Bifurcation theory and applications:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New Jersey [u.a.]
World Scientific
2005
|
| Series: | World Scientific series on nonlinear science : Series A
53 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIII, 375 S. graph. Darst. |
| ISBN: | 9812562877 9812563520 |
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| 020 | |a 9812563520 |c pbk. |9 981-256-352-0 | ||
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| 084 | |a SK 540 |0 (DE-625)143245: |2 rvk | ||
| 100 | 1 | |a Ma, Tian |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Bifurcation theory and applications |c Tian Ma ; Shouhong Wang |
| 264 | 1 | |a New Jersey [u.a.] |b World Scientific |c 2005 | |
| 300 | |a XIII, 375 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a World Scientific series on nonlinear science : Series A |v 53 | |
| 650 | 4 | |a Bifurcation theory | |
| 650 | 0 | 7 | |a Verzweigung |g Mathematik |0 (DE-588)4078889-1 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Verzweigung |g Mathematik |0 (DE-588)4078889-1 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Wang, Shouhong |e Verfasser |4 aut | |
| 810 | 2 | |a World Scientific series on nonlinear science |t Series A |v 53 |w (DE-604)BV009051753 |9 53 | |
| 856 | 4 | 2 | |m Digitalisierung UB Augsburg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015446331&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
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| _version_ | 1804136224820035584 |
|---|---|
| adam_text | Contents
Preface vii
2.
Introduction
to Steady
State
Bifurcation
Theory
1
1.1
Implicit Function Theorem
.................. 1
1.2
Basics of Topological Degree Theory
............. 2
1.2.1
Brouwer
degree
..................... 2
1.2.2
Basic theorems of
Brouwer
degree
........... 4
1.2.3
Leray-Schauder degree
................. 5
1.2.4
Indices of isolated singularities
............. 7
1.3
Lyapunov-Schmidt Method
.................. 8
1.3.1
Preliminaries
...................... 8
1.3.2
Lyapunov-Schmidt procedure
............. 9
1.3.3
Normalization
...................... 12
1.4
Krasnosel ski Bifurcation Theorems
.............. 13
1.4.1
Bifurcation from eigenvalues with odd multiplicity
. 13
1.4.2
Krasnosel ski theorem for potential operators
.... 14
1.5
Rabinowitz Global Bifurcation Theorem
........... 17
1.6
Notes
.............................. 19
2.
Introduction to Dynamic Bifurcation
21
2.1
Motivation
........................... 21
2.2
Semi-groups of Linear Operators
............... 23
2.2.1
Introduction
....................... 23
2.2.2
Strongly continuous semi-groups
........... 25
2.2.3
Sectorial
operators and analytic semi-groups
..... 26
2.2.4
Powers of linear operators
............... 28
x
Bifurcation Theory and Applications
2.3
Dissipative Dynamical Systems
................ 29
2.4
Center Manifold Theorems
.................. 32
2.4.1
Center and stable manifolds in Rn
.......... 32
2.4.2
Center manifolds for infinite dimensional systems
. . 34
2.4.3
Construction of center manifolds
........... 37
2.5 Hopf
Bifurcation
........................ 38
2.6
Notes
.............................. 40
3.
Reduction Procedures and Stability
41
3.1
Spectrum Theory of Linear Completely Continuous Fields
. 41
3.1.1
Eigenvalues of linear completely continuous fields
. . 41
3.1.2
Spectral theorems
.................... 44
3.1.3
Asymptotic properties of eigenvalues
......... 50
3.1.4
Generic properties
................... 53
3.2
Reduction Methods
....................... 56
3.2.1
Reduction procedures
.................. 56
3.2.2
Morse index of
nondegenerate
singular points
.... 62
3.3
Asymptotic Stability at Critical States
............ 66
3.3.1
Introduction to the Lyapunov stability
........ 66
3.3.2
Finite dimensional cases
................ 67
3.3.3
An alternative principle for stability
......... 70
3.3.4
Dimension reduction
.................. 72
3.4
Notes
.............................. 74
4.
Steady State Bifurcations
75
4.1
Bifurcations from Higher-Order
Nondegenerate
Singularities
75
4.1.1
Even-order
nondegenerate
singularities
........ 75
4.1.2
Bifurcation at geometric simple eigenvalues:
r
= 1 . 83
4.1.3
Bifurcation with
r = k =
2............... 85
4.1.4
Reduction to potential operators
........... 90
4.2
Alternative Method
....................... 92
4.2.1
Introduction
....................... 92
4.2.2
Alternative bifurcation theorems
........... 94
4.2.3
General principle
.................... 98
4.3
Bifurcation from Homogeneous Terms
............ 100
4.4
Notes
.............................. 103
5.
Dynamic Bifurcation Theory: Finite Dimensional Case
105
Contents xi
5.1
Introduction
........................... 105
5.1.1
Pendulum in a symmetric magnetic field
....... 105
5.1.2
Business cycles for Kaldor s model
.......... 110
5.1.3
Basic principle of attractor bifurcation
........ 112
5.2
Attractor Bifurcation
...................... 114
5.2.1
Main theorems
..................... 114
5.2.2
Stability of attractors
.................. 116
5.2.3
Proof of Theorems
5.2
and
5.3............. 119
5.2.4
Structure of bifurcated attractors
........... 123
5.2.5
Generalized
Hopf
bifurcation
............. 127
5.3
Invariant Closed Manifolds
.................. 129
5.3.1
Hyperbolic invariant manifolds
............ 129
5.3.2
S1 attractor bifurcation
................ 132
5.4
Stability of Dynamic Bifurcation
............... 138
5.5
Notes
.............................. 149
6.
Dynamic Bifurcation Theory: Infinite Dimensional Case
151
6.1
Attractor Bifurcation
...................... 152
6.1.1
Equations with first-order in time
........... 152
6.1.2
Equations with second-order in time
......... 154
6.2
Bifurcation from Simple Eigenvalues
............. 160
6.2.1
Structure of dynamic bifurcation
........... 160
6.2.2
Saddle-node bifurcation
................ 163
6.3
Bifurcation from Eigenvalues with Multiplicity Two
.... 165
6.3.1
An index formula
.................... 165
6.3.2
Main theorems
..................... 169
6.3.3
Proof of main theorems
................ 172
6.3.4
Case where k>3
.................... 184
6.3.5
Bifurcation to periodic solutions
............ 184
6.4
Stability for Perturbed Systems
................ 188
6.4.1
General case
....................... 188
6.4.2
Perturbation at simple eigenvalues
.......... 191
6.5
Notes
.............................. 194
7.
Bifurcations for Nonlinear Elliptic Equations
197
7.1
Preliminaries
.......................... 197
7.1.1
Sobolev spaces
..................... 197
7.1.2
Regularity estimates
.................. 200
xii
Bifurcation
Theory and Applications
7.1.3
Maximum principle
................... 201
7.2
Bifurcation of
Semilinear
Elliptic Equations
......... 202
7.2.1
Transcritical bifurcations
................ 202
7.2.2
Saddle-node bifurcation
................ 207
7.3
Bifurcation from Homogenous Terms
............. 209
7.3.1 Superlinear
case
..................... 209
7.3.2
Sublinear case
...................... 210
7.4
Bifurcation of Positive Solutions of Second Order Elliptic
Equations
............................ 213
7.4.1
Bifurcation in exponent parameter
. ......... 214
7.4.2
Local bifurcation
.................... 222
7.4.3
Global bifurcation from the
sublinear
terms
..... 231
7.4.4
Global bifurcation from the linear terms
........ 236
7.5
Notes
.............................. 240
8.
Reaction-Diffusion Equations
241
8.1
Introduction
........................... 241
8.1.1
Equations and their mathematical setting
...... 241
8.1.2
Examples from Physics, Chemistry and Biology
. . . 243
8.2
Bifurcation of Reaction-Diffusion Systems
.......... 246
8.2.1
Periodic solutions
.................... 246
8.2.2
Attractor bifurcation
.................. 248
8.3
Singularity Sphere in 5>m-Attractors
............. 251
8.3.1
Dirichlet boundary condition
............. 251
8.3.2
Periodic boundary condition
.............. 256
8.3.3
Invariant homological spheres
............. 258
8.4
Belousov-Zhabotinsky Reaction Equations
.......... 259
8.4.1
Set-up
.......................... 259
8.4.2
Bifurcated attractor
.................. 260
8.5
Notes
.............................. 265
9.
Pattern Formation and Wave Equations
267
9.1
Kuramoto-Sivashinsky Equation
............... 267
9.1.1
Set-up
.......................... 267
9.Ï.2
Symmetric case
..................... 268
9.1.3
General case
....................... 271
9.1.4
S^-invariant sets
.................... 273
9.2
Cahn-Hillard Equation
..................... 275
Contents xiii
9.2.1
Set-up
.......................... 275
9.2.2
Neumann boundary condition
............. 276
9.2.3
Periodic boundary condition
.............. 286
9.2.4
Saddle-node bifurcation
................ 290
9.3
Complex Ginzburg-Landau Equation
............. 291
9.3.1
Set-up
.......................... 291
9.3.2
Dirichlet boundary condition
............. 293
9.3.3
Periodic boundary condition
.............. 296
9.4
Ginzburg-Landau Equations of Superconductivity
...... 297
9.4.1
The model
........................ 297
9.4.2
Attractor bifurcation
.................. 302
9.4.3
Physical remarks
.................... 315
9.5
Wave Equations
......................... 322
9.5.1
Wave equations with damping
............. 322
9.5.2
System of wave equations
............... 324
9.6
Notes
.............................. 325
10.
Fluid Dynamics
327
10.1
Geometric Theory for 2-D Incompressible Flows
....... 327
10.1.1
Introduction and preliminaries
............. 327
10.1.2
Structural stability theorems
.............. 327
10.2
Rayleigh-Bénard
Convection
.................. 330
10.2.1
Benard
problem
..................... 330
10.2.2
Boussinesq equations
.................. 331
10.2.3
Attractor bifurcation of the
Rayleigh-Bénard
problem
335
10.2.4
2-D
Rayleigh-Bénard
convection
............ 341
10.3
Taylor Problem
......................... 343
10.3.1
Taylor s experiments and Taylor vortices
....... 343
10.3.2
Governing equations
.................. 343
10.3.3
Stability of secondary flows
.............. 349
10.3.4
Taylor vortices
..................... 354
10.4
Notes
............;................. 365
Bibliography
367
Index
373
|
| adam_txt |
Contents
Preface vii
2.
Introduction
to Steady
State
Bifurcation
Theory
1
1.1
Implicit Function Theorem
. 1
1.2
Basics of Topological Degree Theory
. 2
1.2.1
Brouwer
degree
. 2
1.2.2
Basic theorems of
Brouwer
degree
. 4
1.2.3
Leray-Schauder degree
. 5
1.2.4
Indices of isolated singularities
. 7
1.3
Lyapunov-Schmidt Method
. 8
1.3.1
Preliminaries
. 8
1.3.2
Lyapunov-Schmidt procedure
. 9
1.3.3
Normalization
. 12
1.4
Krasnosel'ski Bifurcation Theorems
. 13
1.4.1
Bifurcation from eigenvalues with odd multiplicity
. 13
1.4.2
Krasnosel'ski theorem for potential operators
. 14
1.5
Rabinowitz Global Bifurcation Theorem
. 17
1.6
Notes
. 19
2.
Introduction to Dynamic Bifurcation
21
2.1
Motivation
. 21
2.2
Semi-groups of Linear Operators
. 23
2.2.1
Introduction
. 23
2.2.2
Strongly continuous semi-groups
. 25
2.2.3
Sectorial
operators and analytic semi-groups
. 26
2.2.4
Powers of linear operators
. 28
x
Bifurcation Theory and Applications
2.3
Dissipative Dynamical Systems
. 29
2.4
Center Manifold Theorems
. 32
2.4.1
Center and stable manifolds in Rn
. 32
2.4.2
Center manifolds for infinite dimensional systems
. . 34
2.4.3
Construction of center manifolds
. 37
2.5 Hopf
Bifurcation
. 38
2.6
Notes
. 40
3.
Reduction Procedures and Stability
41
3.1
Spectrum Theory of Linear Completely Continuous Fields
. 41
3.1.1
Eigenvalues of linear completely continuous fields
. . 41
3.1.2
Spectral theorems
. 44
3.1.3
Asymptotic properties of eigenvalues
. 50
3.1.4
Generic properties
. 53
3.2
Reduction Methods
. 56
3.2.1
Reduction procedures
. 56
3.2.2
Morse index of
nondegenerate
singular points
. 62
3.3
Asymptotic Stability at Critical States
. 66
3.3.1
Introduction to the Lyapunov stability
. 66
3.3.2
Finite dimensional cases
. 67
3.3.3
An alternative principle for stability
. 70
3.3.4
Dimension reduction
. 72
3.4
Notes
. 74
4.
Steady State Bifurcations
75
4.1
Bifurcations from Higher-Order
Nondegenerate
Singularities
75
4.1.1
Even-order
nondegenerate
singularities
. 75
4.1.2
Bifurcation at geometric simple eigenvalues:
r
= 1 . 83
4.1.3
Bifurcation with
r = k =
2. 85
4.1.4
Reduction to potential operators
. 90
4.2
Alternative Method
. 92
4.2.1
Introduction
. 92
4.2.2
Alternative bifurcation theorems
. 94
4.2.3
General principle
. 98
4.3
Bifurcation from Homogeneous Terms
. 100
4.4
Notes
. 103
5.
Dynamic Bifurcation Theory: Finite Dimensional Case
105
Contents xi
5.1
Introduction
. 105
5.1.1
Pendulum in a symmetric magnetic field
. 105
5.1.2
Business cycles for Kaldor's model
. 110
5.1.3
Basic principle of attractor bifurcation
. 112
5.2
Attractor Bifurcation
. 114
5.2.1
Main theorems
. 114
5.2.2
Stability of attractors
. 116
5.2.3
Proof of Theorems
5.2
and
5.3. 119
5.2.4
Structure of bifurcated attractors
. 123
5.2.5
Generalized
Hopf
bifurcation
. 127
5.3
Invariant Closed Manifolds
. 129
5.3.1
Hyperbolic invariant manifolds
. 129
5.3.2
S1 attractor bifurcation
. 132
5.4
Stability of Dynamic Bifurcation
. 138
5.5
Notes
. 149
6.
Dynamic Bifurcation Theory: Infinite Dimensional Case
151
6.1
Attractor Bifurcation
. 152
6.1.1
Equations with first-order in time
. 152
6.1.2
Equations with second-order in time
. 154
6.2
Bifurcation from Simple Eigenvalues
. 160
6.2.1
Structure of dynamic bifurcation
. 160
6.2.2
Saddle-node bifurcation
. 163
6.3
Bifurcation from Eigenvalues with Multiplicity Two
. 165
6.3.1
An index formula
. 165
6.3.2
Main theorems
. 169
6.3.3
Proof of main theorems
. 172
6.3.4
Case where k>3
. 184
6.3.5
Bifurcation to periodic solutions
. 184
6.4
Stability for Perturbed Systems
. 188
6.4.1
General case
. 188
6.4.2
Perturbation at simple eigenvalues
. 191
6.5
Notes
. 194
7.
Bifurcations for Nonlinear Elliptic Equations
197
7.1
Preliminaries
. 197
7.1.1
Sobolev spaces
. 197
7.1.2
Regularity estimates
. 200
xii
Bifurcation
Theory and Applications
7.1.3
Maximum principle
. 201
7.2
Bifurcation of
Semilinear
Elliptic Equations
. 202
7.2.1
Transcritical bifurcations
. 202
7.2.2
Saddle-node bifurcation
. 207
7.3
Bifurcation from Homogenous Terms
. 209
7.3.1 Superlinear
case
. 209
7.3.2
Sublinear case
. 210
7.4
Bifurcation of Positive Solutions of Second Order Elliptic
Equations
. 213
7.4.1
Bifurcation in exponent parameter
. . 214
7.4.2
Local bifurcation
. 222
7.4.3
Global bifurcation from the
sublinear
terms
. 231
7.4.4
Global bifurcation from the linear terms
. 236
7.5
Notes
. 240
8.
Reaction-Diffusion Equations
241
8.1
Introduction
. 241
8.1.1
Equations and their mathematical setting
. 241
8.1.2
Examples from Physics, Chemistry and Biology
. . . 243
8.2
Bifurcation of Reaction-Diffusion Systems
. 246
8.2.1
Periodic solutions
. 246
8.2.2
Attractor bifurcation
. 248
8.3
Singularity Sphere in 5>m-Attractors
. 251
8.3.1
Dirichlet boundary condition
. 251
8.3.2
Periodic boundary condition
. 256
8.3.3
Invariant homological spheres
. 258
8.4
Belousov-Zhabotinsky Reaction Equations
. 259
8.4.1
Set-up
. 259
8.4.2
Bifurcated attractor
. 260
8.5
Notes
. 265
9.
Pattern Formation and Wave Equations
267
9.1
Kuramoto-Sivashinsky Equation
. 267
9.1.1
Set-up
. 267
9.Ï.2
Symmetric case
. 268
9.1.3
General case
. 271
9.1.4
S^-invariant sets
. 273
9.2
Cahn-Hillard Equation
. 275
Contents xiii
9.2.1
Set-up
. 275
9.2.2
Neumann boundary condition
. 276
9.2.3
Periodic boundary condition
. 286
9.2.4
Saddle-node bifurcation
. 290
9.3
Complex Ginzburg-Landau Equation
. 291
9.3.1
Set-up
. 291
9.3.2
Dirichlet boundary condition
. 293
9.3.3
Periodic boundary condition
. 296
9.4
Ginzburg-Landau Equations of Superconductivity
. 297
9.4.1
The model
. 297
9.4.2
Attractor bifurcation
. 302
9.4.3
Physical remarks
. 315
9.5
Wave Equations
. 322
9.5.1
Wave equations with damping
. 322
9.5.2
System of wave equations
. 324
9.6
Notes
. 325
10.
Fluid Dynamics
327
10.1
Geometric Theory for 2-D Incompressible Flows
. 327
10.1.1
Introduction and preliminaries
. 327
10.1.2
Structural stability theorems
. 327
10.2
Rayleigh-Bénard
Convection
. 330
10.2.1
Benard
problem
. 330
10.2.2
Boussinesq equations
. 331
10.2.3
Attractor bifurcation of the
Rayleigh-Bénard
problem
335
10.2.4
2-D
Rayleigh-Bénard
convection
. 341
10.3
Taylor Problem
. 343
10.3.1
Taylor's experiments and Taylor vortices
. 343
10.3.2
Governing equations
. 343
10.3.3
Stability of secondary flows
. 349
10.3.4
Taylor vortices
. 354
10.4
Notes
.;. 365
Bibliography
367
Index
373 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Ma, Tian Wang, Shouhong |
| author_facet | Ma, Tian Wang, Shouhong |
| author_role | aut aut |
| author_sort | Ma, Tian |
| author_variant | t m tm s w sw |
| building | Verbundindex |
| bvnumber | BV022235314 |
| callnumber-first | Q - Science |
| callnumber-label | QA380 |
| callnumber-raw | QA380 |
| callnumber-search | QA380 |
| callnumber-sort | QA 3380 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 540 |
| ctrlnum | (OCoLC)61504323 (DE-599)BVBBV022235314 |
| dewey-full | 515.392 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.392 |
| dewey-search | 515.392 |
| dewey-sort | 3515.392 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV022235314 |
| illustrated | Illustrated |
| index_date | 2024-07-02T16:33:55Z |
| indexdate | 2024-07-09T20:53:00Z |
| institution | BVB |
| isbn | 9812562877 9812563520 |
| language | English |
| lccn | 2006295789 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015446331 |
| oclc_num | 61504323 |
| open_access_boolean | |
| owner | DE-703 DE-384 DE-11 |
| owner_facet | DE-703 DE-384 DE-11 |
| physical | XIII, 375 S. graph. Darst. |
| publishDate | 2005 |
| publishDateSearch | 2005 |
| publishDateSort | 2005 |
| publisher | World Scientific |
| record_format | marc |
| series2 | World Scientific series on nonlinear science : Series A |
| spelling | Ma, Tian Verfasser aut Bifurcation theory and applications Tian Ma ; Shouhong Wang New Jersey [u.a.] World Scientific 2005 XIII, 375 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier World Scientific series on nonlinear science : Series A 53 Bifurcation theory Verzweigung Mathematik (DE-588)4078889-1 gnd rswk-swf Verzweigung Mathematik (DE-588)4078889-1 s DE-604 Wang, Shouhong Verfasser aut World Scientific series on nonlinear science Series A 53 (DE-604)BV009051753 53 Digitalisierung UB Augsburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015446331&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Ma, Tian Wang, Shouhong Bifurcation theory and applications Bifurcation theory Verzweigung Mathematik (DE-588)4078889-1 gnd |
| subject_GND | (DE-588)4078889-1 |
| title | Bifurcation theory and applications |
| title_auth | Bifurcation theory and applications |
| title_exact_search | Bifurcation theory and applications |
| title_exact_search_txtP | Bifurcation theory and applications |
| title_full | Bifurcation theory and applications Tian Ma ; Shouhong Wang |
| title_fullStr | Bifurcation theory and applications Tian Ma ; Shouhong Wang |
| title_full_unstemmed | Bifurcation theory and applications Tian Ma ; Shouhong Wang |
| title_short | Bifurcation theory and applications |
| title_sort | bifurcation theory and applications |
| topic | Bifurcation theory Verzweigung Mathematik (DE-588)4078889-1 gnd |
| topic_facet | Bifurcation theory Verzweigung Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015446331&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV009051753 |
| work_keys_str_mv | AT matian bifurcationtheoryandapplications AT wangshouhong bifurcationtheoryandapplications |