Table Of ContentUndergraduate Texts in Mathematics
Edirors
S. Axler
F. W. Gehring
P. R. Halmos
Springer-Verlag Berlin Heidelberg GmbH
Undergraduate Texts in Mathematics
Anglin: Mathematics: A Concise History Devlin: The Joy of Sets: Fundamentals
and Philosophy. of Contemporary Set Theory.
Readings in Mathematics. Second edition.
AnglinILambek: The Heritage of Dixmier: General Topology.
Thales. Driver: Why Math?
Readings in Mathematics. EbbinghauslFlumlfhomas:
Apostol: Introduction to Analytic Mathematical Logic. Second edition.
Number Theory. Second edition. Edgar: Measure, Topology, and Fractal
Armstrong: Basic Topology. Geometry.
Armstrong: Groups and Symmetry. Elaydi: Introduction to Difference
AxIer: Linear Algebra Done Right. Equations.
Beardon: Limits: A New Approach to Exner: An Accompaniment to Higher
Real Analysis. Mathematics.
BakINewman: Complex Analysis. FineJRosenberger: The Fundamental
Second edition. Theory of Algebra.
BanchoffIWermer: Linear Algebra Fischer: Intermediate Real Analysis.
Through Geometry. Second edition. Flanigan/Kazdan: Calculus Two: Linear
Berberian: A First Course in Real and Nonlinear Functions. Second
Analysis. edition.
Bremaud: An Introduction to Fleming: Functions of Several Variables.
Probabilistic Modeling. Second edition.
Bressoud: Factorization and Primality Foulds: Combinatorial Optimization for
Testing. Undergraduates.
Bressoud: Second Year Calculus. Foulds: Optimization Techniques: An
Readings in Mathematics. Introduction.
Brickman: Mathematical Introduction Franklin: Methods of Mathematical
to Linear Programming and Game Economics.
Theory. Gordon: Discrete Probability.
Browder: Mathematical Analysis: HairerIWanner: Analysis by Its History.
An Introduction. Readings in Mathematics.
Buskeslvan Rooij: Topological Spaces: Halmos: Finite-Dimensional Vector
From Distance to Neighborhood. Spaces. Second edition.
Cederberg: A Course in Modem Halmos: Naive Set Theory.
Geometries. HiimmerlinlHoffmann: Numerical
Childs: A Concrete Introduction to Mathematics.
Higher Algebra. Second edition. Readings in Mathematics.
Chung: Elementary Probability Theory Hijab: Introduction to Calculus and
with Stochastic Processes. Third Classical Analysis.
edition. HiltonIHoltonIPedersen: Mathematical
Cox/LittleJO'Shea: Ideals, Varieties, Reflections: In a Room with Many
and Algorithms. Second edition. Mirrors.
Croom: Basic Concepts of Algebraic Iooss/Joseph: Elementary Stability and
Topology. Bifurcation Theory. Second edition.
Curtis: Linear Algebra: An Introductory Isaac: The Pleasures of Probability.
Approach. Fourth edition. Readings in Mathematics.
(continued after index)
Gerard Iooss
Daniel D. Joseph
Elementary Stability and
Bifurcation Theory
Second Edition
With 58 Illustrations
, Springer
GCrard looss Daniel D. Joseph
InstituI Non Lineairc, de Nicc Dcpartment of Aerospacc Enginccring
1361 roule des Lucioles and Mechanics
Sophia-Antipolis Vniversily of Minnesota
06560 Valbonnc Minneapolis. MN 55455
Francc V.S.A.
Editorial BOClrd
S. Axler F. W. Gchring P. R. HaJmos
Dcpartment of Dcpartment of Dcpartment of
Mathematics Malhematics Malhematics
Michigan State Vniversity University of Michigan Santa Clara University
Easl Lansing, MI 48824 Ann Arbor, MI 48019 Santa Clara, CA 95053
USA U.S.A. U.S.A.
Mathematics Subjcct CJassilication (l99I~ 34-01, 34, A34, 34030, 34099, 34C99
Library of Congress Cataloging_in_Publication Oala
10000, Gerard.
Elementary stability and bifurcation thCQry f Gerard looss, Daniel
D. Jostph. - 2nd ed.
p. CJIl, - (Undergraduate tcxls in mathematics)
Includcs bibliographical referern;es.
1. Oiffcrential equalions- Numcrical SOlulions. 2. Evolution
cquations-NumericaI5OIutions. 3. Stabilily. 4. Bifurcation
thCQry. 1. loscph. Daniel D. II. Titlc. III. Scrics.
Q1\372.168 1989
o51f.Jo5- dc20 89·21765
Printcd on acid·free papcr
C 1980, 1990 bySpringer.Verlag Berlin Hcidclbcrg
Originally publisbed by Springer-Veriag Berlin Heidelbcrg New York in 1990
Softcover reprint of tbe hardcovcr 2nd cdition 1990
AII rights rcscrvcd. This work may nOI tic lranslatcd or copicd in whole or in pari wilhout the
wriuen permission of the publi$hcr (Springer-Verlag Berlin Heidelberg GmbH).
ucept for brid ucerpts in conneelion wilh leviews OI scholarly analysis. Usc in connee-
tion with any fOlm of information storagc: and retrieva1. electronic adaptation. computer software,
or by similar or dissimilar mcthodology now known ar hereafter devcloped is forbidden.
The usc of general descriptive namcs, trade names, trademarks, etc., in this publication, even if
the former are not cspecially identilied, is not 10 bc laken as a sign that such name •. as ,.mdcrstood
by Ihe Trade Marks and Merchandisc Marks Act. may accordingly tic uscd rreely by anyonc.
9 8 7 6 5 4 1 2 tCorrectcd second printin" 1997)
ISBN 978-1-4612-6977-9 ISBN 978-1-4612-0997-3 (eBook)
DOI 10.1007/978-1-4612-0997-3
Everything should be made as simple as possible,
but not simpler.
ALBERT EINSTEIN
Contents
List of Frequently Used Symbols Xlll
Introduction XVll
Preface to the Second Edition XXI
CHAPTER I
Asymptotic Solutions of Evolution Problems
I.1 One-Dimensional, Two-Dimensional, n-Dimensional, and
Infinite-Dimensional Interpretations of (Ll)
1.2 Forced Solutions; Steady Forcing and T-Periodic Forcing;
Autonomous and Nonautonomous Problems 3
I.3 Reduction to Local Form 4
1.4 Asymptotic Solutions 5
1.5 Asymptotic Solutions and Bifurcating Solutions 5
1.6 Bifurcating Solutions and the Linear Theory of Stability 6
I.7 Notation for the Functional Expansion of F(t, /1, U) 7
Notes 8
CHAPTER II
Bifurcation and Stability of Steady Solutions of Evolution
Equations in One Dimension 10
11.1 The Implicit Function Theorem 10
11.2 Classification of Points on Solution Curves 11
11.3 The Characteristic Quadratic. Double Points, Cusp Points, and
Conjugate Points 12
11.4 Double-Point Bifurcation and the Implicit Function Theorem 13
U.5 Cusp-Point Bifurcation 14
VII
VI11 Contents
11.6 Triple-Point Bifurcation 15
11.7 Conditional Stability Theorem IS
11.8 The Factorization Theorem in One Dimension 19
11.9 Equivalence of Strict Loss of Stability and Double-Point Bifurcation 20
II.10 Exchange of Stability at a Double Point 20
ILII Exchange of Stability at a Double Point for Problems Reduced to
Local Form 22
II.12 Exchange of Stability at a Cusp Point 25
11.13 Exchange of Stability at a Triple Point 26
11.14 Global Properties of Stability of Isolated Solutions 26
CHAPTER III
Imperfection Theory and Isolated Solutions Which Perturb
Bifurcation 29
III. I The Structure of Problems Which Break Double-Point Bifurcation 30
I1I.2 The Implicit Function Theorem and the Saddle Surface Breaking
Bifurcation 31
111.3 Examples of Isolated Solutions Which Break Bifurcation 33
IlIA Iterative Procedures for Finding Solutions 34
III.S Stability of Solutions Which Break Bifurcation 37
III.6 Isolas 39
Exercise 39
Notes 40
CHAPTER IV
Stability of Steady Solutions of Evolution Equations in Two
Dimensions and n Dimensions 42
IV.1 Eigenvalues and Eigenvectors of an n x n Matrix 43
IV.2 Algebraic and Geometric Multiplicity-The Riesz Index 43
IV.3 The Adjoint Eigenvalue Problem 44
IVA Eigenvalues and Eigenvectors of a 2 x 2 Matrix 45
4.1 Eigenvalues 45
4.2 Eigenvectors 46
4.3 Algebraically Simple Eigenvalues 46
4A Algebraically Double Eigenvalues 46
4A.I Riesz Index I 46
4A.2 Riesz Index 2 47
IV.S The Spectral Problem and Stability of the Solution u = 0 in ~n 48
IV.6 Nodes, Saddles, and Foci 49
IV.7 Criticality and Strict Loss of Stability 50
Appendix IV.I
Biorthogonality for Generalized Eigenvectors 52
Appendix IV.2
Projections 55
Contents IX
CHAPTER V
Bifurcation of Steady Solutions in Two Dimensions and the
Stability of the Bifurcating Solutions 59
V.l The Form of Steady Bifurcating Solutions and Their Stability 59
V.2 Necessary Conditions for the Bifurcation of Steady Solutions 62
V.3 Bifurcation at a Simple Eigenvalue 63
V.4 Stability of the Steady Solution Bifurcating at a Simple Eigenvalue 64
V.5 Bifurcation at a Double Eigenvalue of Index Two 64
V.6 Stability of the Steady Solution Bifurcating at a Double Eigenvalue
of Index Two 66
V.7 Bifurcation and Stability of Steady Solutions in the Form (V.2) at a
Double Eigenvalue ofIndex One (Semi-Simple) 67
V.8 Bifurcation and Stability of Steady Solutions (V.3) at a Semi-Simple
Double Eigenvalue 70
V.9 Examples of Stability Analysis at a Double Semi-Simple
(Index-One) Eigenvalue 72
V.lO Saddle-Node Bifurcation 77
Appendix V.I
Implicit Function Theorem for a System of Two Equations
in Two Unknown Functions of One Variable 80
Exercises 82
CHAPTER VI
Methods of Projection for General Problems of Bifurcation
into Steady Solutions 87
VI.! The Evolution Equation and the Spectral Problem 87
VI.2 Construction of Steady Bifurcating Solutions as Power Series
in the Amplitude 88
VI.3 [Rl and [Rl in Projection 90
VI.4 Stability of the Bifurcating Solution 91
VI.5 The Extra Little Part for [Rl in Projection 92
VI.6 Projections of Higher-Dimensional Problems 94
VI. 7 The Spectral Problem for the Stability of u = 0 96
VI.8 The Spectral Problem and the Laplace Transform 98
VI.9 Projections into [Rl 101
VI.lO The Method of Projection for Isolated Solutions Which Perturb
Bifurcation at a Simple Eigenvalue (Imperfection Theory) 102
VI.II The Method of Projection at a Double Eigenvalue of Index Two 104
VI.I2 The Method of Projection at a Double Semi-Simple Eigenvalue 107
VI.l3 Examples of the Method of Projection 111
VI.14 Symmetry and Pitchfork Bifurcation 136
x Contents
CHAPTER VII
Bifurcation of Periodic Solutions from Steady Ones (Hopf
Bifurcation) in Two Dimensions 139
VII. I The Structure of the Two-Dimensional Problem Governing
Hopf Bifurcation 139
VII.2 Amplitude Equation for Hopf Bifurcation 140
VII.3 Series Solution 141
VIl.4 Equations Governing the Taylor Coefficients 141
VII.5 Solvability Conditions (the Fredholm Alternative) 141
VII.6 Floquet Theory 142
6.1 Floquet Theory in ~ 1 143
6.2 Floquet Theory in ~2 and ~n 145
VII.7 Equations Governing the Stability of the Periodic Solutions 149
VII.8 The Factorization Theorem 149
VII.9 Interpretation of the Stability Result 150
Example 150
CHAPTER VIII
Bifurcation of Periodic Solutions in the General Case 156
VIII. I Eigenprojections of the Spectral Problem 156
VIII.2 Equations Governing the Projection and the Complementary
Projection 157
VIII.3 The Series Solution Using the Fredholm Alternative 159
VIII.4 Stability of the Hopf Bifurcation in the General Case 164
VIII.5 Systems with Rotational Symmetry 165
Examples 167
Notes 175
CHAPTER IX
Subharmonic Bifurcation of Forced T-Periodic Solutions 177
Notation 177
IX.l Definition of the Problem of Subharmonic Bifurcation 178
IX.2 Spectral Problems and the Eigenvalues a(Jl) 180
IX.3 Biorthogonality 181
IX.4 Criticality 181
IX.5 The Fredholm Alternative for J(Jl) - a(Jl) and a Formula
Expressing the Strict Crossing (IX.20) 182
IX.6 Spectral Assumptions 183
IX.7 Rational and Irrational Points of the Frequency Ratio at Criticality 183
IX.8 The Operator JI and its Eigenvectors 185
IX.9 The Adjoint Operator J1*, Biorthogonality, Strict Crossing, and
the Fredholm Alternative for JI 186
IX.lO The Amplitude e and the Biorthogonal Decomposition of
Bifurcating Subharmonic Solutions 187
Contents Xl
IX.ll The Equations Governing the Derivatives of Bifurcating
Subharmonic Solutions with Respect to [; at [; = 0 188
IX.12 Bifurcation and Stability of T-Periodic and 2 T-Perio(jic Solutions 189
IX.13 Bifurcation and Stability of nT-Periodic Solutions with n > 2 192
IX.14 Bifurcation and Stability of 3 T-Periodic Solutions 193
IX.15 Bifurcation of 4T-Periodic Solutions 196
IX.16 Stability of 4T-Periodic Solutions 199
IX.17 Nonexistence of Higher-Order Subharmonic Solutions and
Weak Resonance 203
IX.18 Summary of Results About Subharmonic Bifurcation 204
IX.19 Imperfection Theory with a Periodic Imperfection 204
Exercises 205
IX.20 Saddle-Node Bifurcation of T-Periodic Solutions 206
IX.21 General Remarks About Subharmonic Bifurcations 207
CHAPTER X
Bifurcation of Forced T-Periodic Solutions into Asymptotically
Quasi-Periodic Solutions 208
X.I Decomposition of the Solution and Amplitude Equation 209
Exercise 209
X.2 Derivation of the Amplitude Equation 210
X.3 The Normal Equations in Polar Coordinates 214
X.4 The Torus and Trajectories on the Torus in the Irrational Case 216
X.5 The Torus and Trajectories on the Torus When Wo T/2n Is a
Rational Point of Higher Order (n ;::. 5) 219
X.6 The Form of the Torus in the Case n = 5 221
X.7 Trajectorics on the Torus When n = 5 222
X.8 The Form of the Torus When n > 5 225
X.9 Trajectories on the Torus When n ;::. 5 228
X.lO Asymptotically Quasi-Periodic Solutions 231
X.II Stability of the Bifurcated Torus 232
X.12 Subharmonic Solutions on the Torus 233
X.13 Stability of Subharmonic Solutions on the Torus 236
X.14 Frequency Locking 239
Appendix X.I
Direct Computation of Asymptotically Quasi-Periodic Solutions
Which Bifurcate at Irrational Points Using the Method of Two
Times, Power Series, and the Fredholm Alternative 243
Appendix X.2
Direct Computation of Asymptotically Quasi-Periodic Sollutions
Which Bifurcate at Rational Points of Higher Order
Using the Method of Two Times 247
Exercise 254
No~s 2~
