Hansjörg Kielhöfer

Bifurcation Theory (eBook, PDF)

An Introduction with Applications to Partial Differential Equations

• Format: PDF

Produktbeschreibung

In the past three decades, bifurcation theory has matured into a well-established and vibrant branch of mathematics. This book gives a unified presentation in an abstract setting of the main theorems in bifurcation theory, as well as more recent and lesser known results. It covers both the local and global theory of one-parameter bifurcations for operators acting in infinite-dimensional Banach spaces, and shows how to apply the theory to problems involving partial differential equations. In addition to existence, qualitative properties such as stability and nodal structure of bifurcating solutions are treated in depth. This volume will serve as an important reference for mathematicians, physicists, and theoretically-inclined engineers working in bifurcation theory and its applications to partial differential equations.

The second edition is substantially and formally revised and new material is added. Among this is bifurcation with a two-dimensional kernel with applications, the buckling of the Euler rod, the appearance of Taylor vortices, the singular limit process of the Cahn-Hilliard model, and an application of this method to more complicated nonconvex variational problems.

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Produktdetails

• Verlag: Springer US
• Seitenzahl: 400
• Erscheinungstermin: 13. November 2011
• Englisch
• ISBN-13: 9781461405023
• Artikelnr.: 37362317

Autorenporträt

Hansjörg Kielhöfer is a Professor at the University of Augsburg, Germany.

Inhaltsangabe

Introduction
Appendix I Local Theory
I.1 The Implicit Function Theorem
I.2 The Method of Lyapunov-Schmidt
I.3 The Lyapunov-Schmidt Reduction for Potential Operators
I.4 An Implicit Function Theorem for One-Dimensional Kernels: Turning Points
I.5 Bifurcation with a One-Dimensional Kernel
I.6 Bifurcation Formulas (stationary case)
I.7 The Principle of Exchange of Stability (stationary case)
I.8 Hopf Bifurcation
I.9 Bifurcation Formulas for Hopf Bifurcation
I.10 A Lyapunov Center Theorem
I.11 Constrained Hopf Bifurcation for Hamiltonian, Reversible, and Conservative Systems
I.12 The Principle of Exchange of Stability for Hopf Bifurcation
I.13 Continuation of Periodic Solutions and Their Stability
I.14 Period Doubling Bifurcation and Exchange of Stability
I.15 Newton Polygon
I.16 Degenerate Bifurcation at a Simple Eigenvalue and Stability of Bifurcating Solutions
I.17 Degenerate Hopf Bifurcation and Floquet Exponents of Bifurcating Periodic Orbits
I.18 The Principle of Reduced Stability for Stationary and Periodic Solutions
I.19 Bifurcation with High-Dimensional Kernels, Multiparameter Bifurcation and Application of the Principle of Reduced Stability
I.20 Bifurcation from Infinity
I.21 Bifurcation with High-Dimensional Kernels for Potential Operators: Variational Methods
I.22 Notes and Remarks to Chapter I
Appendix II Global Theory
II.1 The Brouwer Degree
II.2 The Leray Schauder Degree
II.3 Application of the Degree in Bifurcation Theory II.4 Odd Crossing Numbers
II.5 A Degree for a Class of Proper Fredholm Operators and Global Bifurcation Theorems
II.6 A Global Implicit Function Theorem
II.7 Change of Morse Index and Local Bifurcation for Potential Operators
II.8 Notes and Remarks to Chapter II
Appendix III Applications
III.1 The Fredholm Property of Elliptic Operators
III.2 Local Bifurcation for Elliptic Problems
III.3 Free Nonlinear Vibrations
III.4 Hopf Bifurcation for Parabolic Problems
III.5 Global Bifurcation and Continuation for Elliptic Problems
III.6 Preservation of Nodal Structure on Global Branches
III.7 Smoothness and Uniqueness of Global Positive Solution Branches
III.8 Notes and Remarks to Chapter III

Rezensionen

From the reviews of the second edition:

"This book is a valuable resource for mathematicians working in the areas of Nonlinear Analysis and/or Differential Equations. ... This book is intended for advanced graduate students, for specialists in Bifurcation Theory and for researchers in related areas willing to master the subject. ... this is a great reference book on the subject of Bifurcations." (Florin Catrina, MAA Reviews, January, 2013)

"The volume under review gives a unified presentation in an abstract setting of the main theorems in bifurcation theory, in relation with some new results and relevant applications to partial differential equations. ... The book is very well written and the many examples make it an excellent choice for a good course on bifurcation problems." (Vicentiu D. Radulescu, Zentralblatt MATH, Vol. 1230, 2012)