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Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields

  • Book
  • © 1983

Overview

Authors:
  1. John Guckenheimer

    1. Department of Mathematics, Cornell University, Ithaca, USA

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  2. Philip Holmes

    1. Department of Mechanical and Aerospace Engineering and Program in Applied and Computational Mathematics, Princeton University, Princeton, USA

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    You can also search for this author in PubMed   Google Scholar

Part of the book series: Applied Mathematical Sciences (AMS, volume 42)

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About this book

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

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Table of contents (7 chapters)

  1. Front Matter

    Pages i-xvi
    Download chapter PDF

  2. Introduction: Differential Equations and Dynamical Systems

    • John Guckenheimer, Philip Holmes
    Pages 1-65

  3. An Introduction to Chaos: Four Examples

    • John Guckenheimer, Philip Holmes
    Pages 66-116

  4. Local Bifurcations

    • John Guckenheimer, Philip Holmes
    Pages 117-165

  5. Averaging and Perturbation from a Geometric Viewpoint

    • John Guckenheimer, Philip Holmes
    Pages 166-226

  6. Hyperbolic Sets, Symbolic Dynamics, and Strange Attractors

    • John Guckenheimer, Philip Holmes
    Pages 227-288

  7. Global Bifurcations

    • John Guckenheimer, Philip Holmes
    Pages 289-352

  8. Local Codimension Two Bifurcations of Flows

    • John Guckenheimer, Philip Holmes
    Pages 353-420

  9. Back Matter

    Pages 421-462
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Reviews

J. Guckenheimer and P. Holmes

Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields

"The book is rewarding reading . . . The elementary chapters are suitable for an introductory graduate course for mathematicians and physicists . . . Its excellent survey of the mathematical literature makes it a valuable reference."—JOURNAL OF STATISTICAL PHYSICS

Authors and Affiliations

  • Department of Mathematics, Cornell University, Ithaca, USA

    John Guckenheimer

  • Department of Mechanical and Aerospace Engineering and Program in Applied and Computational Mathematics, Princeton University, Princeton, USA

    Philip Holmes

Bibliographic Information

  • Book Title: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields

  • Authors: John Guckenheimer, Philip Holmes

  • Series Title: Applied Mathematical Sciences

  • DOI: https://doi.org/10.1007/978-1-4612-1140-2

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer Science+Business Media New York 1983

  • Hardcover ISBN: 978-0-387-90819-9Published: 01 August 1983

  • Softcover ISBN: 978-1-4612-7020-1Published: 23 November 2013

  • eBook ISBN: 978-1-4612-1140-2Published: 21 November 2013

  • Series ISSN: 0066-5452

  • Series E-ISSN: 2196-968X

  • Edition Number: 1

  • Number of Pages: XVI, 462

  • Topics: Analysis

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