Elements of Applied Bifurcation Theory
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Beschreibung
This is a book on nonlinear dynamical systems and their bifurcations under parameter variation. It provides a reader with a solid basis in dynamical systems theory, as well as explicit procedures for application of general mathematical results to particular problems. Special attention is given to efficient numerical implementations of the developed techniques. Several examples from recent research papers are used as illustrations. The book is designed for advanced undergraduate or graduate students in applied mathematics, as well as for Ph.D. students and researchers in physics, biology, engineering, and economics who use dynamical systems as model tools in their studies. A moderate mathematical background is assumed, and, whenever possible, only elementary mathematical tools are used. This new edition preserves the structure of the previous editions, while updating the context to incorporate recent theoretical and software developments and modern techniques for bifurcation analysis. Reviews of earlier editions: "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 "It is both a toolkit and a primer" - UK Nonlinear News
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.
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.
From Reviews of the First Edition: "I know of no other book that so clearly explains the basic phenomena of bifurcation theory." –Mathematical Reviews
This book provides a student or researcher with a solid basis in nonlinear dynamical systems and their bifurcations, giving them the necessary understanding of the approaches, methods, results and terminology used in the modern applied mathematics literature. It covers the basic topics of the bifurcation theory and can help in composing a course on nonlinear dynamical systems or system theory. This new edition preserves the structure of the previous edition, while updating the context to incorporate recent theoretical and software developments, in particular new and improved numerical methods for bifurcation analysis.
Autor*in
Yuri Kuznetsov
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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)
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