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

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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.

Exploring Nature's Dynamics (Wiley Series in Nonlinear Science)

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Exploring Nature's Dynamics (Wiley Series in Nonlinear Science)
E. Atlee Jackson
Manufacturer: Wiley-Interscience
ProductGroup: Book
Binding: Hardcover

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ASIN: 0471191469

Book Description

Explore the remarkable variety of nature's dynamics
The development of the computer over the latter half of the twentieth century has greatly advanced our ability to explore the complex dynamics that occur in nature. With the aid of the computer, we can now study nonlinear types of dynamics that cannot generally be studied by mathematics. Realistic computer models of natural dynamics can now be developed, and even the simplest have uncovered remarkable and unexpected types of natural dynamics.
Researchers have found that simple dynamic actions over short periods of time can produce long-term dynamics that were never dreamt of in the past. Even the smallest changes in a system's short-time behavior can have enormous effects on the future of the system. Slightly different initial configurations can also produce very different future dynamics, which is now recognized as the important "sensitivity" feature of many dynamic systems.
Exploring Nature's Dynamics offers an introductory opportunity to learn about-and explore for yourself-some of the diverse forms of dynamics that occur in nature's reproduction processes. These forms include the competitive and cooperative interactions between species, neurological behaviors, dynamic spatial organizations, and the amazing constructive contributions of chaotic dynamics to our minds and hearts, the evolution of our solar system, and the human species itself. All you need to explore these ideas is a healthy curiosity and your own imagination. The accompanying disk includes simple Qbasic computer programs that allow you to witness dynamic systems at work. The methods for using these programs-and for making modifications based on your own creativity-are fully explained for people who have never used computer programs before. So come explore!

International Conference On Dynamical Systems (Research Notes in Mathematics Series)

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International Conference On Dynamical Systems (Research Notes in Mathematics Series)
F Ledrappier , Sheldon Newhouse , and Jorge Lewowicz
Manufacturer: Chapman & Hall/CRC
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Binding: Hardcover

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This volume clearly reflects Ricardo Mane's legacy, his contribution to mathematics and the diversity of his mathematical intersts. It contains fifteen refereed research papers on thems including Hamiltonian and Lagrangian dynamics, growth rate of the number of geodesics on a compact manifold, one dimensional complex and real dynamics, and bifurcations and singular cycles. This book also contains two famous sets of notes by Ricardo Mane. One is the seminal paper on Lagrangian dynamics that he had prepared for the conference; the other is on the genericity of zero exponents area preserving diffeomorphisms on surfaces when non Anosov. This book will be of particular interest to researchers and graduate students in mathematics, mechanics and mathematical physics.

Classical Many-Body Problems Amenable to Exact Treatments: (Solvable and/or Integrable and/or Linearizable...) in One-, Two- and Three-Dimensional Space (Lecture Notes in Physics)

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Classical Many-Body Problems Amenable to Exact Treatments: (Solvable and/or Integrable and/or Linearizable...) in One-, Two- and Three-Dimensional Space (Lecture Notes in Physics)
Francesco Calogero
Manufacturer: Springer
ProductGroup: Book
Binding: Hardcover

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ASIN: 3540417648

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This book focuses on exactly treatable classical (i.e. non-quantal non-relativistic) many-body problems, as described by Newton's equation of motion for mutually interacting point particles. Most of the material is based on the author's research and is published here for the first time in book form. One of the main novelties is the treatment of problems in two- and three-dimensional space. Many related techniques are presented, e.g. the theory of generalized Lagrangian-type interpolation in higher-dimensional spaces.This book is written for students as well as for researchers; it works out detailed examples before going on to treat more general cases. Many results are presented via exercises, with clear hints pointing to their solutions.

Methods of Qualitative Theory in Nonlinear Dynamics Part 2

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Methods of Qualitative Theory in Nonlinear Dynamics Part 2
Andrey L. Shilnikov , Dmitry V. Turaev , and Leon O. Chua
Manufacturer: World Scientific Publishing Company
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Binding: Hardcover

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Methods of Qualitative Theory in Nonlinear Dynamics (World Scientific Series on Nonlinear Science, Series a , Part 1)

ASIN: 9810240724

Infinite Dimensonal Dynamical Systems in Mechanics and Physics (Applied Mathematical Sciences)

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Infinite Dimensonal Dynamical Systems in Mechanics and Physics (Applied Mathematical Sciences)
Roger Temam
Manufacturer: Springer
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ASIN: 038794866X

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In this book the author presents the dynamical systems in infinite dimension, especially those generated by dissipative partial differential equations. This book attempts a systematic study of infinite dimensional dynamical systems generated by dissipative evolution partial differential equations arising in mechanics and physics and in other areas of sciences and technology. This second edition has been updated and extended.

Multi-Hamiltonian Theory of Dynamical Systems (Texts and Monographs in Physics)

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Multi-Hamiltonian Theory of Dynamical Systems (Texts and Monographs in Physics)

Manufacturer: Springer
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Binding: Hardcover

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ASIN: 354064251X

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A modern Hamiltonian theory offering a unified treatment of all types of systems (i.e. finite, lattice, and field) is presented. Particular attention is paid to nonlinear systems that have more than one Hamiltonian formulation in a single coordinate system. As this property is closely related to integrability, this book presents an algebraic theory of integrable systems. The book is intended for scientists, lecturers, and students interested in the field.

Applied Asymptotic Methods in Nonlinear Oscillations (Solid Mechanics and Its Applications)

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Compact and well structured

Applied Asymptotic Methods in Nonlinear Oscillations (Solid Mechanics and Its Applications)
Yuri A. Mitropolsky , and Nguyen Van Dao
Manufacturer: Springer
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Binding: Hardcover

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ASIN: 079234605X

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The present volume addresses the application of asymptotic methods in nonlinear oscillations. Such methods see a large variety of applications in physics, mechanics and engineering.
The advantages of using asymptotic methods in solving nonlinear problems are firstly simplicity, especially for computing higher approximations, and secondly their applicability to a large class of quasi-linear systems.
In contrast to the existing literature, this book is concerned with the applied aspects of the methods concerned and also contains problems relevant to the everyday practice of engineers, physicists and mathematicians.
Usually, dynamics systems are classified and examined by their degrees of freedom. This book is constructed from another point of view based upon the originating mechanism of the oscillations: free oscillation, self-excited oscillation, forced oscillation, and parametrically excited oscillation.
The text has been designed to cover material from the elementary to the more advanced, in increasing order of difficulty. It is of considerable interest to both students and researchers in applied mathematics, physical and mechanical sciences, and engineering.

Customer Reviews:

Compact and well structured.......1999-07-23

The book is a more compact version of the 'Asymptotic methods in the theory of nonlinear oscillations' by Mitropolskii and Bogoliobov published years back. Nonlinear equations close to linear ones are solved using the methods suggested by Krylov and Bogoliobov. The presentation is orderly dealing with free oscillations first, then self oscillations leading to forced oscillations. A neat article on the averaging method ends the book in a nice fashion bringing about some of the connections.

Chaos, Resonance and Collective Dynamical Phenomena in the Solar System (International Astronomical Union Symposia)

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Chaos, Resonance and Collective Dynamical Phenomena in the Solar System (International Astronomical Union Symposia)

Manufacturer: Springer
ProductGroup: Book
Binding: Paperback

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ASIN: 0792317823

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This symposium was devoted to a new celestial mechanics whose aim has become the study of such `objects' as the planetary system, planetary rings, the asteroidal belt, meteor swarms, satellite systems, comet families, the zodiacal cloud, the preplanetary nebula, etc. When the three-body problem is considered instead of individual orbits we are, now, looking for the topology of extended regions of its phase space. This Symposium was one step in the effort to close the ties between two scientific families: the observationally-oriented scientists and the theoretically-oriented scientists.

Chaotic Transport in Dynamical Systems (Interdisciplinary Applied Mathematics)

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Chaotic Transport in Dynamical Systems (Interdisciplinary Applied Mathematics)
S. Wiggins
Manufacturer: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
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Binding: Hardcover

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ASIN: 3540975225

Book Description

Provides a new and more realistic framework for describing the dynamics of non-linear systems. A number of issues arising in applied dynamical systems from the viewpoint of problems of phase space transport are raised in this monograph. Illustrating phase space transport problems arising in a variety of applications that can be modeled as time-periodic perturbations of planar Hamiltonian systems, the book begins with the study of transport in the associated two-dimensional Poincaré Map. This serves as a starting point for the further motivation of the transport issues through the development of ideas in a non-perturbative framework with generalizations to higher dimensions as well as more general time dependence. A timely and important contribution to those concerned with the applications of mathematics.

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