Applications of Degree Theory on Dynamical Systems with Symmetry

178 Pages·2017·0.72 MB·English

by  Wu, Hao Pin

APPLICATIONS OF DEGREE THEORY TO
DYNAMICAL SYSTEMS WITH SYMMETRY
(WITH SPECIAL FOCUS ON COMPUTATIONAL ASPECTS
AND ALGEBRAIC CHALLENGES)
by
Hao-pin Wu
APPROVED BY SUPERVISORY COMMITTEE:
Zalman Balanov, Co-Chair
Wieslaw Krawcewicz, Co-Chair
Dmitry Rachinskiy
Qingwen Hu
Copyright ⃝c 2018
Hao-pin Wu
All rights reserved
To My Family
APPLICATIONS OF DEGREE THEORY TO
DYNAMICAL SYSTEMS WITH SYMMETRY
(WITH SPECIAL FOCUS ON COMPUTATIONAL ASPECTS
AND ALGEBRAIC CHALLENGES)
by
HAO-PIN WU, MS
DISSERTATION
Presented to the Faculty of
The University of Texas at Dallas
in Partial Ful(cid:12)llment
of the Requirements
for the Degree of
DOCTOR OF PHILOSOPHY IN
MATHEMATICS
THE UNIVERSITY OF TEXAS AT DALLAS
May 2018
ACKNOWLEDGMENTS
I would like to thank Professor Balanov and Professor Krawcewicz for their advice and
support throughout the PhD program; it would not be possible for me to accomplish
what I have in my dissertation without their help. Thank to Professor Rachinskiy and
Professor Hu for sharing their insights and valuable ideas with me from time to time;
their help is important to my research work. For the research work in Chapter 4 of
my dissertation, I want to thank Profosser Muzychuk; it is a precious experience for
me to work with such an accomplished mathematician. Also, thanks to Dr. Jafeh and
Dr. Li; the efforts they made in their research eventually became the foundation of my
work. Finally, I want to thank my family and all the friends I met in UTD for their support
on my pursuit of the PhD degree.
April 2018
v
APPLICATIONS OF DEGREE THEORY TO
DYNAMICAL SYSTEMS WITH SYMMETRY
(WITH SPECIAL FOCUS ON COMPUTATIONAL ASPECTS
AND ALGEBRAIC CHALLENGES)
Hao-pin Wu, PhD
The University of Texas at Dallas, 2018
Supervising Professors: Zalman Balanov, Co-Chair
Wieslaw Krawcewicz, Co-Chair
The study of dynamical systems with symmetry usually deals with the impact of symmetries
(described by a certain group G) on the existence, multiplicity, stability and topological
structure of solutions to the system, (local/global) bifurcation phenomena, etc. Among
differentapproaches, degreetheory(includingBrouwerdegreeandequivariantdegree), which
involves analysis, topology and algebra, provides an effective tool for the study.
In our research, we focus on three motivating problems related to dynamical systems with
symmetry: (a) bifurcation of periodic solutions in symmetric reversible FDEs; (b) existence
ofperiodicsolutionstoequivariantHamiltoniansystems; (c)existenceofperiodicsolutionsto
systems homogeneous at in(cid:12)nity. In Problem (a), equivariant degree with no free parameters
provides us with the complete description of bifurcating branches of 2(cid:25)-periodic solutions
to reversible systems. In Problem (b), we can predict various symmetric vibrational modes
of the fullerene molecule C using gradient degree. The study of Problem (c) leads to
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several results in algebra; two important results among them are (i) a characterization of
the class of (cid:12)nite solvable groups in terms of lengths of non-trivial orbits in irreducible
vi
representations, and (ii) the existence of an equivariant quadratic map between two non-
equivalent (n (cid:0) 1)-dimensional S -representation spheres (n is odd). Finally, as the result
n
of developing computational tools for equivariant degree, we present the related algorithms
and examples.
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TABLE OF CONTENTS
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Motivation and Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Bifurcation of Periodic Solutions in Symmetric Reversible FDEs . . . 1
1.1.2 Existence of Periodic Solutions to Equivariant Hamiltonian Systems . 3
1.1.3 Existence of Periodic Solutions to Systems Homogeneous at In(cid:12)nity . 3
1.2 Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.1 Brouwer Degree in the Presence of Symmetries . . . . . . . . . . . . . 4
1.2.2 The Equivariant Degree . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Algebraic Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3.1 Computation of Equivariant Degrees . . . . . . . . . . . . . . . . . . 6
1.3.2 Solvable Groups and the Existence of Equivariant Quadratic Maps . . 7
1.4 Summary of Results and Overview . . . . . . . . . . . . . . . . . . . . . . . 8
1.4.1 Bifurcation of Periodic Solutions in Symmetric Reversible FDEs . . . 8
1.4.2 Existence of Periodic Solutions to Equivariant Hamiltonian Systems . 8
1.4.3 Existence of Periodic Solutions to Systems Homogeneous at In(cid:12)nity . 9
1.4.4 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
CHAPTER2 BIFURCATIONOFSPACEPERIODICSOLUTIONSINSYMMETRIC
REVERSIBLE FDES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1 Equivariant Jargon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.2 Subgroups in Direct Product of Groups G (cid:2)G . . . . . . . . . . . . 18
1 2
2.2.3 ((cid:0)(cid:2)O(2))-Representations . . . . . . . . . . . . . . . . . . . . . . . 19
2.2.4 Admissible G-Pairs and Burnside Ring A(G) . . . . . . . . . . . . . . 22
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2.2.5 Equivariant Degree without Parameter . . . . . . . . . . . . . . . . . 24
2.3 Assumptions and Fixed-Point Problem Reformulation . . . . . . . . . . . . . 27
2.3.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.3.2 G-Equivariant Operator Reformulation of (2.1) in Functional Spaces . 29
2.4 ((cid:0)(cid:2)O(2))-Degree Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4.1 Linearization and Necessary Condition for the Bifurcation . . . . . . 32
2.4.2 Sufficient Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.4.3 Computation of !((cid:11) ) . . . . . . . . . . . . . . . . . . . . . . . . . . 38
o
2.5 Computation of !((cid:11) ): Examples . . . . . . . . . . . . . . . . . . . . . . . . 41
0
2.5.1 Space-Reversal Symmetry for Second Order DDEs and IDEs . . . . . 41
2.5.2 Coupling Systems with Octahedral Symmetry and First Order Refor-
mulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.5.3 Equivariant Spectral Information . . . . . . . . . . . . . . . . . . . . 43
2.5.4 Parameter Space and Bifurcation Mechanism . . . . . . . . . . . . . . 47
2.5.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
CHAPTER 3 NONLINEAR VIBRATIONS IN THE FULLERENE MOLECULE C 53
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3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2 Equivariant Degree Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2.1 Euler Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2.2 Burnside Ring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.2.3 Euler Ring Homomorphism . . . . . . . . . . . . . . . . . . . . . . . 61
3.2.4 Twisted Subgroups and Related Modules . . . . . . . . . . . . . . . . 62
3.2.5 Computations of Euler Ring U(G) in Special Cases . . . . . . . . . . 63
3.2.6 Brouwer G-Equivariant Degree . . . . . . . . . . . . . . . . . . . . . 65
3.2.7 G-Equivariant Gradient Degree . . . . . . . . . . . . . . . . . . . . . 67
3.2.8 Degree on the Slice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
3.2.9 Computations of the Gradient G-Equivariant Degree . . . . . . . . . 73
3.2.10 Basic Degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.3 Fullerene Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
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3.3.1 Equations for Carbons . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.3.2 Force Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.3.3 Icosahedral Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . 83
3.3.4 Icosahedral Minimizer . . . . . . . . . . . . . . . . . . . . . . . . . . 85
3.3.5 Isotypical Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . 87
3.4 Equivariant Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
3.4.1 Equivariant Gradient Map . . . . . . . . . . . . . . . . . . . . . . . . 91
3.4.2 Bifurcation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.4.3 Computation of the Gradient Degree . . . . . . . . . . . . . . . . . . 96
3.5 Description of the Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . 98
3.5.1 Standing Waves (Brake Orbits) . . . . . . . . . . . . . . . . . . . . . 100
3.5.2 Traveling Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
CHAPTER 4 ON SOME APPLICATIONS OF GROUP REPRESENTATION THE-
ORY TO ALGEBRAIC PROBLEMS RELATED TO THE CONGRUENCE PRIN-
CIPLE FOR EQUIVARIANT MAPS . . . . . . . . . . . . . . . . . . . . . . . . . 102
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.1.1 Topological Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.1.2 Main Results and Overview . . . . . . . . . . . . . . . . . . . . . . . 105
4.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.2.1 Groups and Their Actions . . . . . . . . . . . . . . . . . . . . . . . . 107
4.2.2 Group Representations . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.3 (cid:11)-Characteristic of G-Representations . . . . . . . . . . . . . . . . . . . . . . 110
4.3.1 (cid:11)-Characteristic of Solvable Group Representations . . . . . . . . . . 113
4.3.2 (cid:11)-Characteristic of Nilpotent Group Representations . . . . . . . . . 117
4.4 (cid:11)-Characteristic of an Augmentation Module Related to 2-transitive Group
Actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.4.1 2-Transitive Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
4.4.2 Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
4.4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
4.5 Irreducible Representations with Trivial (cid:11)-Characteristic . . . . . . . . . . . 122
x