An edition of Applied Equivariant Degree
(2006)
Applied Equivariant Degree
by Wiesław Krawcewicz
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"Applied Equivariant Degree" is a self-contained
comprehensive exposition of the equivariant degree
theory and its applications to a variety of problems
arising in physics, chemistry, biology and engineering.
This monograph presents the theoretical foundations,
construction, and the fundamental properties of the
equivariant degree and its practical variations, which
are applied to a series of examples from (functional)
differential equations. It contains
a) the first thorough and complete introduction up
to the present state of art to equivariant degree
theory including non-abelian actions, and
b) provides for the first time several computer
r o u t i n e s a l l o w i n g a n e f f e c t i v e p r a c t i c a l
computation of the degree, illustrated by
numerous concrete examples and charts.
The exposition of the material is mainly addressed to
researchers and graduate students interested in
applications of equivariant topological methods, or
working with differential equations and their
applications, such as physicists, biologists, chemists
and engineers dealing with nonlinear dynamics with
symmetries.
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Table of Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Motivation and Subject . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Main Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Historical Roots and Topological Aspects of Equivariant Degree 9
1.4 Overview of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5 Other Related Publication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
1.6 Potential for Further Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
1.7 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Part I EQUIVARIANT DEGREE THEORY: TECHNICAL
TOOLS
2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2 Elements of Representation Theory . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2.1 Finite-dimensional G-Representations . . . . . . . . . . . . . . . . 30
2.2.2 Complexification and Conjugation . . . . . . . . . . . . . . . . . . . 33
2.2.3 Character of Representation . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2.4 Haar Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2.5 Isotypical Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.2.6 Decomposition of GLG(V ). . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.2.7 Banach G-Representations . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.3 Equivariant Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.4 Elements of Smooth Equivariant Topology . . . . . . . . . . . . . . . . . . 44
2.4.1 Basic Facts from Differential Topology . . . . . . . . . . . . . . . 44
2.4.2 G-Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.4.3 G-Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
2.4.4 Orientation on G-manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.4.5 Bi-orientable Compact Lie Groups . . . . . . . . . . . . . . . . . . . 52
2.4.6 Local Brouwer Degree. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
XVI Contents
2.5 Numbers n(L,H) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
2.5.2 Interpretation of the Numbers n(L,H) . . . . . . . . . . . . . . . 58
2.6 Elements of Bordism Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.6.1 Oriented Bordism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.6.2 Singular Oriented Bordism and Whitney Theorem . . . . . 60
2.6.3 Singular Oriented Bordism and Homology . . . . . . . . . . . . 61
2.6.4 Framed Bordism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
2.6.5 Framed Equivariant Bordism Relation . . . . . . . . . . . . . . . 63
2.7 Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3 Equivariant Degree Theory: Construction and Basic
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.1 Stable Equivariant Homotopy Groups of Spheres . . . . . . . . . . . . 67
3.1.1 Non-Equivariant Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
3.1.2 Equivariant Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2 Construction of Equivariant Degree . . . . . . . . . . . . . . . . . . . . . . . . 73
3.2.1 Non-Equivariant Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.2.2 Equivariant Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.3 Regular Normal Approximation of Equivariant Maps . . . . . . . . . 76
3.3.1 Normal Maps: Definition and Examples . . . . . . . . . . . . . . 77
3.3.2 Normal Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.3.3 Regular Normal Approximations . . . . . . . . . . . . . . . . . . . . 82
3.4 Structure of the Group ΠG: Pontryagin-Thom Isomorphism . . 87
3.4.1 Equivariant Degree Techniques . . . . . . . . . . . . . . . . . . . . . . 87
3.4.2 Decomposition of the Group ΠG . . . . . . . . . . . . . . . . . . . . 89
3.4.3 Group Π(H) and Equivariant Bordism: Pontryagin-
Thom Isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.5 Computations of Primary Groups Π(H) . . . . . . . . . . . . . . . . . . . . 92
3.5.1 Equivariant Bordism Approach . . . . . . . . . . . . . . . . . . . . . . 93
3.5.2 Fundamental Domain Approach . . . . . . . . . . . . . . . . . . . . . 95
3.6 Computations of the Secondary Groups Π(H) for n = 1 . . . . . . 99
3.6.1 Homomorphism Φ and Short Exact Sequence . . . . . . . . . 99
3.6.2 Homomorphism p∗ and Five Lemma . . . . . . . . . . . . . . . . . 101
3.7 Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
4 S1-Equivariant Degree and Primary Degree for One Free
Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.1 Primary Equivariant Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
4.1.1 Primary Equivariant Degree: Construction . . . . . . . . . . . . 106
4.1.2 Primary Equivariant Degree: Justification . . . . . . . . . . . . 108
4.1.3 Primary Equivariant Degree: Basic Properties . . . . . . . . . 108
4.1.4 Axiomatic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
4.2 Axiomatic Definition of the S1-Equivariant Degree. . . . . . . . . . . 112
4.2.1 Basic Maps and m-Folding . . . . . . . . . . . . . . . . . . . . . . . . . 113
Contents XVII
4.2.2 Formulation of the Main Result and Immediate
Consequences of Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
4.3 Proof of Theorem 4.11: Preliminaries and Central Lemma . . . . 115
4.3.1 Positive Orientation in a Slice and Central Lemma . . . . 115
4.3.2 Proof of Lemma 4.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.4 Proof of Theorem 4.11: Existence and Uniqueness . . . . . . . . . . . 121
4.4.1 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.4.2 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.5 Computation of S1-Degree via Reduction to Basic and
C-Complementing Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.5.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 123
4.5.2 C-Complementing Maps and Suspension Procedure . . . . 124
4.5.3 Homotopy Factorization: Properties of GLG(V ) . . . . . . . 128
4.5.4 Splitting Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
4.5.5 S1-Degree Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
4.6 Recurrence Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
4.6.2 Proof of Theorem 4.25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
4.7 Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
5 Basic Maps and Computations of Twisted Primary Degree 139
5.1 Twisted Subgroups of G = Γ × S1 . . . . . . . . . . . . . . . . . . . . . . . . . 140
5.2 Twisted Primary Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.3 Examples and Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
5.3.1 Quaternionic Units Group Q8 . . . . . . . . . . . . . . . . . . . . . . . 143
5.3.2 Dihedral Group DN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
5.3.3 Tetrahedral Group A4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
5.3.4 Octahedral Group S4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
5.3.5 Icosahedral Group A5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
5.3.6 Orthogonal Group O(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
5.4 Twisted Subgroups of SO(3) × S1 and Numbers n(L,H) . . . . . 162
5.5 Basic Maps for Irreducible Representations of G = Γ × S1 . . . . 165
5.6 Computations of Twisted Degrees of the Basic Maps . . . . . . . . . 170
5.6.1 Computational Formula for Basic Maps with one
Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
5.6.2 Computational Formula for Basic Maps without
Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
5.7 Examples of Computations of the Equivariant Degree for
Basic Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
5.7.1 Degrees of Basic Maps for the Group Q8 . . . . . . . . . . . . . 173
5.7.2 Degrees of Basic Maps for the Dihedral Group DN . . . . 174
5.7.3 Degrees of Basic Maps for the Tetrahedral Group A4 . . 177
5.7.4 Degrees of Basic Maps for the Octahedral Group S4 . . . 178
5.7.5 Degrees of Basic Maps for the Icosahedral Group A5 . . . 181
5.7.6 Degrees of Basic Maps for the Group O(2) . . . . . . . . . . . . 184
XVIII Contents
5.8 Twisted Degree of Basic Maps for the Group SO(3) . . . . . . . . . 185
5.8.1 Irreducible representations of SO(3) × S1 . . . . . . . . . . . . . 185
5.8.2 Computation of Twisted Degrees for SO(3)×S1-Actions189
5.9 Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
6 Algebraic Properties of the Equivariant Degree . . . . . . . . . . . . 191
6.1 Computations of the Burnside Ring . . . . . . . . . . . . . . . . . . . . . . . . 191
6.2 A(Γ)-Module Structure on At
1(Γ × S1) and Multiplication
Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
6.3 Examples of A(Γ)-Modules At
1(Γ × S1) . . . . . . . . . . . . . . . . . . . . 197
6.3.1 A(Q8)-Module At
1(Q8 × S1) . . . . . . . . . . . . . . . . . . . . . . . . 198
6.3.2 A(DN)-Module At
1(DN × S1) . . . . . . . . . . . . . . . . . . . . . . . 198
6.3.3 A(A4)-Module At
1(A4 × S1). . . . . . . . . . . . . . . . . . . . . . . . . 201
6.3.4 A(S4)-Module At
1(S4 × S1) . . . . . . . . . . . . . . . . . . . . . . . . . 203
6.3.5 A(A5)-Module At
1(A5 × S1). . . . . . . . . . . . . . . . . . . . . . . . . 203
6.3.6 A(O(2))-Module At
1(O(2) × S1) . . . . . . . . . . . . . . . . . . . . . 203
6.4 Computation of the A(SO(3))-Module Structure for
At
1(SO(3) × S1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
6.5 Multiplicativity Property for the Twisted Degree . . . . . . . . . . . . 207
6.6 Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
7 Secondary Groups and Secondary Equivariant Degree for
Γ = SO(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
7.1 Preliminary Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
7.2 Computations of Reduced Secondary Groups Π∗(K) for
Γ = SO(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
7.2.1 Simple Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
7.2.2 Representations Vt
j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
7.2.3 Computation of Π∗(K): Regular Cases . . . . . . . . . . . . . . . 219
7.2.4 Computation of Π∗(K): First Typical Exceptional Case 221
7.2.5 Computation of Π∗(K): Second Typical Exceptional
Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
7.2.6 Computation of Π∗(K): Conclusions . . . . . . . . . . . . . . . . . 226
7.3 Some Remarks on Computations of the Secondary Degree . . . . 228
7.3.1 Computation of the Reduced Degree deg∗
Γ (f,Ω) . . . . . . . 228
7.3.2 Computation of the Z2-Component of degs
Γ (f,Ω). . . . . . 231
7.4 Computations of Secondary Degree for Γ = SO(3): Special
Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
7.5 Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
8 Orthogonal G-Equivariant Degree (by H. Ruan) . . . . . . . . . . . . 235
8.1 Equivariant Orthogonal Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
8.2 S1-Normality for G-Orthogonal Maps . . . . . . . . . . . . . . . . . . . . . . 237
8.3 Construction of Orthogonal Degree . . . . . . . . . . . . . . . . . . . . . . . . 240
8.4 Orthogonal Degree of G-Orthogonal Linear Maps . . . . . . . . . . . . 243
Contents XIX
8.5 Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
Part II APPLICATIONS OF EQUIVARIANT DEGREE
TO DIFFERENTIAL EQUATIONS AND BIFURCATION
PROBLEMS
9 Setting for Studying Ordinary Differential Equations with
Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
9.1 Infinite-Dimensional Equivariant Degree . . . . . . . . . . . . . . . . . . . . 252
9.1.1 Leray-Schauder Equivariant Degree . . . . . . . . . . . . . . . . . . 252
9.1.2 Nussbaum-Sadovskii Equivariant Degree . . . . . . . . . . . . . 255
9.2 Hopf Bifurcation Problem for ODEs without Symmetries . . . . . 259
9.2.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 259
9.2.2 S1-Equivariant Reformulation of the Problem . . . . . . . . . 261
9.2.3 S1-Degree Method for Hopf Bifurcation Problem . . . . . . 264
9.2.4 Deformation of the Map Fς . . . . . . . . . . . . . . . . . . . . . . . . . 270
9.2.5 Product Formula for S1-Deg (Fς,Ω) . . . . . . . . . . . . . . . . . 274
9.2.6 Crossing Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275
9.2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
9.3 Hopf Bifurcation Problem for an ODE System with Symmetries280
9.3.1 Symmetric Hopf Bifurcation and Local Bifurcation
Invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280
9.3.2 Computation of Local Bifurcation Invariant:
Reduction to Product Formula . . . . . . . . . . . . . . . . . . . . . . 283
9.3.3 Computation of Local Invariant: Reduction to
Crossing Numbers and Basic Degrees . . . . . . . . . . . . . . . . 284
9.3.4 Summary of the Equivariant Degree Method . . . . . . . . . . 287
9.3.5 An ODE System with SO(3)-Symmetries . . . . . . . . . . . . . 289
9.4 Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
10 Symmetric Hopf Bifurcation for Functional Differential
Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291
10.1 Symmetric Hopf Bifurcation for FDEs: General Framework . . . 292
10.2 Computation of the Local Γ × S1-Invariant . . . . . . . . . . . . . . . . . 297
10.3 Γ-Symmetric FDEs Describing Configurations of Identical
Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
10.3.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
10.3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302
10.3.3 Characteristic Equation for a Symmetric Configuration
of Identical Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305
10.3.4 Applications of the Equivariant Degree . . . . . . . . . . . . . . . 306
10.4 Hopf Bifurcation Results for Concrete Configurations of
Identical Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307
10.4.1 Model and Equivariant Degree Data . . . . . . . . . . . . . . . . . 307
XX Contents
10.4.2 Equivariant Degree: First Coefficients . . . . . . . . . . . . . . . . 310
10.4.3 Usage of Maple c Routines . . . . . . . . . . . . . . . . . . . . . . . . . 311
10.4.4 Hopf Bifurcation in a System with Dihedral Symmetries 314
10.4.5 Hopf Bifurcation in a System with Tetrahedral
Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
10.4.6 Hopf Bifurcation in a System with Octahedral
Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
10.4.7 Hopf Bifurcation in a System with Icosahedral
Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
10.5 Symmetric Hopf Bifurcation for Neutral FDEs . . . . . . . . . . . . . . 326
10.6 Hopf Bifurcation in Transmission Lines Model (by H. Ruan) . . 329
10.6.1 Symmetric Configurations of Lossless Transmission
Line Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
10.6.2 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 338
10.6.3 Characteristic Equation and Isolated Centers . . . . . . . . . 339
10.6.4 Negative Spectrum σ− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
10.6.5 Crossing Numbers tj,1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
10.7 Concrete Results for Selected Symmetry Groups and Usage
of Maple c Package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
10.8 Global Bifurcation Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347
10.8.1 Setting for Symmetric Global Hopf Bifurcation Theory . 348
10.8.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
10.9 Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
11 Symmetric Bifurcation Problems for Parabolic Systems
of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
11.1 Fredholm Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
11.2 Symmetric Bifurcation in Parametrized Equivariant
Coincidence Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364
11.2.1 Functional Setting for Equivariant Coincidence Problems365
11.2.2 Bifurcation Invariant for the Equivariant Coincidence
Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367
11.3 Hopf Bifurcation for FPDEs with Symmetries . . . . . . . . . . . . . . . 370
11.3.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 370
11.3.2 Normalization of the Period . . . . . . . . . . . . . . . . . . . . . . . . 371
11.3.3 Γ × S1-Setting in Functional Spaces . . . . . . . . . . . . . . . . . 372
11.3.4 Γ-Symmetric Steady-State Solutions to (11.18) . . . . . . . . 373
11.3.5 Characteristic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
11.3.6 Local Bifurcation Γ × S1-Invariant and Its Computation377
11.3.7 Local Bifurcation Result . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
11.4 Symmetric System of Hutchinson Model in Population
Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
11.4.1 Hutchinson Model of an n Species Ecosystem . . . . . . . . . 381
11.4.2 Symmetric System of the Hutchinson Model . . . . . . . . . . 382
11.4.3 Characteristic Equation and Isolated Centers . . . . . . . . . 383
Contents XXI
11.4.4 Computations for the Local Bifurcation Γ ×S1-Invariant384
11.4.5 Usage of Maple c Package and Concrete Results for
Selected Symmetry Groups . . . . . . . . . . . . . . . . . . . . . . . . . 387
11.5 Hopf Bifurcation in the Taylor-Couette Problem . . . . . . . . . . . . . 394
11.5.1 Description of the Taylor-Couette Model . . . . . . . . . . . . . 394
11.5.2 Taylor-Couette Problem as a Parametrized
O(2) × S1-Equivariant Coincidence Problem . . . . . . . . . . 397
11.5.3 Characteristic Equation for Taylor-Couette Problem . . . 399
11.5.4 Local Hopf Bifurcation Invariant for the Taylor-
Couette Problem: the Case of (j, j)-Mode
Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
11.6 Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404
12 Systems of Symmetric Van Der Pol Equations . . . . . . . . . . . . . 405
12.1 Systems of Van Der Pol Equations with Symmetries . . . . . . . . . 405
12.2 Reformulation as an Equivariant Fixed-Point Problem with
One Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
12.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407
12.2.2 Setting in Functional Spaces . . . . . . . . . . . . . . . . . . . . . . . . 408
12.2.3 Operator Reformulation of the Problem (12.9): Setting
for the Equivariant Degree Treatment. . . . . . . . . . . . . . . . 409
12.2.4 Hirano-Rybicki Approach: Reduction to a
Computation of the Equivariant Degree . . . . . . . . . . . . . . 411
12.3 Computations of the Equivariant Degree: Reduction to Basic
Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414
12.3.1 Finite-Dimensional Reduction . . . . . . . . . . . . . . . . . . . . . . . 414
12.3.2 Isotypical Decomposition and Basic Maps . . . . . . . . . . . . 416
12.3.3 Product Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417
12.4 Existence of Symmetric Periodic Solutions in Van Der Pol
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
12.4.1 A Priori Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419
12.4.2 Existence Result: Formulation . . . . . . . . . . . . . . . . . . . . . . 422
12.4.3 Proof of Theorem 12.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423
12.5 Conclusions and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425
12.5.1 Usage of Maple c Routines . . . . . . . . . . . . . . . . . . . . . . . . . 426
12.5.2 Conclusions for the Dihedral Group . . . . . . . . . . . . . . . . . . 426
12.5.3 Conclusions for the Tetrahedral Group . . . . . . . . . . . . . . . 427
12.5.4 Conclusions for the Octahedral Group . . . . . . . . . . . . . . . 428
12.5.5 Conclusions for the Icosahedral Group . . . . . . . . . . . . . . . 428
12.6 Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430
13 Variational Problems with Symmetries (by H. Ruan) . . . . . . . . 431
13.1 Orthogonal G-Equivariant Degree for Maps in Hilbert Space . . 431
13.2 Autonomous Newtonian System with Symmetries. . . . . . . . . . . . 434
13.2.1 Setting in Functional Spaces . . . . . . . . . . . . . . . . . . . . . . . . 436
XXII Contents
13.2.2 Existence Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438
13.2.3 Computation of deg1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439
13.2.4 Concrete Existence Results for Selected Symmetries . . . 442
13.3 Bifurcation Problem for Symmetric Variational Equations . . . . 449
13.3.1 Setting in Functional Spaces . . . . . . . . . . . . . . . . . . . . . . . . 449
13.3.2 Local Bifurcation Invariant . . . . . . . . . . . . . . . . . . . . . . . . . 451
13.3.3 Computation of ω(λo) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453
13.3.4 Computational Results for Symmetric Variational
Bifurcation Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455
13.4 Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462
Part III APPENDICES
Justification and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465
A1 Maple c Routines (by A. Biglands) . . . . . . . . . . . . . . . . . . . . . . . . . . 469
A1.1 Preparation of Maple c Routines . . . . . . . . . . . . . . . . . . . . . . . . . . 469
A1.2 Generation of Multiplication Tables . . . . . . . . . . . . . . . . . . . . . . . . 470
A1.3 Generation of Basic Degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474
A1.4 Useful Maple c Procedures for Computations of the
Equivariant Degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476
A1.5 Usage of the Equivariant Degree Library for Maple c . . . . . . . . . 478
A2 Complex Structures in Algebras, Poincar´e Index and
Bounded Solutions to Homogeneous ODEs (by Z. Balanov,
Y. Krasnov and W. Krawcewicz) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483
A2.1 Differential Systems in Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 483
A2.1.1 Algebras: Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 484
A2.1.2 Some Special Classes of Algebras . . . . . . . . . . . . . . . . . . . . 486
A2.1.3 Powers, Monomials and Derivation of Polynomials in
Non-Associative Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 490
A2.1.4 Quadratic ODE’s, Riccati Equation and Algebraic
Insolubility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491
A2.1.5 Examples of Quadratic Systems . . . . . . . . . . . . . . . . . . . . . 492
A2.2 Complex Structures in Two-Dimensional Algebras and
Poincar´e Index of Quadratic Maps . . . . . . . . . . . . . . . . . . . . . . . . . 493
A2.2.1 Complex Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493
A2.2.2 Poincar´e Index of Planar Quadratic Maps and Norms
in Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495
A2.2.3 Index of Division Algebras and Complex Structures . . . . 497
A2.2.4 Index of Algebras with 2-Nilpotents and Complex
Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499
A2.3 Complex Structures in Two-Dimensional Algebras and
Poincar´e Index of Cubic Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502
Contents XXIII
A2.3.1 General Strategy for Computing ind (0, x3) . . . . . . . . . . . 503
A2.3.2 Negative 3-Idempotents in Algebras without both 2-
and 3-Nilpotents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503
A2.3.3 Complex Structures in Regular Algebras with
2-Nilpotents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505
A2.3.4 Complex Structures in Regular Algebras with
3-Nilpotents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506
A2.3.5 Negative 3-Idempotent and ind (0, x3) . . . . . . . . . . . . . . . . 508
A2.4 Complex Structures and Bounded Solutions to the Riccati
and Cubic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508
A2.4.1 Bounded Solutions: Definition and Motivating Examples508
A2.4.2 Bounded Solutions to the Real Two-Dimensional
Riccati Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509
A2.4.3 Bounded Solutions to the Complex Two-Dimensional
Riccati Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512
A2.4.4 n-Dimensional Corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . 513
A2.4.5 System (A2.14) via Complex Structures . . . . . . . . . . . . . . 515
A2.4.6 Complex Structures in Two-Dimensional Algebras
and Bounded Solutions to x˙ = x3 . . . . . . . . . . . . . . . . . . . . 516
A2.5 Systems Homogeneous at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . 517
A2.5.1 Muhamadiyev-type Results . . . . . . . . . . . . . . . . . . . . . . . . . 517
A2.5.2 Constructing Regular Guiding Functions: Statement
of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519
A2.5.3 Derivation in MA and Regular Guiding Functions . . . . . 520
A2.6 Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521
A3 Examples of Equivariant Homogeneous Maps, Symmetric
Powers and Atiyah-Tall Theorem (by Z. Balanov,
W. Krawcewicz and A. Kushkuley) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523
A3.1 Goal and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523
A3.2Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524
A3.3 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526
A3.4 More on Example A3.1 and Atiyah-Tall Theorem . . . . . . . . . . . . 529
A3.5 Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547
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| August 22, 2020 | Edited by ISBNbot2 | normalize ISBN |
| April 20, 2012 | Edited by 99.72.204.121 | Edited without comment. |
| April 20, 2012 | Edited by 99.72.204.121 | Added new cover |
| April 20, 2012 | Created by 99.72.204.121 | Added new book. |