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Elements of Applied Bifurcation Theory

  • Book
  • © 1998

Overview

Authors:
  1. Yuri A. Kuznetsov

    1. Department of Mathematics, Utrecht University, Utrecht, The Netherlands
      Institute of Mathematical Problems of Biology, Russian Academy of Sciences, Pushchino, Moscow Region, Russia

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    You can also search for this author in PubMed   Google Scholar

  • Dynamical systems continues to be a topic of current interest in mathematics, engineering, and physics.
  • This modern approach provides the reader with a solid basis in dynamical systems theory and the approaches, methods, results, and terminology used in the current applied mathematics literature.
  • Makes the theory understandable with illustrations of applications to nonlinear phenomena.
  • Whenever possible, only elementary mathematical tools are used.
  • It will be useful to a broad range of people who use dynamical systems as modeling tools.
  • Special attention given to efficient numerical implementations of the developed techniques.
  • New edition incorporates recent theoretical developments, in particular new and improved numerical methods for bifurcation analysis.

Part of the book series: Applied Mathematical Sciences (AMS, volume 112)

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Table of contents (10 chapters)

  1. Front Matter

    Pages i-xx
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  2. Introduction to Dynamical Systems

    Pages 1-37

  3. Topological Equivalence, Bifurcations, and Structural Stability of Dynamical Systems

    Pages 39-78

  4. One-Parameter Bifurcations of Equilibria in Continuous-Time Dynamical Systems

    Pages 79-112

  5. One-Parameter Bifurcations of Fixed Points in Discrete-Time Dynamical Systems

    Pages 113-150

  6. Bifurcations of Equilibria and Periodic Orbits in n-Dimensional Dynamical Systems

    Pages 151-194

  7. Bifurcations of Orbits Homoclinic and Heteroclinic to Hyperbolic Equilibria

    Pages 195-248

  8. Other One-Parameter Bifurcations in Continuous-Time Dynamical Systems

    Pages 249-291

  9. Two-Parameter Bifurcations of Equilibria in Continuous-Time Dynamical Systems

    Pages 293-392

  10. Two-Parameter Bifurcations of Fixed Points in Discrete-Time Dynamical Systems

    Pages 393-462

  11. Numerical Analysis of Bifurcations

    Pages 463-539

  12. Back Matter

    Pages 541-591
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Keywords

About this book

The favorable reaction to the ?rst edition of this book con?rmed that the publication of such an application-oriented text on bifurcation theory of dynamical systems was well timed. The selected topics indeed cover - jor practical issues of applying the bifurcation theory to ?nite-dimensional problems. This new edition preserves the structure of the ?rst edition while updating the context to incorporate recent theoretical developments, in particular, new and improved numerical methods for bifurcation analysis. The treatment of some topics has been clari?ed. Major additions can be summarized as follows: In Chapter 3, an e- mentary proof of the topological equivalence of the original and truncated normal forms for the fold bifurcation is given. This makes the analysis of codimension-one equilibrium bifurcations of ODEs in the book complete. This chapter also includes an example of the Hopf bifurcation analysis in a planar system using MAPLE, a symbolic manipulation software. Chapter 4 includes a detailed normal form analysis of the Neimark-Sacker bif- cation in the delayed logistic map. In Chapter 5, we derive explicit f- mulas for the critical normal form coe?cients of all codim 1 bifurcations of n-dimensional iterated maps (i. e. , fold, ?ip, and Neimark-Sacker bif- cations). The section on homoclinic bifurcations in n-dimensional ODEs in Chapter 6 is completely rewritten and introduces the Melnikov in- gral that allows us to verify the regularity of the manifold splitting under parameter variations.

Reviews

Review of earlier edition

"I know of no other book that so clearly explains the basic phenomena of bifurcation theory." Math Reviews "The book is a fine addition to the dynamical systems literature. It is good to see, in our modern rush to quick publication, that we, as a mathematical community, still have time to bring together, and in such a readable and considered form, the important results on our subject." Bulletin of the AMS

From the reviews of the third edition:

"In the third edition of this textbook, the material again has been slightly extended while the main structure of the book was kept. … the clear structure of the book allows applied scientists to use it as a reference book. … Kuznetsov’s book on applied bifurcation theory is still very useful both as a textbook and as a reference work for researchers from the natural sciences, engineering or economics." (Jörg Härterich, Zentralblatt MATH, Vol. 1082, 2006)

Authors and Affiliations

  • Department of Mathematics, Utrecht University, Utrecht, The Netherlands

    Yuri A. Kuznetsov

  • Institute of Mathematical Problems of Biology, Russian Academy of Sciences, Pushchino, Moscow Region, Russia

    Yuri A. Kuznetsov

Bibliographic Information

  • Book Title: Elements of Applied Bifurcation Theory

  • Authors: Yuri A. Kuznetsov

  • Series Title: Applied Mathematical Sciences

  • DOI: https://doi.org/10.1007/b98848

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag New York 1998

  • eBook ISBN: 978-0-387-22710-8Published: 10 January 2008

  • Series ISSN: 0066-5452

  • Series E-ISSN: 2196-968X

  • Edition Number: 2

  • Number of Pages: XIX, 594

  • Topics: Analysis

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