during this time I demonstrated and published that thermal convection at infinite Prandtl number--a condition that pervades many planetary interiors including our own--is indeed chaotic in the absence of external forcing.

Prior to this, planetary convection codes primarily looked for steady state solutions. Since, numerical analysts in the field have upgraded to time-dependent models. The source of chaos at infinite Prandtle number I identified--the heat advection term--is now widely accepted as the source of what is now called "Thermal Turbulence" in planetary interiors.

The defense at Chicago was quite an event. Since my new advisors were flown in from Ithaca, you might say my thesis--The Nonlinear Dynamics of Thermal Convection at Infinite Prandtl Number--passed with flying colors. Someone at Chicago might disagree, but his opinion is irrelevant.

Demonstrating the many possible solutions to a single set of equations and showing how the choice of solution depends very sensitively on the rather poorly-constrained initial conditions of the earth--does render mantle modeling itself rather superfluous and indeed, scientifically suspect. However, many important professors who stayed in the field nonetheless continue to run their time-dependent mantle convection codes, and never cease to wonder at the fact that they all get different results. It's rather amusing, really.

When all that too has passed away, the truths so beautifully put forth in Guckenheimer and Holmes will remain. Like I said, it's a classic. Furthermore, being number 42 in its series, it's got to be the answer to the ultimate question of life, the universe and everything. Was for me, anyway.

Basic and clasic.......1999-08-22

For the moment it is "the" book on Dynamical Systems, through the world. Its first chapter is a good introduction on the mathematics needed to aboard the subject. The second introduces chaos, and the rest is for a good understanding of the newest and prolific science.

Nonlinear Oscillations (Wiley Classics Library)

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A perfect book

Nonlinear Oscillations (Wiley Classics Library)
Ali H. Nayfeh , and Dean T. Mook
Manufacturer: Wiley-Interscience
ProductGroup: Book
Binding: Paperback

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ASIN: 0471121428

Book Description

Nonlinear Oscillations is a self-contained and thorough treatment of the vigorous research that has occurred in nonlinear mechanics since 1970. The book begins with fundamental concepts and techniques of analysis and progresses through recent developments and provides an overview that abstracts and introduces main nonlinear phenomena. It treats systems having a single degree of freedom, introducing basic concepts and analytical methods, and extends concepts and methods to systems having degrees of freedom. Most of this material cannot be found in any other text. Nonlinear Oscillations uses simple physical examples to explain nonlinear dispersive and nondispersive waves. The notation is unified and the analysis modified to conform to discussions. Solutions are worked out in detail for numerous examples, results are plotted and explanations are couched in physical terms. The book contains an extensive bibliography.

Customer Reviews:

A perfect book.......2000-01-22

This is a very important book for my study. In my research field, everyone regard that this book must be read. Today, I have seen some books written by the same authors, but I can't have opportunity to read them. If possible I want to know the authors address. Thank you. sundajun Tianjin University,China

Nonlinear Oscillations in Physical Systems

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Nonlinear Oscillations in Physical Systems
Chihiro Hayashi
Manufacturer: Princeton Univ Pr
ProductGroup: Book
Binding: Paperback

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ASIN: 0691083835

Introduction to Control of Oscillations and Chaos (World Scientific Series on Nonlinear Science. Series a, Monographs and Treatises, V. 35)

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Introduction to Control of Oscillations and Chaos (World Scientific Series on Nonlinear Science. Series a, Monographs and Treatises, V. 35)
A. L. Fradkov , and Alexander L. Fradkov
Manufacturer: World Scientific Publishing Company
ProductGroup: Book
Binding: Hardcover

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ASIN: 9810230699

Book Description

This book gives an exposition of the exciting field of control of oscillatory and chaotic systems, which has numerous potential applications in mechanics, laser and chemical technologies, communications, biology and medicine, economics, ecology, etc.

A novelty of the book is its systematic application of modern nonlinear and adaptive control theory to the new class of problems. The proposed control design methods are based on the concepts of Lyapunov functions, Poincare maps, speed-gradient and gradient algorithms. The conditions which ensure such control goals as an excitation or suppression of oscillations, synchronization and transformation from chaotic mode to the periodic one or vice versa, are established. The performance and robustness of control systems under disturbances and uncertainties are evaluated.

The described methods and algorithms are illustrated by a number of examples, including classical models of oscillatory and chaotic systems: coupled pendula, brusselator, Lorenz, Van der Pol, Duffing, Henon and Chua systems. Practical examples from different fields of science and technology such as communications, growth of thin films, synchronization of chaotic generators based on tunnel diods, stabilization of swings in power systems, increasing predictability of business-cycles are also presented.

The book includes many results on nonlinear and adaptive control published previously in Russian and therefore were not known to the West.

Book Description

"A good book for a nice price!" (Monatshefte für Mathematik)

"... for lecture courses that cover the classical theory of nonlinear differential equations associated with Poincaré and Lyapunov and introduce the student to the ideas of bifurcation theory and chaos this is an ideal text ..." (Mathematika)

"The pedagogical style is excellent, consisting typically of an insightful overview followed by theorems, illustrative examples and exercises." (Choice)

Customer Reviews:

Springer dropped the ball.......2003-09-03

Over all, Verhulst does a ressonable job of conveying the central ideas of the topic. I found it to be slow going for a couple of reasons: First, it's clearly a translation and Springer should have done a better proofreading job ... many of the sentence constructions are unusual for English which is, at the very least, distracting. Second, there are substantial gaps in some of the arguments ... maybe ok for a book at this level, but it often creates considerable frustration.

In contrast, the book by Jordan and Smith (Nonlinear Ordinary Differential Equations) has few flaws and, in my view, should be read first.

This book is not even reasonable.......2001-07-09

This book was not written for an student of ODE, he was written to the author himself!

This book was used in a course of PhD here in Brazil and the results were very negative. I strongly don't recomend this book... Avoid it...

This book takes you by hand through dynamical systems theory.......2000-04-18

What I like more of this work is tha the autor explains all theconcepts he is using, so it is ideal for people who is a naturalscientist but not necesarilly knows all the formalism of modern mathematics. It is not hard to read and covers the basics for study recent research papers. END

This book takes you by hand through dynamical systems theory.......2000-04-18

Regular and Chaotic Dynamics (Applied Mathematical Sciences)

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Regular and Chaotic Dynamics (Applied Mathematical Sciences)
A.J. Lichtenberg , and M.A. Lieberman
Manufacturer: Springer
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Binding: Hardcover

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ASIN: 0387977457

Book Description

This book treats nonlinear dynamics in both Hamiltonian and dissipative systems. The emphasis is on the mechanics for generating chaotic motion, methods of calculating the transitions from regular to chaotic motion, and the dynamical and statistical properties of the dynamics when it is chaotic. The book is intended as a self consistent treatment of the subject at the graduate level and as a reference for scientists already working in the field. It emphasizes both methods of calculation and results. It is accessible to physicists and engineers without training in modern mathematics. The new edition brings the subject matter in a rapidly expanding field up to date, and has greatly expanded the treatment of dissipative dynamics to include most important subjects. It can be used as a graduate text for a two semester course covering both Hamiltonian and dissipative dynamics.

Nonlinearities in Action: Oscillations, Chaos, Order, Fractals

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Nonlinearities in Action: Oscillations, Chaos, Order, Fractals
Andrei V. Gaponov-Grekhov , and Mikhail I. Rabinovich
Manufacturer: Springer
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Binding: Hardcover

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Book Description

This concise and comprehensive overview of nonlinear processes addresses all those interested in natural sciences and mathematics. It also contains a beautifully illustrated color insert easily accessible to the interested layperson. Thus it is suitable for leisure reading and also for an introductory (under)graduate course in nonlinear physics. Both well-established and more recent new results are discussed, outlining the relation between classical aspects of nonlinear physics and important current problems like the birth of chaos in simple deterministic systems and the emergence of order out of disorder and turbulence. Keywords: Chaos, fractals, strange attractors, turbulence.

An Introduction to Nonlinear Chemical Dynamics: Oscillations, Waves, Patterns, and Chaos (Topics in Physical Chemistry)

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An Introduction to Nonlinear Chemical Dynamics: Oscillations, Waves, Patterns, and Chaos (Topics in Physical Chemistry)
Irving R. Epstein , and John A. Pojman
Manufacturer: Oxford University Press, USA
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Binding: Hardcover

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ASIN: 0195096703

Book Description

Just a few decades ago, chemical oscillations were thought to be exotic reactions of only theoretical interest. Now known to govern an array of physical and biological processes, including the regulation of the heart, these oscillations are being studied by a diverse group across the sciences. This book is the first introduction to nonlinear chemical dynamics written specifically for chemists. It covers oscillating reactions, chaos, and chemical pattern formation, and includes numerous practical suggestions on reactor design, data analysis, and computer simulations. Assuming only an undergraduate knowledge of chemistry, the book is an ideal starting point for research in the field. The book begins with a brief history of nonlinear chemical dynamics and a review of the basic mathematics and chemistry. The authors then provide an extensive overview of nonlinear dynamics, starting with the flow reactor and moving on to a detailed discussion of chemical oscillators. Throughout the authors emphasize the chemical mechanistic basis for self-organization. The overview is followed by a series of chapters on more advanced topics, including complex oscillations, biological systems, polymers, interactions between fields and waves, and Turing patterns. Underscoring the hands-on nature of the material, the book concludes with a series of classroom-tested demonstrations and experiments appropriate for an undergraduate laboratory.

Asymptotic Approaches in Nonlinear Dynamics: New Trends and Applications (Springer Series in Synergetics)

Average customer rating:

Very good book, far too expensive.

Asymptotic Approaches in Nonlinear Dynamics: New Trends and Applications (Springer Series in Synergetics)
Jan Awrejcewicz , Leonid I. Manevitch , and I. V. Andrianov
Manufacturer: Springer
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Binding: Hardcover

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Book Description

This book covers developments in the field of the theory of oscillations from the diverse viewpoints, reflecting the field's multidisciplinary nature. Addressing researchers in mechanics, physics, applied mathematics, and engineering, as well as students, the book gives an introduction to the state of the art in this area and to various applications. For the first time a treatment of the asymptotic and homogenization methods in the theory of oscillations in combination with Pad approximations is presented. Because of its wealth of interesting examples this book will prove useful as an introduction to the field for novices and a reference for specialists.

Customer Reviews:

Very good book, far too expensive........2004-04-28

The book covers an impressive number of problems, from the most simple to the more complex. There are very few errors in the book, just a few small typos. Most of these topics were previously available only in the specialized literature. This is one of the few attempts at presenting them in a monograph. The authors have done a good job.

My main complaint is the cost ($119 on Amazon) of this book, which in my opinion, is completely obscene. I bought a copy of the hardcover edition 2 years ago for $49 brand new at a university bookstore. How could the price possibly have gone from $49 to $119 in 2 years?

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