Bifurcation Theory: An Introduction with Applications to PDEs

352 Pages·2004·3.106 MB·English

by  Hansjörg Kielhöfer (auth.)

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
Hansjo¨rg Kielho¨fer
Bifurcation Theory
An Introduction with Applications to PDEs
With 38 Figures
Springer
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
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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
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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
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
2 Chapter 0. Introduction
twonewstableequilibria.Thatexchangeofstability,however,isnotrestricted
to variational problems, but is typical for all “generic” bifurcations.
state
stable bifurcating state
stable unstable ground state (trivial solution)
parameter
Figure 1
subcritical supercritical potential
Figure 2
Bifurcation Theory provides the mathematical existence of bifurcation
scenarios observed in various systems and experiments. A necessary condi-
tion is obviously the failure of the Implicit Function Theorem.
In this book we present some sufficient conditions for “one-parameter bi-
furcation,” which means that the bifurcation parameter is a real scalar. We
do not treat “multiparameter bifurcation theory.”
Wedistinguishalocaltheory,whichdescribesthebifurcationdiagramina
neighborhood of the bifurcation point, and a global theory, where the contin-
uation of local solution branches beyond that neighborhood is investigated.
In applications we also prove specific qualitative properties of solutions on
global branches, which, in turn, help to separate global branches, to decide
on their unboundedness, and, in special cases, to establish their smoothness
and asymptotic behavior.
Chapter 0. Introduction 3
As mentioned before, bifurcation is often related to a breaking of symme-
try. We sometimes make use of symmetry in the applications in investigating
the qualitative properties of solutions on global branches. However, we typi-
cally exploit symmetry in an ad hoc manner. For a systematic treatment of
symmetry and bifurcation, we refer to the monographs [16], [54], [55], [146].
Symmetry ideas do not play a dominant role in this book.
We present the results of Chapter I and Chapter II in an abstract way,
and we apply these abstract results to concrete problems for partial differen-
tial equations only in Chapter III. The theory is separated from applications
for the following reasons: It is our opinion that mathematical understanding
can be reached only via abstraction and not by examples or applications.
Moreover,onlyanabstractresultissuitabletobeadaptedtoanewproblem.
Therefore, we resisted mixing the general theory with our personal selection
of applications.
The general theory of Chapters I and II is formulated for operators act-
ing in infinite-dimensional spaces. This lays the groundwork for Chapter III,
where detailed applications to concrete partial differential equations are pro-
vided.TheabstractversionsoftheHopfBifurcationTheoreminChapterIare
directly applicable to ODEs, RFDEs, and Hamiltonian or reversible systems.
For stability considerations we employ throughout the principle of linearized
stability, which means, in turn, that stability is determined by the perturba-
tion of the critical eigenvalue or Floquet exponent.
The motivation to write this book came from many questions of students
and colleagues about bifurcation theorems. Most of the results contained
herein are not new. But many are apparently known only to a few experts,
and a unified presentation was not available. Indeed, while there exist many
good books treating various aspects of bifurcation theory, e.g., [11], [16], [17],
[33], [54], [55], [56], [60], [75], [146], [153], there is precious little analysis of
problemsgovernedbypartialdifferentialequationsavailableintextbookform.
This work addresses that gap. We apologize to all who have obtained similar
or better results that are not mentioned here. During the last thirty years a
vastliteratureonbifurcationtheoryhasbeenpublished,andwehavenotbeen
able to write a survey. A reason for this limitation is that we feel competent
only in fields where we have worked ourselves.
In many of the above-mentioned books we find the “basic” or “generic”
bifurcations in simple settings illustrating the geometric ideas behind them,
mostlyfromadynamicalviewpoint,cf.[60].Inviewofthatexcellentheuristic
literature, we think that there is no need to repeat these ideas but that it is
necessary to give the calculations in a most general setting. This might be
hard for beginners, but we hope that it is useful to advanced students.
Apart from the Cahn−Hilliard model (serving as a paradigm), our appli-
cations to partial differential equations are motivated only by, but are not di-
rectlyrelatedto,mathematicalphysics.Theformulationofaspecificproblem
of physics and the verification of all hypotheses are typically quite involved,
and such an expenditure might disguise the essence of Bifurcation Theory.
4 Chapter 0. Introduction
For these reasons we believe that a detailed presentation of the cascade of
bifurcations appearing in the Taylor model, for instance, is not appropriate
here; rather, we refer to the literature, [15], for example. On the other hand,
we hope that our choice of mathematical applications offers a broad selection
of techniques illustrating the use of the abstract theory without getting lost
intoomanytechnicalities.Finally,ifnecessary,theanalysiscanbecompleted
by numerical analysis as expounded in [4], [81], and [142].
I am indebted to Rita Moeller for having typed the entire text in LATEX.
And in particular, I thank my friend Tim Healey for the encouragement and
his help in writing this book: Many of the results obtained in a fruitful col-
laboration with him are presented here.