Elementary stability and bifurcation theory:
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York [u.a.]
Springer
1990
|
| Ausgabe: | 2. ed. |
| Schriftenreihe: | Undergraduate texts in mathematics
|
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | XXIII, 324 S. graph. Darst. |
| ISBN: | 0387970681 3540970681 |
Internformat
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| 100 | 1 | |a Iooss, Gérard |d 1944- |e Verfasser |0 (DE-588)138607931 |4 aut | |
| 245 | 1 | 0 | |a Elementary stability and bifurcation theory |c Gerard Iooss ; Daniel D. Joseph |
| 250 | |a 2. ed. | ||
| 264 | 1 | |a New York [u.a.] |b Springer |c 1990 | |
| 300 | |a XXIII, 324 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Undergraduate texts in mathematics | |
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| 650 | 7 | |a Equations différentielles - Solutions numériques |2 ram | |
| 650 | 7 | |a Evolutionaire vergelijkingen |2 gtt | |
| 650 | 7 | |a Stabilité |2 ram | |
| 650 | 4 | |a Bifurcation theory | |
| 650 | 4 | |a Differential equations |x Numerical solutions | |
| 650 | 4 | |a Evolution equations |x Numerical solutions | |
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Datensatz im Suchindex
| DE-BY-863_location | 1911 |
|---|---|
| DE-BY-FWS_call_number | 1911/1999:1265 |
| DE-BY-FWS_katkey | 62264 |
| DE-BY-FWS_media_number | 083100652361 |
| _version_ | 1824555397234556928 |
| adam_text | Contents
List of Frequently Used Symbols xiii
Introduction xvii
Preface to the Second Edition xxi
CHAPTER I
Asymptotic Solutions of Evolution Problems 1
1.1 One Dimensional, Two Dimensional, n Dimensional, and
Infinite Dimensional Interpretations of (I.I) 1
1.2 Forced Solutions; Steady Forcing and T Periodic Forcing;
Autonomous and Nonautonomous Problems 3
1.3 Reduction to Local Form 4
1.4 Asymptotic Solutions 5
1.5 Asymptotic Solutions and Bifurcating Solutions 5
1.6 Bifurcating Solutions and the Linear Theory of Stability 6
1.7 Notation for the Functional Expansion of F((, fi, U) 7
Notes 8
CHAPTER II
Bifurcation and Stability of Steady Solutions of Evolution
Equations in One Dimension 10
II. 1 The Implicit Function Theorem 10
11.2 Classification of Points on Solution Curves 11
11.3 The Characteristic Quadratic. Double Points, Cusp Points, and
Conjugate Points 12
11.4 Double Point Bifurcation and the Implicit Function Theorem 13
11.5 Cusp Point Bifurcation 14
vii
viii Contents
11.6 Triple Point Bifurcation 15
11.7 Conditional Stability Theorem 15
11.8 The Factorization Theorem in One Dimension 19
11.9 Equivalence of Strict Loss of Stability and Double Point Bifurcation 20
II. 10 Exchange of Stability at a Double Point 20
II. 11 Exchange of Stability at a Double Point for Problems Reduced to
Local Form 22
II. 12 Exchange of Stability at a Cusp Point 25
11.13 Exchange of Stability at a Triple Point 26
11.14 Global Properties of Stability of Isolated Solutions 26
CHAPTER III
Imperfection Theory and Isolated Solutions Which Perturb
Bifurcation 29
111.1 The Structure of Problems Which Break Double Point Bifurcation 30
111.2 The Implicit Function Theorem and the Saddle Surface Breaking
Bifurcation 31
111.3 Examples of Isolated Solutions Which Break Bifurcation 33
111.4 Iterative Procedures for Finding Solutions 34
111.5 Stability of Solutions Which Break Bifurcation 37
111.6 Isolas 39
Exercise 39
Notes 40
CHAPTER IV
Stability of Steady Solutions of Evolution Equations in Two
Dimensions and n Dimensions 42
IV. 1 Eigenvalues and Eigenvectors of an n x n Matrix 43
IV.2 Algebraic and Geometric Multiplicity—The Riesz Index 43
IV. 3 The Adjoint Eigenvalue Problem 44
IV.4 Eigenvalues and Eigenvectors of a 2 x 2 Matrix 45
4.1 Eigenvalues 45
4.2 Eigenvectors 46
4.3 Algebraically Simple Eigenvalues 46
4.4 Algebraically Double Eigenvalues 46
4.4.1 Riesz Index 1 46
4.4.2 Riesz Index 2 47
IV. 5 The Spectral Problem and Stability of the Solution u = 0 in W 48
IV.6 Nodes, Saddles, and Foci 49
IV.7 Criticality and Strict Loss of Stability 50
Appendix IV. 1
Biorthogonality for Generalized Eigenvectors 52
Appendix IV.2
Projections 55
Contents jx
CHAPTER V
Bifurcation of Steady Solutions in Two Dimensions and the
Stability of the Bifurcating Solutions 59
V. 1 The Form of Steady Bifurcating Solutions and Their Stability 59
V.2 Necessary Conditions for the Bifurcation of Steady Solutions 62
V.3 Bifurcation at a Simple Eigenvalue 63
V.4 Stability of the Steady Solution Bifurcating at a Simple Eigenvalue 64
V.5 Bifurcation at a Double Eigenvalue of Index Two 64
V.6 Stability of the Steady Solution Bifurcating at a Double Eigenvalue
of Index Two 66
V.7 Bifurcation and Stability of Steady Solutions in the Form (V.2) at a
Double Eigenvalue of Index One (Semi Simple) 67
V.8 Bifurcation and Stability of Steady Solutions (V.3) at a Semi Simple
Double Eigenvalue 70
V.9 Examples of Stability Analysis at a Double Semi Simple
(Index One) Eigenvalue 72
V. 10 Saddle Node Bifurcation 77
Appendix V. 1
Implicit Function Theorem for a System of Two Equations
in Two Unknown Functions of One Variable 80
Exercises 82
CHAPTER VI
Methods of Projection for General Problems of Bifurcation
into Steady Solutions 87
VI. 1 The Evolution Equation and the Spectral Problem 87
VI.2 Construction of Steady Bifurcating Solutions as Power Series
in the Amplitude 88
VI.3 1R1 and IR1 in Projection 90
VI.4 Stability of the Bifurcating Solution 91
VI.5 The Extra Little Part for R1 in Projection 92
VI.6 Projections of Higher Dimensional Problems 94
VI.7 The Spectral Problem for the Stability of u = 0 96
VI.8 The Spectral Problem and the Laplace Transform 98
VI.9 Projections into Ul 101
VI. 10 The Method of Projection for Isolated Solutions Which Perturb
Bifurcation at a Simple Eigenvalue (Imperfection Theory) 102
VI. 11 The Method of Projection at a Double Eigenvalue of Index Two 104
VI. 12 The Method of Projection at a Double Semi Simple Eigenvalue 107
VI. 13 Examples of the Method of Projection 111
VI. 14 Symmetry and Pitchfork Bifurcation 136
X Contents
CHAPTER VII
Bifurcation of Periodic Solutions from Steady Ones (Hopf
Bifurcation) in Two Dimensions 139
VII. 1 The Structure of the Two Dimensional Problem Governing
Hopf Bifurcation 139
VII.2 Amplitude Equation for Hopf Bifurcation 140
VII.3 Series Solution 141
VII.4 Equations Governing the Taylor Coefficients 141
VII.5 Solvability Conditions (the Fredholm Alternative) 141
VII.6 Floquet Theory 142
6.1 Floquet Theory in M 143
6.2 Floquet Theory in R2 and U 145
VII.7 Equations Governing the Stability of the Periodic Solutions 149
VII.8 The Factorization Theorem 149
VII.9 Interpretation of the Stability Result 150
Example 150
CHAPTER VIII
Bifurcation of Periodic Solutions in the General Case 156
VIII. 1 Eigenprojections of the Spectral Problem 156
VIII.2 Equations Governing the Projection and the Complementary
Projection 157
VIII. 3 The Series Solution Using the Fredholm Alternative 159
VIII.4 Stability of the Hopf Bifurcation in the General Case 164
VIII.5 Systems with Rotational Symmetry 165
Examples 167
Notes 175
CHAPTER IX
Subharmonic Bifurcation of Forced ^ Periodic Solutions 177
Notation 177
IX. 1 Definition of the Problem of Subharmonic Bifurcation 178
IX.2 Spectral Problems and the Eigenvalues o(n) 180
IX. 3 Biorthogonality 181
IX.4 Criticality 181
IX.5 The Fredholm Alternative for J(fi) — o(n) and a Formula
Expressing the Strict Crossing (IX.20) 182
IX.6 Spectral Assumptions 183
IX.7 Rational and Irrational Points of the Frequency Ratio at Criticality 183
IX.8 The Operator J and its Eigenvectors 185
IX.9 The Adjoint Operator J)*, Biorthogonality, Strict Crossing, and
the Fredholm Alternative for J 186
IX. 10 The Amplitude e and the Biorthogonal Decomposition of
Bifurcating Subharmonic Solutions 187
Contents xi
IX. 11 The Equations Governing the Derivatives of Bifurcating
Subharmonic Solutions with Respect to e at e = 0 188
IX. 12 Bifurcation and Stability of ^ Periodic and 27 Periodic Solutions 189
IX. 13 Bifurcation and Stability of nT Periodic Solutions with m 2 192
IX.14 Bifurcation and Stability of 37 Periodic Solutions 193
IX.15 Bifurcation of 47 Periodic Solutions 196
IX.16 Stability of 4T Periodic Solutions 199
IX. 17 Nonexistence of Higher Order Subharmonic Solutions and
Weak Resonance 203
IX. 18 Summary of Results About Subharmonic Bifurcation 204
IX. 19 Imperfection Theory with a Periodic Imperfection 204
Exercises 205
IX.20 Saddle Node Bifurcation of T Periodic Solutions 206
IX.21 General Remarks About Subharmonic Bifurcations 207
CHAPTER X
Bifurcation of Forced T Periodic Solutions into Asymptotically
Quasi Periodic Solutions 208
X. 1 Decomposition of the Solution and Amplitude Equation 209
Exercise 209
X.2 Derivation of the Amplitude Equation 210
X.3 The Normal Equations in Polar Coordinates 214
X.4 The Torus and Trajectories on the Torus in the Irrational Case 216
X.5 The Torus and Trajectories on the Torus When w0 T/2n Is a
Rational Point of Higher Order (n 5) 219
X.6 The Form of the Torus in the Case n = 5 221
X.7 Trajectories on the Torus When n = 5 222
X.8 The Form of the Torus When n 5 225
X.9 Trajectories on the Torus When n 5 228
X.10 Asymptotically Quasi Periodic Solutions 231
X.ll Stability of the Bifurcated Torus 232
X.I2 Subharmonic Solutions on the Torus 233
X. 13 Stability of Subharmonic Solutions on the Torus 236
X.14 Frequency Locking 239
Appendix X.I
Direct Computation of Asymptotically Quasi Periodic Solutions
Which Bifurcate at Irrational Points Using the Method of Two
Times, Power Series, and the Fredholm Alternative 243
Appendix X.2
Direct Computation of Asymptotically Quasi Periodic Solutions
Which Bifurcate at Rational Points of Higher Order
Using the Method of Two Times 247
Exercise 254
Notes 254
xii Contents
CHAPTER XI
Secondary Subharmonic and Asymptotically Quasi Periodic
Bifurcation of Periodic Solutions (of Hopf s Type) in the
Autonomous Case 256
Notation 258
XI. 1 Spectral Problems 258
XI.2 Criticality and Rational Points 260
XI.3 Spectral Assumptions About Jo 261
XI.4 Spectral Assumptions About J in the Rational Case 261
XI.5 Strict Loss of Stability at a Simple Eigenvalue of Jo 263
XI.6 Strict Loss of Stability at a Double Semi Simple Eigenvalue of Jo 265
XI.7 Strict Loss of Stability at a Double Eigenvalue of Index Two 266
XI.8 Formulation of the Problem of Subharmonic Bifurcation of
Periodic Solutions of Autonomous Problems 268
XI.9 The Amplitude of the Bifurcating Solution 269
XI. 10 Power Series Solutions of the Bifurcation Problem 270
XI. 11 Subharmonic Bifurcation When n = 2 272
XI. 12 Subharmonic Bifurcation When n 2 275
XI. 13 Subharmonic Bifurcation When n = 1 in the Semi Simple Case 278
XI. 14 Subharmonic Bifurcation When n = 1 in the Case When
Zero is an Index Two Double Eigenvalue of Jo 279
XI. 15 Stability of Subharmonic Solutions 281
XI.16 Summary of Results About Subharmonic Bifurcation in the
Autonomous Case 285
XI. 17 Amplitude Equations 286
XI. 18 Amplitude Equations for the Cases n 3 or rjo/a o Irrational 287
XI. 19 Bifurcating Tori. Asymptotically Quasi Periodic Solutions 291
XI.20 Period Doubling, n = 2 293
XI.21 Pitchfork Bifurcation of Periodic Orbits in the Presence of
Symmetry, n = 1 296
Exercises 298
XI.22 Rotationally Symmetric Problems 299
Exercise 302
CHAPTER XII
Stability and Bifurcation in Conservative Systems 303
XII.l The Rolling Ball 304
XII.2 Euler Buckling 305
Exercises 308
XII.3 Some Remarks About Spectral Problems for Conservative Systems 309
XII.4 Stability and Bifurcation of Rigid Rotation of Two Immiscible Liquids 311
Steady Rigid Rotation of Two Fluids 312
Index 319
|
| any_adam_object | 1 |
| author | Iooss, Gérard 1944- Joseph, Daniel D. 1929-2011 |
| author_GND | (DE-588)138607931 (DE-588)1082460168 |
| author_facet | Iooss, Gérard 1944- Joseph, Daniel D. 1929-2011 |
| author_role | aut aut |
| author_sort | Iooss, Gérard 1944- |
| author_variant | g i gi d d j dd ddj |
| building | Verbundindex |
| bvnumber | BV002615876 |
| callnumber-first | Q - Science |
| callnumber-label | QA372 |
| callnumber-raw | QA372 |
| callnumber-search | QA372 |
| callnumber-sort | QA 3372 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 520 SK 920 |
| classification_tum | MAT 587f MAT 673f |
| ctrlnum | (OCoLC)20262914 (DE-599)BVBBV002615876 |
| dewey-full | 515/.35 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.35 |
| dewey-search | 515/.35 |
| dewey-sort | 3515 235 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 2. ed. |
| format | Book |
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| id | DE-604.BV002615876 |
| illustrated | Illustrated |
| indexdate | 2025-02-20T07:07:01Z |
| institution | BVB |
| isbn | 0387970681 3540970681 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-001681952 |
| oclc_num | 20262914 |
| open_access_boolean | |
| owner | DE-91 DE-BY-TUM DE-91G DE-BY-TUM DE-384 DE-739 DE-20 DE-29T DE-703 DE-863 DE-BY-FWS DE-898 DE-BY-UBR DE-634 DE-188 DE-83 |
| owner_facet | DE-91 DE-BY-TUM DE-91G DE-BY-TUM DE-384 DE-739 DE-20 DE-29T DE-703 DE-863 DE-BY-FWS DE-898 DE-BY-UBR DE-634 DE-188 DE-83 |
| physical | XXIII, 324 S. graph. Darst. |
| publishDate | 1990 |
| publishDateSearch | 1990 |
| publishDateSort | 1990 |
| publisher | Springer |
| record_format | marc |
| series2 | Undergraduate texts in mathematics |
| spellingShingle | Iooss, Gérard 1944- Joseph, Daniel D. 1929-2011 Elementary stability and bifurcation theory Bifurcatie gtt Bifurcation, théorie de la ram Differentiaalvergelijkingen gtt Equations d'évolution - Solutions numériques ram Equations différentielles - Solutions numériques ram Evolutionaire vergelijkingen gtt Stabilité ram Bifurcation theory Differential equations Numerical solutions Evolution equations Numerical solutions Stability Stabilität (DE-588)4056693-6 gnd Evolutionsgleichung (DE-588)4129061-6 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Verzweigung Mathematik (DE-588)4078889-1 gnd Differentialgleichung (DE-588)4012249-9 gnd Nichtlineare Differentialgleichung (DE-588)4205536-2 gnd Asymptotisches Lösungsverhalten (DE-588)4134367-0 gnd Nichtlineare Evolutionsgleichung (DE-588)4221363-0 gnd Numerische Mathematik (DE-588)4042805-9 gnd |
| subject_GND | (DE-588)4056693-6 (DE-588)4129061-6 (DE-588)4128130-5 (DE-588)4078889-1 (DE-588)4012249-9 (DE-588)4205536-2 (DE-588)4134367-0 (DE-588)4221363-0 (DE-588)4042805-9 |
| title | Elementary stability and bifurcation theory |
| title_auth | Elementary stability and bifurcation theory |
| title_exact_search | Elementary stability and bifurcation theory |
| title_full | Elementary stability and bifurcation theory Gerard Iooss ; Daniel D. Joseph |
| title_fullStr | Elementary stability and bifurcation theory Gerard Iooss ; Daniel D. Joseph |
| title_full_unstemmed | Elementary stability and bifurcation theory Gerard Iooss ; Daniel D. Joseph |
| title_short | Elementary stability and bifurcation theory |
| title_sort | elementary stability and bifurcation theory |
| topic | Bifurcatie gtt Bifurcation, théorie de la ram Differentiaalvergelijkingen gtt Equations d'évolution - Solutions numériques ram Equations différentielles - Solutions numériques ram Evolutionaire vergelijkingen gtt Stabilité ram Bifurcation theory Differential equations Numerical solutions Evolution equations Numerical solutions Stability Stabilität (DE-588)4056693-6 gnd Evolutionsgleichung (DE-588)4129061-6 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Verzweigung Mathematik (DE-588)4078889-1 gnd Differentialgleichung (DE-588)4012249-9 gnd Nichtlineare Differentialgleichung (DE-588)4205536-2 gnd Asymptotisches Lösungsverhalten (DE-588)4134367-0 gnd Nichtlineare Evolutionsgleichung (DE-588)4221363-0 gnd Numerische Mathematik (DE-588)4042805-9 gnd |
| topic_facet | Bifurcatie Bifurcation, théorie de la Differentiaalvergelijkingen Equations d'évolution - Solutions numériques Equations différentielles - Solutions numériques Evolutionaire vergelijkingen Stabilité Bifurcation theory Differential equations Numerical solutions Evolution equations Numerical solutions Stability Stabilität Evolutionsgleichung Numerisches Verfahren Verzweigung Mathematik Differentialgleichung Nichtlineare Differentialgleichung Asymptotisches Lösungsverhalten Nichtlineare Evolutionsgleichung Numerische Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001681952&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT ioossgerard elementarystabilityandbifurcationtheory AT josephdanield elementarystabilityandbifurcationtheory |
Inhaltsverzeichnis
THWS Würzburg Magazin
| Signatur: |
1911 1999:1265 |
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