Applied Equivariant Degree
Publisher: AIMS
Published: 2006
Place of publication: Springfield, MO 65801-2604, USA
Format: Hardcover
Number of pages: 574
Description: "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.
Table of contents: title: 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1; title: 1.1 Motivation and Subject . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1; title: 1.2 Main Goal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5; title: 1.3 Historical Roots and Topological Aspects of Equivariant Degree 9; title: 1.4 Overview of the Book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14; title: 1.5 Other Related Publication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18; title: 1.6 Potential for Further Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19; title: 1.7 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20; title: Part I EQUIVARIANT DEGREE THEORY: TECHNICAL; title: TOOLS; title: 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25; title: 2.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25; title: 2.2 Elements of Representation Theory . . . . . . . . . . . . . . . . . . . . . . . . 30; title: 2.2.1 Finite-dimensional G-Representations . . . . . . . . . . . . . . . . 30; title: 2.2.2 Complexification and Conjugation . . . . . . . . . . . . . . . . . . . 33; title: 2.2.3 Character of Representation . . . . . . . . . . . . . . . . . . . . . . . . 35; title: 2.2.4 Haar Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35; title: 2.2.5 Isotypical Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 35; title: 2.2.6 Decomposition of GLG(V ). . . . . . . . . . . . . . . . . . . . . . . . . . 37; title: 2.2.7 Banach G-Representations . . . . . . . . . . . . . . . . . . . . . . . . . . 37; title: 2.3 Equivariant Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40; title: 2.4 Elements of Smooth Equivariant Topology . . . . . . . . . . . . . . . . . . 44; title: 2.4.1 Basic Facts from Differential Topology . . . . . . . . . . . . . . . 44; title: 2.4.2 G-Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46; title: 2.4.3 G-Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47; title: 2.4.4 Orientation on G-manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 49; title: 2.4.5 Bi-orientable Compact Lie Groups . . . . . . . . . . . . . . . . . . . 52; title: 2.4.6 Local Brouwer Degree. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54; title: XVI Contents; title: 2.5 Numbers n(L,H) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56; title: 2.5.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56; title: 2.5.2 Interpretation of the Numbers n(L,H) . . . . . . . . . . . . . . . 58; title: 2.6 Elements of Bordism Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59; title: 2.6.1 Oriented Bordism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59; title: 2.6.2 Singular Oriented Bordism and Whitney Theorem . . . . . 60; title: 2.6.3 Singular Oriented Bordism and Homology . . . . . . . . . . . . 61; title: 2.6.4 Framed Bordism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62; title: 2.6.5 Framed Equivariant Bordism Relation . . . . . . . . . . . . . . . 63; title: 2.7 Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66; title: 3 Equivariant Degree Theory: Construction and Basic; title: Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67; title: 3.1 Stable Equivariant Homotopy Groups of Spheres . . . . . . . . . . . . 67; title: 3.1.1 Non-Equivariant Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68; title: 3.1.2 Equivariant Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71; title: 3.2 Construction of Equivariant Degree . . . . . . . . . . . . . . . . . . . . . . . . 73; title: 3.2.1 Non-Equivariant Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73; title: 3.2.2 Equivariant Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74; title: 3.3 Regular Normal Approximation of Equivariant Maps . . . . . . . . . 76; title: 3.3.1 Normal Maps: Definition and Examples . . . . . . . . . . . . . . 77; title: 3.3.2 Normal Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79; title: 3.3.3 Regular Normal Approximations . . . . . . . . . . . . . . . . . . . . 82; title: 3.4 Structure of the Group ΠG: Pontryagin-Thom Isomorphism . . 87; title: 3.4.1 Equivariant Degree Techniques . . . . . . . . . . . . . . . . . . . . . . 87; title: 3.4.2 Decomposition of the Group ΠG . . . . . . . . . . . . . . . . . . . . 89; title: 3.4.3 Group Π(H) and Equivariant Bordism: Pontryagin-; title: Thom Isomorphism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91; title: 3.5 Computations of Primary Groups Π(H) . . . . . . . . . . . . . . . . . . . . 92; title: 3.5.1 Equivariant Bordism Approach . . . . . . . . . . . . . . . . . . . . . . 93; title: 3.5.2 Fundamental Domain Approach . . . . . . . . . . . . . . . . . . . . . 95; title: 3.6 Computations of the Secondary Groups Π(H) for n = 1 . . . . . . 99; title: 3.6.1 Homomorphism Φ and Short Exact Sequence . . . . . . . . . 99; title: 3.6.2 Homomorphism p∗ and Five Lemma . . . . . . . . . . . . . . . . . 101; title: 3.7 Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102; title: 4 S1-Equivariant Degree and Primary Degree for One Free; title: Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105; title: 4.1 Primary Equivariant Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106; title: 4.1.1 Primary Equivariant Degree: Construction . . . . . . . . . . . . 106; title: 4.1.2 Primary Equivariant Degree: Justification . . . . . . . . . . . . 108; title: 4.1.3 Primary Equivariant Degree: Basic Properties . . . . . . . . . 108; title: 4.1.4 Axiomatic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112; title: 4.2 Axiomatic Definition of the S1-Equivariant Degree. . . . . . . . . . . 112; title: 4.2.1 Basic Maps and m-Folding . . . . . . . . . . . . . . . . . . . . . . . . . 113; title: Contents XVII; title: 4.2.2 Formulation of the Main Result and Immediate; title: Consequences of Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . 113; title: 4.3 Proof of Theorem 4.11: Preliminaries and Central Lemma . . . . 115; title: 4.3.1 Positive Orientation in a Slice and Central Lemma . . . . 115; title: 4.3.2 Proof of Lemma 4.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117; title: 4.4 Proof of Theorem 4.11: Existence and Uniqueness . . . . . . . . . . . 121; title: 4.4.1 Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121; title: 4.4.2 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121; title: 4.5 Computation of S1-Degree via Reduction to Basic and; title: C-Complementing Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123; title: 4.5.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 123; title: 4.5.2 C-Complementing Maps and Suspension Procedure . . . . 124; title: 4.5.3 Homotopy Factorization: Properties of GLG(V ) . . . . . . . 128; title: 4.5.4 Splitting Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130; title: 4.5.5 S1-Degree Formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132; title: 4.6 Recurrence Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134; title: 4.6.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134; title: 4.6.2 Proof of Theorem 4.25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136; title: 4.7 Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138; title: 5 Basic Maps and Computations of Twisted Primary Degree 139; title: 5.1 Twisted Subgroups of G = Γ × S1 . . . . . . . . . . . . . . . . . . . . . . . . . 140; title: 5.2 Twisted Primary Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142; title: 5.3 Examples and Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143; title: 5.3.1 Quaternionic Units Group Q8 . . . . . . . . . . . . . . . . . . . . . . . 143; title: 5.3.2 Dihedral Group DN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145; title: 5.3.3 Tetrahedral Group A4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150; title: 5.3.4 Octahedral Group S4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153; title: 5.3.5 Icosahedral Group A5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156; title: 5.3.6 Orthogonal Group O(2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161; title: 5.4 Twisted Subgroups of SO(3) × S1 and Numbers n(L,H) . . . . . 162; title: 5.5 Basic Maps for Irreducible Representations of G = Γ × S1 . . . . 165; title: 5.6 Computations of Twisted Degrees of the Basic Maps . . . . . . . . . 170; title: 5.6.1 Computational Formula for Basic Maps with one; title: Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171; title: 5.6.2 Computational Formula for Basic Maps without; title: Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172; title: 5.7 Examples of Computations of the Equivariant Degree for; title: Basic Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172; title: 5.7.1 Degrees of Basic Maps for the Group Q8 . . . . . . . . . . . . . 173; title: 5.7.2 Degrees of Basic Maps for the Dihedral Group DN . . . . 174; title: 5.7.3 Degrees of Basic Maps for the Tetrahedral Group A4 . . 177; title: 5.7.4 Degrees of Basic Maps for the Octahedral Group S4 . . . 178; title: 5.7.5 Degrees of Basic Maps for the Icosahedral Group A5 . . . 181; title: 5.7.6 Degrees of Basic Maps for the Group O(2) . . . . . . . . . . . . 184; title: XVIII Contents; title: 5.8 Twisted Degree of Basic Maps for the Group SO(3) . . . . . . . . . 185; title: 5.8.1 Irreducible representations of SO(3) × S1 . . . . . . . . . . . . . 185; title: 5.8.2 Computation of Twisted Degrees for SO(3)×S1-Actions189; title: 5.9 Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190; title: 6 Algebraic Properties of the Equivariant Degree . . . . . . . . . . . . 191; title: 6.1 Computations of the Burnside Ring . . . . . . . . . . . . . . . . . . . . . . . . 191; title: 6.2 A(Γ)-Module Structure on At; title: 1(Γ × S1) and Multiplication; title: Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195; title: 6.3 Examples of A(Γ)-Modules At; title: 1(Γ × S1) . . . . . . . . . . . . . . . . . . . . 197; title: 6.3.1 A(Q8)-Module At; title: 1(Q8 × S1) . . . . . . . . . . . . . . . . . . . . . . . . 198; title: 6.3.2 A(DN)-Module At; title: 1(DN × S1) . . . . . . . . . . . . . . . . . . . . . . . 198; title: 6.3.3 A(A4)-Module At; title: 1(A4 × S1). . . . . . . . . . . . . . . . . . . . . . . . . 201; title: 6.3.4 A(S4)-Module At; title: 1(S4 × S1) . . . . . . . . . . . . . . . . . . . . . . . . . 203; title: 6.3.5 A(A5)-Module At; title: 1(A5 × S1). . . . . . . . . . . . . . . . . . . . . . . . . 203; title: 6.3.6 A(O(2))-Module At; title: 1(O(2) × S1) . . . . . . . . . . . . . . . . . . . . . 203; title: 6.4 Computation of the A(SO(3))-Module Structure for; title: At; title: 1(SO(3) × S1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203; title: 6.5 Multiplicativity Property for the Twisted Degree . . . . . . . . . . . . 207; title: 6.6 Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211; title: 7 Secondary Groups and Secondary Equivariant Degree for; title: Γ = SO(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213; title: 7.1 Preliminary Computations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213; title: 7.2 Computations of Reduced Secondary Groups Π∗(K) for; title: Γ = SO(3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216; title: 7.2.1 Simple Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216; title: 7.2.2 Representations Vt; title: j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217; title: 7.2.3 Computation of Π∗(K): Regular Cases . . . . . . . . . . . . . . . 219; title: 7.2.4 Computation of Π∗(K): First Typical Exceptional Case 221; title: 7.2.5 Computation of Π∗(K): Second Typical Exceptional; title: Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224; title: 7.2.6 Computation of Π∗(K): Conclusions . . . . . . . . . . . . . . . . . 226; title: 7.3 Some Remarks on Computations of the Secondary Degree . . . . 228; title: 7.3.1 Computation of the Reduced Degree deg∗; title: Γ (f,Ω) . . . . . . . 228; title: 7.3.2 Computation of the Z2-Component of degs; title: Γ (f,Ω). . . . . . 231; title: 7.4 Computations of Secondary Degree for Γ = SO(3): Special; title: Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232; title: 7.5 Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234; title: 8 Orthogonal G-Equivariant Degree (by H. Ruan) . . . . . . . . . . . . 235; title: 8.1 Equivariant Orthogonal Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235; title: 8.2 S1-Normality for G-Orthogonal Maps . . . . . . . . . . . . . . . . . . . . . . 237; title: 8.3 Construction of Orthogonal Degree . . . . . . . . . . . . . . . . . . . . . . . . 240; title: 8.4 Orthogonal Degree of G-Orthogonal Linear Maps . . . . . . . . . . . . 243; title: Contents XIX; title: 8.5 Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247; title: Part II APPLICATIONS OF EQUIVARIANT DEGREE; title: TO DIFFERENTIAL EQUATIONS AND BIFURCATION; title: PROBLEMS; title: 9 Setting for Studying Ordinary Differential Equations with; title: Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251; title: 9.1 Infinite-Dimensional Equivariant Degree . . . . . . . . . . . . . . . . . . . . 252; title: 9.1.1 Leray-Schauder Equivariant Degree . . . . . . . . . . . . . . . . . . 252; title: 9.1.2 Nussbaum-Sadovskii Equivariant Degree . . . . . . . . . . . . . 255; title: 9.2 Hopf Bifurcation Problem for ODEs without Symmetries . . . . . 259; title: 9.2.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 259; title: 9.2.2 S1-Equivariant Reformulation of the Problem . . . . . . . . . 261; title: 9.2.3 S1-Degree Method for Hopf Bifurcation Problem . . . . . . 264; title: 9.2.4 Deformation of the Map Fς . . . . . . . . . . . . . . . . . . . . . . . . . 270; title: 9.2.5 Product Formula for S1-Deg (Fς,Ω) . . . . . . . . . . . . . . . . . 274; title: 9.2.6 Crossing Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275; title: 9.2.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279; title: 9.3 Hopf Bifurcation Problem for an ODE System with Symmetries280; title: 9.3.1 Symmetric Hopf Bifurcation and Local Bifurcation; title: Invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 280; title: 9.3.2 Computation of Local Bifurcation Invariant:; title: Reduction to Product Formula . . . . . . . . . . . . . . . . . . . . . . 283; title: 9.3.3 Computation of Local Invariant: Reduction to; title: Crossing Numbers and Basic Degrees . . . . . . . . . . . . . . . . 284; title: 9.3.4 Summary of the Equivariant Degree Method . . . . . . . . . . 287; title: 9.3.5 An ODE System with SO(3)-Symmetries . . . . . . . . . . . . . 289; title: 9.4 Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290; title: 10 Symmetric Hopf Bifurcation for Functional Differential; title: Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291; title: 10.1 Symmetric Hopf Bifurcation for FDEs: General Framework . . . 292; title: 10.2 Computation of the Local Γ × S1-Invariant . . . . . . . . . . . . . . . . . 297; title: 10.3 Γ-Symmetric FDEs Describing Configurations of Identical; title: Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300; title: 10.3.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300; title: 10.3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302; title: 10.3.3 Characteristic Equation for a Symmetric Configuration; title: of Identical Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305; title: 10.3.4 Applications of the Equivariant Degree . . . . . . . . . . . . . . . 306; title: 10.4 Hopf Bifurcation Results for Concrete Configurations of; title: Identical Oscillators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307; title: 10.4.1 Model and Equivariant Degree Data . . . . . . . . . . . . . . . . . 307; title: XX Contents; title: 10.4.2 Equivariant Degree: First Coefficients . . . . . . . . . . . . . . . . 310; title: 10.4.3 Usage of Maple c Routines . . . . . . . . . . . . . . . . . . . . . . . . . 311; title: 10.4.4 Hopf Bifurcation in a System with Dihedral Symmetries 314; title: 10.4.5 Hopf Bifurcation in a System with Tetrahedral; title: Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321; title: 10.4.6 Hopf Bifurcation in a System with Octahedral; title: Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322; title: 10.4.7 Hopf Bifurcation in a System with Icosahedral; title: Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322; title: 10.5 Symmetric Hopf Bifurcation for Neutral FDEs . . . . . . . . . . . . . . 326; title: 10.6 Hopf Bifurcation in Transmission Lines Model (by H. Ruan) . . 329; title: 10.6.1 Symmetric Configurations of Lossless Transmission; title: Line Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329; title: 10.6.2 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 338; title: 10.6.3 Characteristic Equation and Isolated Centers . . . . . . . . . 339; title: 10.6.4 Negative Spectrum σ− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341; title: 10.6.5 Crossing Numbers tj,1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342; title: 10.7 Concrete Results for Selected Symmetry Groups and Usage; title: of Maple c Package . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344; title: 10.8 Global Bifurcation Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347; title: 10.8.1 Setting for Symmetric Global Hopf Bifurcation Theory . 348; title: 10.8.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350; title: 10.9 Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355; title: 11 Symmetric Bifurcation Problems for Parabolic Systems; title: of Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357; title: 11.1 Fredholm Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357; title: 11.2 Symmetric Bifurcation in Parametrized Equivariant; title: Coincidence Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364; title: 11.2.1 Functional Setting for Equivariant Coincidence Problems365; title: 11.2.2 Bifurcation Invariant for the Equivariant Coincidence; title: Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 367; title: 11.3 Hopf Bifurcation for FPDEs with Symmetries . . . . . . . . . . . . . . . 370; title: 11.3.1 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 370; title: 11.3.2 Normalization of the Period . . . . . . . . . . . . . . . . . . . . . . . . 371; title: 11.3.3 Γ × S1-Setting in Functional Spaces . . . . . . . . . . . . . . . . . 372; title: 11.3.4 Γ-Symmetric Steady-State Solutions to (11.18) . . . . . . . . 373; title: 11.3.5 Characteristic Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374; title: 11.3.6 Local Bifurcation Γ × S1-Invariant and Its Computation377; title: 11.3.7 Local Bifurcation Result . . . . . . . . . . . . . . . . . . . . . . . . . . . 380; title: 11.4 Symmetric System of Hutchinson Model in Population; title: Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381; title: 11.4.1 Hutchinson Model of an n Species Ecosystem . . . . . . . . . 381; title: 11.4.2 Symmetric System of the Hutchinson Model . . . . . . . . . . 382; title: 11.4.3 Characteristic Equation and Isolated Centers . . . . . . . . . 383; title: Contents XXI; title: 11.4.4 Computations for the Local Bifurcation Γ ×S1-Invariant384; title: 11.4.5 Usage of Maple c Package and Concrete Results for; title: Selected Symmetry Groups . . . . . . . . . . . . . . . . . . . . . . . . . 387; title: 11.5 Hopf Bifurcation in the Taylor-Couette Problem . . . . . . . . . . . . . 394; title: 11.5.1 Description of the Taylor-Couette Model . . . . . . . . . . . . . 394; title: 11.5.2 Taylor-Couette Problem as a Parametrized; title: O(2) × S1-Equivariant Coincidence Problem . . . . . . . . . . 397; title: 11.5.3 Characteristic Equation for Taylor-Couette Problem . . . 399; title: 11.5.4 Local Hopf Bifurcation Invariant for the Taylor-; title: Couette Problem: the Case of (j, j)-Mode; title: Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403; title: 11.6 Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404; title: 12 Systems of Symmetric Van Der Pol Equations . . . . . . . . . . . . . 405; title: 12.1 Systems of Van Der Pol Equations with Symmetries . . . . . . . . . 405; title: 12.2 Reformulation as an Equivariant Fixed-Point Problem with; title: One Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407; title: 12.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407; title: 12.2.2 Setting in Functional Spaces . . . . . . . . . . . . . . . . . . . . . . . . 408; title: 12.2.3 Operator Reformulation of the Problem (12.9): Setting; title: for the Equivariant Degree Treatment. . . . . . . . . . . . . . . . 409; title: 12.2.4 Hirano-Rybicki Approach: Reduction to a; title: Computation of the Equivariant Degree . . . . . . . . . . . . . . 411; title: 12.3 Computations of the Equivariant Degree: Reduction to Basic; title: Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414; title: 12.3.1 Finite-Dimensional Reduction . . . . . . . . . . . . . . . . . . . . . . . 414; title: 12.3.2 Isotypical Decomposition and Basic Maps . . . . . . . . . . . . 416; title: 12.3.3 Product Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417; title: 12.4 Existence of Symmetric Periodic Solutions in Van Der Pol; title: Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419; title: 12.4.1 A Priori Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 419; title: 12.4.2 Existence Result: Formulation . . . . . . . . . . . . . . . . . . . . . . 422; title: 12.4.3 Proof of Theorem 12.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423; title: 12.5 Conclusions and Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425; title: 12.5.1 Usage of Maple c Routines . . . . . . . . . . . . . . . . . . . . . . . . . 426; title: 12.5.2 Conclusions for the Dihedral Group . . . . . . . . . . . . . . . . . . 426; title: 12.5.3 Conclusions for the Tetrahedral Group . . . . . . . . . . . . . . . 427; title: 12.5.4 Conclusions for the Octahedral Group . . . . . . . . . . . . . . . 428; title: 12.5.5 Conclusions for the Icosahedral Group . . . . . . . . . . . . . . . 428; title: 12.6 Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 430; title: 13 Variational Problems with Symmetries (by H. Ruan) . . . . . . . . 431; title: 13.1 Orthogonal G-Equivariant Degree for Maps in Hilbert Space . . 431; title: 13.2 Autonomous Newtonian System with Symmetries. . . . . . . . . . . . 434; title: 13.2.1 Setting in Functional Spaces . . . . . . . . . . . . . . . . . . . . . . . . 436; title: XXII Contents; title: 13.2.2 Existence Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438; title: 13.2.3 Computation of deg1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439; title: 13.2.4 Concrete Existence Results for Selected Symmetries . . . 442; title: 13.3 Bifurcation Problem for Symmetric Variational Equations . . . . 449; title: 13.3.1 Setting in Functional Spaces . . . . . . . . . . . . . . . . . . . . . . . . 449; title: 13.3.2 Local Bifurcation Invariant . . . . . . . . . . . . . . . . . . . . . . . . . 451; title: 13.3.3 Computation of ω(λo) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453; title: 13.3.4 Computational Results for Symmetric Variational; title: Bifurcation Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455; title: 13.4 Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462; title: Part III APPENDICES; title: Justification and Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465; title: A1 Maple c Routines (by A. Biglands) . . . . . . . . . . . . . . . . . . . . . . . . . . 469; title: A1.1 Preparation of Maple c Routines . . . . . . . . . . . . . . . . . . . . . . . . . . 469; title: A1.2 Generation of Multiplication Tables . . . . . . . . . . . . . . . . . . . . . . . . 470; title: A1.3 Generation of Basic Degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474; title: A1.4 Useful Maple c Procedures for Computations of the; title: Equivariant Degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476; title: A1.5 Usage of the Equivariant Degree Library for Maple c . . . . . . . . . 478; title: A2 Complex Structures in Algebras, Poincar´e Index and; title: Bounded Solutions to Homogeneous ODEs (by Z. Balanov,; title: Y. Krasnov and W. Krawcewicz) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483; title: A2.1 Differential Systems in Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 483; title: A2.1.1 Algebras: Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 484; title: A2.1.2 Some Special Classes of Algebras . . . . . . . . . . . . . . . . . . . . 486; title: A2.1.3 Powers, Monomials and Derivation of Polynomials in; title: Non-Associative Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . 490; title: A2.1.4 Quadratic ODE’s, Riccati Equation and Algebraic; title: Insolubility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491; title: A2.1.5 Examples of Quadratic Systems . . . . . . . . . . . . . . . . . . . . . 492; title: A2.2 Complex Structures in Two-Dimensional Algebras and; title: Poincar´e Index of Quadratic Maps . . . . . . . . . . . . . . . . . . . . . . . . . 493; title: A2.2.1 Complex Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493; title: A2.2.2 Poincar´e Index of Planar Quadratic Maps and Norms; title: in Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 495; title: A2.2.3 Index of Division Algebras and Complex Structures . . . . 497; title: A2.2.4 Index of Algebras with 2-Nilpotents and Complex; title: Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499; title: A2.3 Complex Structures in Two-Dimensional Algebras and; title: Poincar´e Index of Cubic Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . 502; title: Contents XXIII; title: A2.3.1 General Strategy for Computing ind (0, x3) . . . . . . . . . . . 503; title: A2.3.2 Negative 3-Idempotents in Algebras without both 2-; title: and 3-Nilpotents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503; title: A2.3.3 Complex Structures in Regular Algebras with; title: 2-Nilpotents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505; title: A2.3.4 Complex Structures in Regular Algebras with; title: 3-Nilpotents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506; title: A2.3.5 Negative 3-Idempotent and ind (0, x3) . . . . . . . . . . . . . . . . 508; title: A2.4 Complex Structures and Bounded Solutions to the Riccati; title: and Cubic Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508; title: A2.4.1 Bounded Solutions: Definition and Motivating Examples508; title: A2.4.2 Bounded Solutions to the Real Two-Dimensional; title: Riccati Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509; title: A2.4.3 Bounded Solutions to the Complex Two-Dimensional; title: Riccati Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512; title: A2.4.4 n-Dimensional Corollaries . . . . . . . . . . . . . . . . . . . . . . . . . . 513; title: A2.4.5 System (A2.14) via Complex Structures . . . . . . . . . . . . . . 515; title: A2.4.6 Complex Structures in Two-Dimensional Algebras; title: and Bounded Solutions to x˙ = x3 . . . . . . . . . . . . . . . . . . . . 516; title: A2.5 Systems Homogeneous at Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . 517; title: A2.5.1 Muhamadiyev-type Results . . . . . . . . . . . . . . . . . . . . . . . . . 517; title: A2.5.2 Constructing Regular Guiding Functions: Statement; title: of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519; title: A2.5.3 Derivation in MA and Regular Guiding Functions . . . . . 520; title: A2.6 Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521; title: A3 Examples of Equivariant Homogeneous Maps, Symmetric; title: Powers and Atiyah-Tall Theorem (by Z. Balanov,; title: W. Krawcewicz and A. Kushkuley) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523; title: A3.1 Goal and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523; title: A3.2Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 524; title: A3.3 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 526; title: A3.4 More on Example A3.1 and Atiyah-Tall Theorem . . . . . . . . . . . . 529; title: A3.5 Bibliographical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532; title: References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533; title: Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 547