Ebook: Elementary Stability and Bifurcation Theory

In its most general form bifurcation theory is a theory of asymptotic solutions of nonlinear equations. By asymptotic solutions we mean, for example, steady solutions, time-periodic solutions, and quasi-periodic solutions. The purpose of this book is to teach the theory of bifurcation of asymptotic solutions of evolution problems governed by nonlinear differential equations. We have written this book for the broadest audience of potentially interested learners: engineers, biologists, chemists, physicists, mathematicians, economists, and others whose work involves understanding asymptotic solutions of nonlinear differential equations. To accomplish our aims, we have thought it necessary to make the analysis: (1) general enough to apply to the huge variety of applications which arise in science and technology; and (2) simple enough so that it can be understood by persons whose mathe­ matical training does not extend beyond the classical methods of analysis which were popular in the nineteenth century. Of course, it is not possible to achieve generality and simplicity in a perfect union but, in fact, the general theory is simpler than the detailed theory required for particular applications. The general theory abstracts from the detailed problems only the essential features and provides the student with the skeleton on which detailed structures of the applications must rest. lt is generally believed that the mathematical theory of bifurcation requires some functional analysis and some ofthe methods of topology and dynamics.

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This second edition has been substantially revised. Its purpose is to teach the theory of bifurcation of asymptotic solutions of evolution problems governed by nonlinear differential equations. It is written not only for mathematicians, but for the broadest audience of potentially interested learners, including engineers, biologists, chemists, physicists and economists. For this reason, it uses only well-known methods of classical analysis at a foundation level. Applications and examples are stressed throughout, and these were chosen to be as varied as possible.

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This second edition has been substantially revised. Its purpose is to teach the theory of bifurcation of asymptotic solutions of evolution problems governed by nonlinear differential equations. It is written not only for mathematicians, but for the broadest audience of potentially interested learners, including engineers, biologists, chemists, physicists and economists. For this reason, it uses only well-known methods of classical analysis at a foundation level. Applications and examples are stressed throughout, and these were chosen to be as varied as possible.
Content:
Front Matter....Pages i-xxiii
Asymptotic Solutions of Evolution Problems....Pages 1-9
Bifurcation and Stability of Steady Solutions of Evolution Equations in One Dimension....Pages 10-28
Imperfection Theory and Isolated Solutions Which Perturb Bifurcation....Pages 29-41
Stability of Steady Solutions of Evolution Equations in Two Dimensions and n Dimensions....Pages 42-58
Bifurcation of Steady Solutions in Two Dimensions and the Stability of the Bifurcating Solutions....Pages 59-86
Methods of Projection for General Problems of Bifurcation into Steady Solutions....Pages 87-138
Bifurcation of Periodic Solutions from Steady Ones (Hopf Bifurcation) in Two Dimensions....Pages 139-155
Bifurcation of Periodic Solutions in the General Case....Pages 156-176
Subharmonic Bifurcation of Forced T-Periodic Solutions....Pages 177-207
Bifurcation of Forced T-Periodic Solutions into Asymptotically Quasi-Periodic Solutions....Pages 208-255
Secondary Subharmonic and Asymptotically Quasi-Periodic Bifurcation of Periodic Solutions (of Hopf’s Type) in the Autonomous Case....Pages 256-302
Stability and Bifurcation in Conservative Systems....Pages 303-318
Back Matter....Pages 319-326

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This second edition has been substantially revised. Its purpose is to teach the theory of bifurcation of asymptotic solutions of evolution problems governed by nonlinear differential equations. It is written not only for mathematicians, but for the broadest audience of potentially interested learners, including engineers, biologists, chemists, physicists and economists. For this reason, it uses only well-known methods of classical analysis at a foundation level. Applications and examples are stressed throughout, and these were chosen to be as varied as possible.
Content:
Front Matter....Pages i-xxiii
Asymptotic Solutions of Evolution Problems....Pages 1-9
Bifurcation and Stability of Steady Solutions of Evolution Equations in One Dimension....Pages 10-28
Imperfection Theory and Isolated Solutions Which Perturb Bifurcation....Pages 29-41
Stability of Steady Solutions of Evolution Equations in Two Dimensions and n Dimensions....Pages 42-58
Bifurcation of Steady Solutions in Two Dimensions and the Stability of the Bifurcating Solutions....Pages 59-86
Methods of Projection for General Problems of Bifurcation into Steady Solutions....Pages 87-138
Bifurcation of Periodic Solutions from Steady Ones (Hopf Bifurcation) in Two Dimensions....Pages 139-155
Bifurcation of Periodic Solutions in the General Case....Pages 156-176
Subharmonic Bifurcation of Forced T-Periodic Solutions....Pages 177-207
Bifurcation of Forced T-Periodic Solutions into Asymptotically Quasi-Periodic Solutions....Pages 208-255
Secondary Subharmonic and Asymptotically Quasi-Periodic Bifurcation of Periodic Solutions (of Hopf’s Type) in the Autonomous Case....Pages 256-302
Stability and Bifurcation in Conservative Systems....Pages 303-318
Back Matter....Pages 319-326
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