Springer New York Berlin Heidelberg Hong Kong London Milan Paris Tokyo. Applied Mathematical Sciences. Volume 156

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1 Applied Mathematical Sciences Volume 156 Editors S.S. Antman J.E. Marsden L. Sirovich Advisors J.K. Hale P. Holmes J. Keener J. Keller B.J. Matkowsky A. Mielke C.S. Peskin K.R. Sreenivasan Springer New York Berlin Heidelberg Hong Kong London Milan Paris Tokyo

2 Applied Mathematical Sciences I. John: Partial Differential Equations, 4th ed. 33. Grenander: Regular Structures: Lectures in 2. Sirovich: Techniques of Asymptotic Analysis. Pattern Theory, Vol. III. 3. Hale: Theory of Functional Differential Equations, 34. KevorkianiCole: Perturbation Methods in Applied 2nded. Mathematics. 4. Percus: Combinatorial Methods. 35. Carr: Applications of Centre Manifold Theory. 5. von Mises/Friedrichs: Fluid Dynamics. 36. BengtssoniGhillKiilLen: Dynamic Meteorology: 6. Freiberger/Grenander: A Short Course in Data Assimilation Methods. Computational Probability and Statistics. 37. Saperstone: Semidynamical Systems in Infmite 7. Pipkin: Lectures on Viscoelasticity Theory. Dimensional Spaces. 8. Giacaglia: Perturbation Methods in Non-linear 38. Lichtenberg/Lieberman: Regular and Chaotic Systems. Dynamics, 2nd ed. 9. Friedrichs: Spectral Theory of Operators in 39. PicciniiStampacchiaiVidossich: Ordinary Hilbert Space. Differential Equations in Rn. 10. Stroud: Nmnerical Quadrature and Solution of 40. Naylor/Sell: Linear Operator Theory in Ordinary Differential Equations. Engineering and Science. II. Wolovich: Linear Multivariable Systems. 41. Sparrow: The Lorenz Equations: Bifurcations, 12. Berkovitz: Optimal Control Theory. Chaos, and Strange Attractors. 13. BlumaniCole: Similarity Methods for Differential 42. GuckenheimeriHolmes: Nonlinear Oscillations, Equations. Dynamical Systems, and Bifurcations of Vector 14. Yoshizawa: Stability Theory and the Existence of Fields. Periodic Solution and Almost Periodic Solutions. 43. OckendoniTaylor: Inviscid Fluid Flows. 15. Braun: Differential Equations and Their 44. Pazy: Semigroups of Linear Operators and Applications, 3rd ed. Applications to Partial Differential Equations. 16. Lefschetz: Applications of Algebraic Topology. 45. GlashofflGustafson: Linear Operations and 17. Collatz/Wetterling: Optimization Problems. Approximation: An Introduction to the Theoretical 18. Grenander: Pattern Synthesis: Lectures in Pattern Analysis and Numerical Treatment of Semi- Theory, Vol. l. Infinite Programs. 19. Marsden/McCracken: HopfBifurcation and Its 46. Wilcox: Scattering Theory for Diffraction Applications. Gratings. 20. Driver: Ordinary and Delay Differential 47. HaleiMagalhiies/Oliva: Dynamics in Infinite Equations. Dimensions, 2nd ed. 21. Courant/Friedrichs: Supersonic Flow and Shock 48. Murray: Asymptotic Analysis. Waves. 49. Ladyzhenskaya: The Boundary-Value Problems of 22. Rouche/Habets/Laloy: Stability Theory by Mathematical Physics. Liapunov's Direct Method. 50. Wilcox: Sound Propagation in Stratified Fluids. 23. Lamperti: Stochastic Processes: A Survey of the 51. Golubitsky/SchaefJer: Bifurcation and Groups in Mathematical Theory. Bifurcation Theory, Vol. l. 24. Grenander: Pattern Analysis: Lectures in Pattern 52. Chipot: Variational Inequalities and Flow in Theory, Vol. II. Porous Media. 25. Davies: Integral Transforms and Their 53. Majda: Compressible Fluid Flow and Systems of Applications, 2nd ed. Conservation Laws in Several Space Variables. 26. Kushner/Clark: Stochastic Approximation 54. Wasow: Linear Turning Point Theory. Methods for Constrained and Unconstrained 55. Yosida: Operational Calculus: A Theory of Systems. Hyperfunctions. 27. de Boor: A Practical Guide to Splines: Revised 56. Chang/Howes: Nonlinear Singular Perturbation Edition. Phenomena: Theory and Applications. 28. Keilson: Markov Chain Models-Rarity and 57. Reinhardt: Analysis of Approximation Methods Exponentiality. for Differential and Integral Equations. 29. de Veubeke: A Course in Elasticity. 58. Dwoyer/HussainilVoigt (eds): Theoretical 30. Sniatycki: Geometric Quantization and Quantum Approaches to Turbulence. Mechanics. 59. SanderslVerhulst: Averaging Methods in 31. Reid: Sturrnian Theory for Ordinary Differential Nonlinear Dynamical Systems. Equations. 32. Meis/Markowitz: Numerical Solution of Partial Differential Equations. (continued following index)

3 Hansjorg Kielhofer Bifurcation Theory An Introduction with Applications to PDEs With 38 Figures, Springer

4 Hansjorg Kielhofer Institute for Mathematics University of Augsburg Universitatsstrasse 14, Raum 2011 D Augsburg Germany Editors: S.S. Antman Department of Mathematics and Institute for Physical Science and Technology University of Maryland College Park, MD USA J.E. Marsden Control and Dynamical Systems, California Institute of Technology Pasadena, CA USA L. Sirovich Division of Applied Mathematics Brown University Providence, RI USA Mathematics SUbject Classification (2000): 35B32, 35P30, 37K50, 37Gxx, 47N20 Library of Congress Cataloging-in-Publication Data KielhOfer, Hansjdrg. Bifurcation theory: an introduction with applications to POEs / Hansjdrg KielhOfer. p. cm. - (Applied mathematical sciences; 156) Includes index. [SBN Bifurcation theory. I. Title. II. Applied mathematical sciences (Springer-Verlag New York Inc.) ; v QAI.A647 vol. 156 [QA380] 510 s-dc21 [515'.35] ISBN ISBN (ebook) DOI / Printed on acid-free paper Springer-Verlag New York, Inc. Softcover reprint of the hardcover 1st edition 2004 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights I SPIN Typesetting: Pages created by the author using Springer's SVMono.cls U.TJ3X 2e macro package. Springer-Verlag New York Berlin Heidelberg A member of BertelsmannSpringer Science+Business Media GmbH

5 Contents o Introduction I Local Theory The Implicit Function Theorem The Method of Lyapunov-Schmidt The Lyapunov-Schmidt Reduction for Potential Operators An Implicit Function Theorem for One-Dimensional Kernels: Turning Points Bifurcation with a One-Dimensional Kernel Bifurcation Formulas (Stationary Case) The Principle of Exchange of Stability (Stationary Case) Hopf Bifurcation Bifurcation Formulas for Hopf Bifurcation A Lyapunov Center Theorem Constrained Hopf Bifurcation for Hamiltonian, Reversible, and Conservative Systems Hamiltonian Systems: Lyapunov Center Theorem and Hamiltonian Hopf Bifurcation I.l1.2 Reversible Systems I.l1.3 Nonlinear Oscillations Conservative Systems I.l2 The Principle of Exchange of Stability for Hopf Bifurcation Continuation of Periodic Solutions and Their Stability I.l3.1 Exchange of Stability at a Turning Point I.l4 Period-Doubling Bifurcation and Exchange of Stability... 97

6 VI Contents I.l4.1 The Principle of Exchange of Stability for a Period-Doubling Bifurcation The Newton Polygon Degenerate Bifurcation at a Simple Eigenvalue and Stability of Bifurcating Solutions I.l6.1 The Principle of Exchange of Stability for Degenerate Bifurcation Degenerate Hopf Bifurcation and Floquet Exponents of Bifurcating Periodic Orbits I.l7.1 The Principle of Exchange of Stability for Degenerate Hopf Bifurcation I.lS The Principle of Reduced Stability for Stationary and Periodic Solutions I.lS.1 The Principle of Reduced Stability for Periodic Solutions Bifurcation with High-Dimensional Kernels, Multiparameter Bifurcation, and Application of the Principle of Reduced Stability A Multiparameter Bifurcation Theorem with a High-Dimensional Kernel Bifurcation from Infinity Bifurcation with High-Dimensional Kernels for Potential Operators: Variational Methods Notes and Remarks to Chapter I II Global Theory The Brouwer Degree The Leray-Schauder Degree... 17S 11.3 Application of the Degree in Bifurcation Theory... ls Odd Crossing Numbers... ls6 II.4.1 Local Bifurcation via Odd Crossing Numbers A Degree for a Class of Proper Fredholm Operators and Global Bifurcation Theorems II.5.l Global Bifurcation via Odd Crossing Numbers II.5.2 Global Bifurcation with One-Dimensional Kernel II.6 A Global Implicit Function Theorem II. 7 Change of Morse Index and Local Bifurcation for Potential Operators II.7.1 Local Bifurcation for Potential Operators II.S Notes and Remarks to Chapter II III Applications IIL1 The Fredholm Property of Elliptic Operators III.l.1 Elliptic Operators on a Lattice III.l.2 Spectral Properties of Elliptic Operators

7 Contents VII III.2 Local Bifurcation for Elliptic Problems III.2.1 Bifurcation with a One-Dimensional Kernel III.2.2 Bifurcation with High-Dimensional Kernels III.2.3 Variational Methods I III.2.4 Variational Methods II III.2.5 An Example III.3 Free Nonlinear Vibrations III.3.1 Variational Methods III.3.2 Bifurcation with a One-Dimensional Kernel III.4 Hopf Bifurcation for Parabolic Problems III.5 Global Bifurcation and Continuation for Elliptic Problems III.5.1 An Example (Continued) III.5.2 Global Continuation III.6 Preservation of Nodal Structure on Global Branches III.6.1 A Maximum Principle III.6.2 Global Branches of Positive Solutions III.6.3 Unbounded Branches of Positive Solutions III.6.4 Separation of Branches III.6.5 An Example (Continued) III.6.6 Global Branches of Positive Solutions via Continuation IIL7 Smoothness and Uniqueness of Global Positive Solution Branches III. 7.1 Bifurcation from Infinity III.7.2 Local Parameterization of Positive Solution Branches over Symmetric Domains III.7.3 Global Parameterization of Positive Solution Branches over Symmetric Domains and Uniqueness III.7.4 Asymptotic Behavior at Ilull oo = 0 and Ilull oo = III.7.5 Stability of Positive Solution Branches III.8 Notes and Remarks to Chapter III References Index