Bifurcation theory and catastrophe theory
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Bifurcation theory and catastrophe theory
Springer-Verlag, c1999
- : pbk
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Itogi nauki i tekhniki, sovremennye problemy matematiki
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Includes bibliographical references and indexes
"Title of the Russian edition: ... Dinamicheskie sistemy 5, publisher VINITI, Moscow 1986"--T.p. verso
"Second printing 1999 of the first edition 1994, which was originally published as Dynamical systems V, volumes 5 of the Encyclopaedia of mathematical sciences"--T.p. verso
Description and Table of Contents
Description
Bifurcation theory and catastrophe theory are two well-known areas within the field of dynamical systems. Both are studies of smooth systems, focusing on properties that seem to be manifestly non-smooth. Bifurcation theory is concerned with the sudden changes that occur in a system when one or more parameters are varied. Examples of such are familiar to students of differential equations, from phase portraits. Understanding the bifurcations of the differential equations that describe real physical systems provides important information about the behavior of the systems. Catastrophe theory became quite famous during the 1970's, mostly because of the sensation caused by the usually less than rigorous applications of its principal ideas to "hot topics", such as the characterization of personalities and the difference between a "genius" and a "maniac". Catastrophe theory is accurately described as singularity theory and its (genuine) applications. The authors of this book, previously published as Volume 5 of the Encyclopaedia, have given a masterly exposition of these two theories, with penetrating insight.
Table of Contents
I. Bifurcation Theory by V.I.Arnol'd, V.S.Afrajmovich, Yu.S.Il'yashenko, L.P.Shil'nikov: Preface.- 1. Bifurcations of Equilibria.- 2. Bifurcations of Limit Cycles.- 3. Nonlocal Bifurcations.- 4. Relaxation Oscillations.- Recommended Literature.- References II. Catastrophe Theory by V.I.Arnol'd: 1. Basic Concepts.- 2. The Theory of Catastrophes Before Poincare.- 3. The Theory of Bifurcations in the Work of Poincare.- 4. The Theory of Bifurcations in the Work of A.A.Andronov.- 5. Physicists' Treatment of Catastrophes Before Catastrophe Theory.- 6. Thom's Conjecture.- 7. Classifications of Singularities and Catastrophes.- Recommended Literature.- References
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