Analytic Theory of Global Bifurcation.

By:

Buffoni, Boris, 1965- [Author]

Material type: Text

text

Media type:

computer

Carrier type:

online resource

ISBN:

9781400884339

Subject(s):

Bifurcation theory

Genre/Form:

Electronic Books.

LOC classification:

QA380 .A535 2016

Online resources:

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Available additional physical forms:

Contents:

Contents -- Preface -- Chapter 1. Introduction -- Part 1. Linear and Nonlinear Functional Analysis -- Chapter 2. Linear Functional Analysis -- Chapter 3. Calculus in Banach Spaces -- Chapter 4. Multilinear and Analytic Operators -- Part 2. Analytic Varieties -- Chapter 5. Analytic Functions on F -- Chapter 6. Polynomials -- Chapter 7. Analytic Varieties -- Part 3. Bifurcation Theory -- Chapter 8. Local Bifurcation Theory -- Chapter 9. Global Bifurcation Theory -- Part IV. Stokes Waves -- Chapter 10. Steady Periodic Water Waves -- Chapter 11. Global Existence of Stokes Waves -- Bibliography -- Index

Subject: Rabinowitz's classical global bifurcation theory, which concerns the study in-the-large of parameter-dependent families of nonlinear equations, uses topological methods that address the problem of continuous parameter dependence of solutions by showing that there are connected sets of solutions of global extent. Even when the operators are infinitely differentiable in all the variables and parameters, connectedness here cannot in general be replaced by path-connectedness. However, in the context of real-analyticity there is an alternative theory of global bifurcation due to Dancer, which offers a much stronger notion of parameter dependence. This book aims to develop from first principles Dancer's global bifurcation theory for one-parameter families of real-analytic operators in Banach spaces. It shows that there are globally defined continuous and locally real-analytic curves of solutions. In particular, in the real-analytic setting, local analysis can lead to global consequences--for example, as explained in detail here, those resulting from bifurcation from a simple eigenvalue. Included are accounts of analyticity and implicit function theorems in Banach spaces, classical results from the theory of finite-dimensional analytic varieties, and the links between these two and global existence theory. Laying the foundations for more extensive studies of real-analyticity in infinite-dimensional problems and illustrating the theory with examples, Analytic Theory of Global Bifurcation is intended for graduate students and researchers in pure and applied analysis.

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Holdings
Item type	Current library	Collection	Call number	URL	Status	Date due	Barcode
Online Book (LOGIN USING YOUR MY CIU LOGIN AND PASSWORD)	G. Allen Fleece Library ONLINE	Non-fiction	QA380 (Browse shelf(Opens below))	Link to resource	Available		ocn957655684

Includes bibliographies and index.

Frontmatter -- Contents -- Preface -- Chapter 1. Introduction -- Part 1. Linear and Nonlinear Functional Analysis -- Chapter 2. Linear Functional Analysis -- Chapter 3. Calculus in Banach Spaces -- Chapter 4. Multilinear and Analytic Operators -- Part 2. Analytic Varieties -- Chapter 5. Analytic Functions on F -- Chapter 6. Polynomials -- Chapter 7. Analytic Varieties -- Part 3. Bifurcation Theory -- Chapter 8. Local Bifurcation Theory -- Chapter 9. Global Bifurcation Theory -- Part IV. Stokes Waves -- Chapter 10. Steady Periodic Water Waves -- Chapter 11. Global Existence of Stokes Waves -- Bibliography -- Index

Rabinowitz's classical global bifurcation theory, which concerns the study in-the-large of parameter-dependent families of nonlinear equations, uses topological methods that address the problem of continuous parameter dependence of solutions by showing that there are connected sets of solutions of global extent. Even when the operators are infinitely differentiable in all the variables and parameters, connectedness here cannot in general be replaced by path-connectedness. However, in the context of real-analyticity there is an alternative theory of global bifurcation due to Dancer, which offers a much stronger notion of parameter dependence. This book aims to develop from first principles Dancer's global bifurcation theory for one-parameter families of real-analytic operators in Banach spaces. It shows that there are globally defined continuous and locally real-analytic curves of solutions. In particular, in the real-analytic setting, local analysis can lead to global consequences--for example, as explained in detail here, those resulting from bifurcation from a simple eigenvalue. Included are accounts of analyticity and implicit function theorems in Banach spaces, classical results from the theory of finite-dimensional analytic varieties, and the links between these two and global existence theory. Laying the foundations for more extensive studies of real-analyticity in infinite-dimensional problems and illustrating the theory with examples, Analytic Theory of Global Bifurcation is intended for graduate students and researchers in pure and applied analysis.

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