Bifurcation theory: an introduction with applications to partial differential equations
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
2012
|
| Edition: | 2. ed. |
| Series: | Applied mathematical sciences
156 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | VII, 398 S. graph. Darst. |
| ISBN: | 9781461405016 |
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MARC
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| 245 | 1 | 0 | |a Bifurcation theory |b an introduction with applications to partial differential equations |c Hansjörg Kielhöfer |
| 250 | |a 2. ed. | ||
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Record in the Search Index
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|---|---|
| adam_text | Titel: Bifurcation theory
Autor: Kielhöfer, Hansjörg
Jahr: 2012
Contents
0 Introduction.............................................. 3
I Local Theory............................................. 7
I.1 The Implicit Function Theorem.......................... 7
I.2 The Method of Lyapunov-Schmidt ....................... 8
I.3 The Lyapunov-Schmidt Reduction for Potential Operators . . 11
I.4 An Implicit Function Theorem for One-Dimensional Kernels:
Turning Points......................................... 13
I.5 Bifurcation with a One-Dimensional Kernel................ 17
I.6 Bifurcation Formulas (Stationary Case) ................... 21
I.7 The Principle of Exchange of Stability (Stationary Case) .... 23
I.8 Hopf Bifurcation....................................... 34
I.9 Bifurcation Formulas for Hopf Bifurcation................. 44
I.10 A Lyapunov Center Theorem............................ 51
I.11 Constrained Hopf Bifurcation for Hamiltonian, Reversible,
and Conservative Systems............................... 57
I.11.1 Hamiltonian Systems: Lyapunov Center Theorem
and Hamiltonian Hopf Bifurcation.................. 63
I.11.2 Reversible Systems............................... 72
I.11.3 Nonlinear Oscillations............................. 76
I.11.4 Conservative Systems............................. 79
I.12 The Principle of Exchange of Stability for Hopf Bifurcation .. 82
I.13 Continuation of Periodic Solutions and Their Stability...... 90
I.13.1 Exchange of Stability at a Turning Point............101
I.14 Period-Doubling Bifurcation and Exchange of Stability......105
I.14.1 The Principle of Exchange of Stability for a
Period-Doubling Bifurcation.......................114
I.15 The Newton Polygon ...................................120
I.16 Degenerate Bifurcation at a Simple Eigenvalue and Stability
of Bifurcating Solutions.................................125
I.16.1 The Principle of Exchange of Stability for Degenerate
Bifurcation......................................133
I.17 Degenerate Hopf Bifurcation and Floquet Exponents of
Bifurcating Periodic Orbits..............................139
I.17.1 The Principle of Exchange of Stability for Degenerate
Hopf Bifurcation.................................146
I.18 The Principle of Reduced Stability for Stationary and
Periodic Solutions......................................153
I.18.1 The Principle of Reduced Stability for Periodic
Solutions........................................160
I.19 Bifurcation with High-Dimensional Kernels, Multiparameter
Bifurcation, and Application of the Principle of Reduced
Stability...............................................166
I.19.1 Bifurcation with a Two-Dimensional Kernel .........173
I.19.2 A Multiparameter Bifurcation Theorem with a
High-Dimensional Kernel..........................179
I.20 Bifurcation from Infinity................................182
I.21 Bifurcation with High-Dimensional Kernels for Potential
Operators: Variational Methods..........................184
I.22 Notes and Remarks to Chapter I.........................192
II Global Theory............................................195
II. 1 The Brouwer Degree....................................195
II.2 The Leray-Schauder Degree.............................199
II.3 Application of the Degree in Bifurcation Theory............203
II.4 Odd Crossing Numbers for Fredholm Operators and Local
Bifurcation............................................206
II.4.1 Local Bifurcation via Odd Crossing Numbers........211
II.5 A Degree for a Class of Proper Fredholm Operators and
Global Bifurcation Theorems ............................216
II.5.1 Global Bifurcation via Odd Crossing Numbers.......226
II.5.2 Global Bifurcation with One-Dimensional Kernel.....227
II.6 A Global Implicit Function Theorem......................231
II.7 Change of Morse Index and Local Bifurcation for Potential
Operators.............................................233
II. 7.1 Local Bifurcation for Potential Operators............236
II.8 Notes and Remarks to Chapter II ........................239
III Applications..............................................241
111.1 The Fredholm Property of Elliptic Operators..............241
III. 1.1 Elliptic Operators on a Lattice.....................248
III. 1.2 Spectral Properties of Elliptic Operators............254
III.2 Local Bifurcation for Elliptic Problems....................255
III.2.1 Bifurcation with a One-Dimensional Kernel..........257
III.2.2 Bifurcation with a Two-Dimensional Kernel .........264
III.2.3 Bifurcation with High-Dimensional Kernels..........267
III.2.4 Variational Methods I.............................208
III.2.5 Variational Methods II............................273
III.2.6 An Example.....................................275
III.3 Free Nonlinear Vibrations...............................288
III.3.1 Variational Methods..............................297
III.3.2 Bifurcation with a One-Dimensional Kernel..........299
III.4 Hopf Bifurcation for Parabolic Problems ..................300
III.5 Global Bifurcation and Continuation for Elliptic Problems .. . 314
III.5.1 An Example (Continued)..........................320
III.5.2 Global Continuation..............................321
III.6 Preservation of Nodal Structure on Global Branches........323
III.0.1 A Maximum Principle ............................323
III.0.2 Global Branches of Positive Solutions on a Domain
or on a Lattice...................................325
III.6.3 Unbounded Branches of Positive Solutions...........331
III.6.4 Separation of Branches............................333
III.6.5 An Example (Continued)..........................334
III.6.0 Global Branches of Positive Solutions via Continuation 348
III. 7 Smoothness and Uniqueness of Global Positive Solution
Branches..............................................351
III.7.1 Bifurcation from Infinity..........................358
III.7.2 Local Parameterization of Positive Solution Branches
over Symmetric Domains..........................304
III. 7.3 Global Parameterization of Positive Solution
Branches over Symmetric Domains and Uniqueness ... 371
III.7.4 Asymptotic Behavior at ||u||oo = 0 and ||u||oo = oo .... 376
III.7.5 Stability of Positive Solution Branches..............381
III.8 Notes and Remarks to Chapter III........................385
References....................................................387
Index.........................................................395
|
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| author | Kielhöfer, Hansjörg 1945- |
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| dewey-full | 510 515.35 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 510 - Mathematics 515 - Analysis |
| dewey-raw | 510 515.35 |
| dewey-search | 510 515.35 |
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| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| institution | BVB |
| isbn | 9781461405016 |
| language | English |
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| spelling | Kielhöfer, Hansjörg 1945- Verfasser (DE-588)108216675 aut Bifurcation theory an introduction with applications to partial differential equations Hansjörg Kielhöfer 2. ed. New York [u.a.] Springer 2012 VII, 398 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Applied mathematical sciences 156 Partielle Differentialgleichung - Verzweigung <Mathematik> Bifurcation theory Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Verzweigung Mathematik (DE-588)4078889-1 gnd rswk-swf Verzweigung Mathematik (DE-588)4078889-1 s Partielle Differentialgleichung (DE-588)4044779-0 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4614-0502-3 Applied mathematical sciences 156 (DE-604)BV000005274 156 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024591807&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Kielhöfer, Hansjörg 1945- Bifurcation theory an introduction with applications to partial differential equations Applied mathematical sciences Partielle Differentialgleichung - Verzweigung <Mathematik> Bifurcation theory Partielle Differentialgleichung (DE-588)4044779-0 gnd Verzweigung Mathematik (DE-588)4078889-1 gnd |
| subject_GND | (DE-588)4044779-0 (DE-588)4078889-1 |
| title | Bifurcation theory an introduction with applications to partial differential equations |
| title_auth | Bifurcation theory an introduction with applications to partial differential equations |
| title_exact_search | Bifurcation theory an introduction with applications to partial differential equations |
| title_full | Bifurcation theory an introduction with applications to partial differential equations Hansjörg Kielhöfer |
| title_fullStr | Bifurcation theory an introduction with applications to partial differential equations Hansjörg Kielhöfer |
| title_full_unstemmed | Bifurcation theory an introduction with applications to partial differential equations Hansjörg Kielhöfer |
| title_short | Bifurcation theory |
| title_sort | bifurcation theory an introduction with applications to partial differential equations |
| title_sub | an introduction with applications to partial differential equations |
| topic | Partielle Differentialgleichung - Verzweigung <Mathematik> Bifurcation theory Partielle Differentialgleichung (DE-588)4044779-0 gnd Verzweigung Mathematik (DE-588)4078889-1 gnd |
| topic_facet | Partielle Differentialgleichung - Verzweigung <Mathematik> Bifurcation theory Partielle Differentialgleichung Verzweigung Mathematik |
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