Bifurcation Theory: An Introduction With Applications to PDEs

355 Pages·2003·2.057 MB·English

by  Hansjörg Kielhöfer

Bifurcation Theory:
An Introduction with
Applications to PDEs
Hansjörg Kielhöfer
Springer
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
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Hansjo¨rg Kielho¨fer
Bifurcation Theory
An Introduction with Applications to PDEs
With 38 Figures
Hansjo¨rgKielho¨fer
InstituteforMathematics
UniversityofAugsburg
Universita¨tsstrasse14,Raum2011
D-86135Augsburg
Germany
[email protected]
Editors:
S.S.Antman J.E.Marsden L.Sirovich
DepartmentofMathematics ControlandDynamical DivisionofApplied
and Systems,107-81 Mathematics
InstituteforPhysicalScience CaliforniaInstituteof BrownUniversity
andTechnology Technology Providence,RI02912
UniversityofMaryland Pasadena,CA91125 USA
CollegePark,MD20742-4015 USA [email protected]
USA [email protected]
[email protected]
MathematicsSubjectClassification(2000):35B32,35P30,37K50,37Gxx,47N20
LibraryofCongressCataloging-in-PublicationData
Kielho¨fer,Hansjo¨rg.
Bifurcationtheory:anintroductionwithapplicationstoPDEs/Hansjo¨rgKielho¨fer.
p.cm.—(Appliedmathematicalsciences;156)
Includesindex.
ISBN0-387-40401-5(alk.paper)
1.Bifurcationtheory. I.Title. II.Appliedmathematicalsciences(Springer-Verlag
NewYorkInc.);v.156.
QA1.A647vol.156
[QA380]
510s—dc21
[515′.35] 2003054793
ISBN0-387-40401-5 Printedonacid-freepaper.
2004Springer-VerlagNewYork,Inc.
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Contents
0 Introduction............................................... 1
I Local Theory .............................................. 5
I.1 The Implicit Function Theorem ........................... 5
I.2 The Method of Lyapunov–Schmidt ........................ 6
I.3 The Lyapunov–Schmidt Reduction
for Potential Operators................................... 9
I.4 An Implicit Function Theorem for
One-Dimensional Kernels: Turning Points .................. 11
I.5 Bifurcation with a One-Dimensional Kernel................. 15
I.6 Bifurcation Formulas (Stationary Case) .................... 18
I.7 The Principle of Exchange of Stability
(Stationary Case) ....................................... 20
I.8 Hopf Bifurcation ........................................ 30
I.9 Bifurcation Formulas for Hopf Bifurcation .................. 40
I.10 A Lyapunov Center Theorem ............................. 46
I.11 Constrained Hopf Bifurcation for Hamiltonian,
Reversible, and Conservative Systems ...................... 51
I.11.1 Hamiltonian Systems: Lyapunov Center Theorem
and Hamiltonian Hopf Bifurcation................... 57
I.11.2 Reversible Systems ................................ 66
I.11.3 Nonlinear Oscillations.............................. 70
I.11.4 Conservative Systems .............................. 73
I.12 The Principle of Exchange of Stability
for Hopf Bifurcation ..................................... 76
I.13 Continuation of Periodic Solutions
and Their Stability ...................................... 84
I.13.1 Exchange of Stability at a Turning Point ............. 94
I.14 Period-Doubling Bifurcation
and Exchange of Stability ................................ 97
VI Contents
I.14.1 The Principle of Exchange of Stability
for a Period-Doubling Bifurcation ...................105
I.15 The Newton Polygon ....................................112
I.16 Degenerate Bifurcation at a Simple Eigenvalue
and Stability of Bifurcating Solutions ......................116
I.16.1 The Principle of Exchange of Stability
for Degenerate Bifurcation..........................123
I.17 Degenerate Hopf Bifurcation and Floquet
Exponents of Bifurcating Periodic Orbits ...................129
I.17.1 The Principle of Exchange of Stability
for Degenerate Hopf Bifurcation.....................136
I.18 The Principle of Reduced Stability
for Stationary and Periodic Solutions ......................143
I.18.1 The Principle of Reduced Stability
for Periodic Solutions ..............................149
I.19 Bifurcation with High-Dimensional Kernels,
Multiparameter Bifurcation, and
Application of the Principle of Reduced Stability ............155
I.19.1 A Multiparameter Bifurcation Theorem
with a High-Dimensional Kernel.....................161
I.20 Bifurcation from Infinity .................................163
I.21 Bifurcation with High-Dimensional Kernels
for Potential Operators: Variational Methods ...............166
I.22 Notes and Remarks to Chapter I ..........................173
II Global Theory.............................................175
II.1 The Brouwer Degree .....................................175
II.2 The Leray–Schauder Degree ..............................178
II.3 Application of the Degree in Bifurcation Theory.............182
II.4 Odd Crossing Numbers...................................186
II.4.1 Local Bifurcation via Odd Crossing Numbers .........190
II.5 A Degree for a Class of Proper Fredholm
Operators and Global Bifurcation Theorems ................195
II.5.1 Global Bifurcation via Odd Crossing Numbers ........205
II.5.2 Global Bifurcation with One-Dimensional Kernel ......206
II.6 A Global Implicit Function Theorem.......................210
II.7 Change of Morse Index and Local Bifurcation
for Potential Operators...................................211
II.7.1 Local Bifurcation for Potential Operators.............214
II.8 Notes and Remarks to Chapter II .........................217
III Applications ...............................................219
III.1 The Fredholm Property of Elliptic Operators ...............219
III.1.1 Elliptic Operators on a Lattice ......................225
III.1.2 Spectral Properties of Elliptic Operators .............230
Contents VII
III.2 Local Bifurcation for Elliptic Problems.....................232
III.2.1 Bifurcation with a One-Dimensional Kernel...........233
III.2.2 Bifurcation with High-Dimensional Kernels ...........238
III.2.3 Variational Methods I..............................239
III.2.4 Variational Methods II.............................244
III.2.5 An Example ......................................245
III.3 Free Nonlinear Vibrations ................................251
III.3.1 Variational Methods ...............................260
III.3.2 Bifurcation with a One-Dimensional Kernel...........261
III.4 Hopf Bifurcation for Parabolic Problems ...................268
III.5 Global Bifurcation and Continuation
for Elliptic Problems.....................................275
III.5.1 An Example (Continued)...........................280
III.5.2 Global Continuation ...............................281
III.6 Preservation of Nodal Structure
on Global Branches......................................283
III.6.1 A Maximum Principle .............................284
III.6.2 Global Branches of Positive Solutions ................285
III.6.3 Unbounded Branches of Positive Solutions............290
III.6.4 Separation of Branches.............................293
III.6.5 An Example (Continued)...........................293
III.6.6 Global Branches of Positive Solutions
via Continuation ..................................300
III.7 Smoothness and Uniqueness of
Global Positive Solution Branches .........................302
III.7.1 Bifurcation from Infinity ...........................309
III.7.2 Local Parameterization of Positive Solution Branches
over Symmetric Domains ...........................313
III.7.3 Global Parameterization of Positive Solution Branches
over Symmetric Domains and Uniqueness.............320
III.7.4 Asymptotic Behavior at (cid:1)u(cid:1)∞ =0 and (cid:1)u(cid:1)∞ =∞.....325
III.7.5 Stability of Positive Solution Branches ...............329
III.8 Notes and Remarks to Chapter III.........................333
References.....................................................335
Index..........................................................343
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0
Introduction
Bifurcation Theory attempts to explain various phenomena that have been
discoveredanddescribedinthenaturalsciencesoverthecenturies.Thebuck-
ling of the Euler rod, the appearance of Taylor vortices, and the onset of
oscillations in an electric circuit, for instance, all have a common cause: A
specific physical parameter crosses a threshold, and that event forces the sys-
tem to the organization of a new state that differs considerably from that
observed before.
Mathematically speaking, the following occurs: The observed states of a
systemcorrespondtosolutionsofnonlinearequationsthatmodelthephysical
system.Astatecanbeobservedifitisstable,anintuitivenotionthatismade
precise for a mathematical solution. One expects that a slight change of a
parameter in a system should not have a big influence, but rather that stable
solutionschangecontinuouslyinauniqueway.Thatexpectationisverifiedby
theImplicitFunctionTheorem.Consequently,aslongasacontinuousbranch
of solutions preserves its stability, no dramatic change is observed when the
parameter is varied. However, if that “ground state” loses its stability when
the parameter reaches a critical value, then the state is no longer observed,
and the system itself organizes a new stable state that “bifurcates” from the
ground state.
Bifurcation is a paradigm for nonuniqueness in Nonlinear Analysis.
We sketch that scenario in Figure 1, which is referred to as a “pitchfork
bifurcation.” The solutions bifurcate in pairs which describe typically one
state in two possible representations. Also typically, the bifurcating state has
less symmetry than the ground state (also called “trivial solution”), in which
case one calls it “symmetry breaking bifurcation.” In Figure 1 we show the
solution set of the odd “bifurcation equation,” λx−x3 = 0, where x ∈ R
represents the state and λ∈R is the parameter.
Inthecaseinwhichsolutionscorrespondtocriticalpointsofaparameter-
dependent functional, Figure 2 shows how a slight change of the potential
turns a stable equilibrium into an unstable one and creates at the same time