Numerical methods for bifurcations of dynamical equilibria:
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Philadelphia, PA
SIAM
2000
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXII, 362 S. graph. Darst. |
| ISBN: | 0898714427 |
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| 245 | 1 | 0 | |a Numerical methods for bifurcations of dynamical equilibria |c Willy J. F. Govaerts |
| 264 | 1 | |a Philadelphia, PA |b SIAM |c 2000 | |
| 300 | |a XXII, 362 S. |b graph. Darst. | ||
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| adam_text | Contents
Preface xiii
Notation xv
Introduction xvii
1 Examples and Motivation 1
1.1 Nonlinear Equations and Dynamical Systems 1
1.2 Examples from Population Dynamics 3
1.2.1 Stable and Unstable Equilibria 3
1.2.2 A Set of Bifurcation Points 4
1.2.3 A Cusp Catastrophe 7
1.2.4 A Hopf Bifurcation 10
1.3 An Example from Combustion Theory 15
1.3.1 Finite Element Discretization 15
1.3.2 Finite Difference Discretization 19
1.3.3 Numerical Continuation: Motivation by an Example 20
1.4 An Example of Symmetry Breaking 21
1.5 Linear and Nonlinear Stability 24
1.6 Exercises 27
2 Manifolds and Numerical Continuation 29
2.1 Manifolds 29
2.1.1 Definitions 29
2.1.2 The Tangent Space 30
2.1.3 Examples 31
2.2 Branches and Limit Points 32
2.3 Numerical Continuation 34
2.3.1 Natural Parameterization 34
2.3.2 Pseudoarclength Continuation 36
2.3.3 Steplength Control 40
2.3.4 Convergence of Newton Iterates 42
2.3.5 Some Practical Considerations 44
2.4 Notes and Further Reading 44
vii
viii Contents
2.5 Exercises 44
3 Bordered Matrices 49
3.1 Introduction: Motivation by Cramer s Rule 49
3.2 The Construction of Nonsingular Bordered Matrices 50
3.3 The Singular Value Inequality 52
3.4 The Schur Inverse as Denning System for Rank Deficiency 57
3.5 Invariant Subspaces of Parameter Dependent Matrices 59
3.6 Numerical Methods for Bordered Linear Systems 61
3.6.1 Backward Stability 61
3.6.2 Algorithm BEM for One Bordered Systems 63
3.6.3 Algorithm BEMW for Wider Bordered Systems 65
3.7 Notes and Further Reading 67
3.8 Exercises 67
4 Generic Equilibrium Bifurcations in One Parameter Problems 71
4.1 Limit Points 71
4.1.1 The Moore Spence System for Quadratic Turning Points 72
4.1.2 Quadratic Turning Points by Direct Bordering Methods 73
4.1.3 Detection of Quadratic Turning Points 74
4.1.4 Continuation of Limit Points 75
4.2 Example: A One Dimensional Continuous Brusselator 75
4.2.1 The Model and Its Discretization 75
4.2.2 Turning Points in the Brusselator Model 78
4.3 Classical Methods for the Computation of Hopf Points 79
4.3.1 Hopf Points 79
4.3.2 Regular Systems with 3N + 2 Equations 81
4.3.3 Regular Systems with 2N + 2 Equations 83
4.3.4 Regular Systems with N + 2 Equations 84
4.3.5 Zero Sum Pairs of Real Eigenvalues 85
4.3.6 Hopf Points by Complex Arithmetic 87
4.4 Tensor Products and Bialternate Products 88
4.4.1 Tensor Products 88
4.4.2 Condensed Tensor Products 89
4.4.3 The Natural Involution in C x C 92
4.4.4 The Bialternate Product of Matrices 92
4.4.5 The Jordan Structure of the Bialternate Product Matrix 95
4.5 Hopf Points with Bialternate Product Methods 101
4.5.1 Reconstruction of the Eigenstructure 103
4.5.2 Double Borders and Detection of Double Hopf Points 104
4.6 Computation of Hopf Points: Examples 105
4.6.1 Zero Sum Pairs of Eigenvalues in the Catalytic Oscillator Model . 105
4.6.2 The Clamped Hodgkin Huxley Equations 1°6
4.6.3 Discretization and Generalized Eigenvalue Problems 1°7
4.7 Notes and Further Reading 11°
Contents be
4.8 Exercises Ill
5 Bifurcations Determined by the Jordan Form of the Jacobian 117
5.1 Bogdanov Takens Points and Their Generalizations 117
5.1.1 Introduction 117
5.1.2 Numerical Computation of BT Points 118
5.1.3 Local Analysis of BT Matrices 121
5.1.4 Transversality and Genericity 125
5.1.5 Test Functions for BT Points 127
5.1.6 Example: A Curve of BT Points in the Catalytic Oscillator Model 127
5.2 ZH Points and Their Generalizations 127
5.2.1 Transversality and Genericity for Simple Hopf 128
5.2.2 Transversality and Genericity for ZH 131
5.2.3 Detection of ZH Points 131
5.3 DH Points and Resonant DH Points 132
5.3.1 Introduction 132
5.3.2 Defining Functions for Multiple Hopf Points 132
5.3.3 Branch Switching at a DH Point 136
5.3.4 Resonant DH Points 137
5.3.5 The Stratified Set of Hopf Points Near a Point with One to One
Resonance 142
5.4 Example: The Lateral Pyloric Neuron 146
5.5 Notes and Further Reading 150
5.6 Exercises 150
6 Singularity Theory 155
6.1 Contact Equivalence of Nonlinear Mappings 155
6.2 The Numerical Lyapunov Schmidt Reduction 156
6.3 Classification of Singularities by Codimension 163
6.3.1 Introduction and Basic Properties 163
6.3.2 Singularities from R into R 165
6.3.3 Singularities from R2 into R 165
6.3.4 Singularities from R2 into R2 172
6.3.5 A Table of ^ Singularities 173
6.3.6 Example: Intersection of a Surface with Its Tangent Plane 174
6.3.7 Example: A Point on a Rolling Wheel 175
6.4 Unfolding Theory 176
6.5 Example: The Continuous Flow Stirred Tank Reactor 185
6.5.1 Description of the Model 186
6.5.2 Numerical Computation of a Cusp Point 187
6.5.3 The Universal Unfolding of a Cusp Point 189
6.5.4 Example: Unfolding a Cusp in the CSTR 192
6.5.5 Pairs of Nondegeneracy Conditions: An Example 195
6.6 Numerical Methods for /C Singularities 195
6.6.1 The Codimension 1 Singularity from R into R 196
x Contents
6.6.2 Singularities from R into R with Codimension Higher than 1 ... 201
6.6.3 Singularities from R2 into R 204
6.6.4 Singularities from R2 into R2 206
6.7 Notes and Further Reading 209
6.8 Exercises 209
7 Singularity Theory with a Distinguished Bifurcation Parameter 213
7.1 Singularities with a Distinguished Bifurcation Parameter 214
7.2 Classification of (A /(^ Singularities from R into R 214
7.3 Classification of (A /^ Singularities from R2 into R2 216
7.4 Numerical Methods for (A — /C) Singularities 219
7.4.1 Numerical Methods for (A /(^ Singularities with Corank 1 .... 220
7.4.2 Numerical Methods for (A — /^ Singularities with Corank 2 .... 222
7.5 Interpretation of Simple Singularities with Corank 1 222
7.6 Examples in Low Dimensional Spaces 225
7.6.1 Winged Cusps in the CSTR 225
7.6.2 An Eutrophication Model 226
7.7 Example: The One Dimensional Brusselator 229
7.7.1 Computational Study of a Curve of Equilibria 229
7.7.2 Computational Study of a Curve of Turning Points 231
7.7.3 Computational Study of a Curve of Hysteresis Points 234
7.7.4 Computational Study of a Curve of Transcritical Bifurcation
Points 236
7.7.5 A Winged Cusp on a Curve of Pitchfork Bifurcations 237
7.7.6 A Degenerate Pitchfork on a Curve of Pitchfork
Bifurcations 239
7.7.7 Computation of Branches of Cusp Points and Quartic Turning
Points 240
7.8 Numerical Branching 242
7.8.1 Simple Bifurcation Point and Isola Center 243
7.8.2 Cusp Points in /C Singularity Theory 243
7.8.3 Transcritical and Pitchfork Bifurcations in (A — /C) Singularity
Theory 247
7.8.4 Branching Point on a Curve of Equilibria 248
7.9 Exercises 249
8 Symmetry Breaking Bifurcations 253
8.1 The Z2 Case: Corank 1 and Symmetry Breaking 254
8.1.1 Basic Results on Z2 Equivariance 254
8.1.2 Symmetry Breaking on a Branch of Equilibria:
Generic Scenario 256
8.1.3 The Lyapunov Schmidt Reduction with Symmetry Adapted
Bordering 257
8.1.4 The Classification of Z2 Equivariant Germs 258
8.1.5 Numerical Detection, Computation, and Continuation 260
Contents xi
8.1.6 Branching and Numerical Study of a Nonsymmetric Branch .... 262
8.2 The Z2 Case: Corank 2 and Mode Interaction 263
8.2.1 Numerical Example: A Corank 2 Point on a Curve of Turning
Points 264
8.2.2 Continuation of Turning Points by Double Bordering 265
8.2.3 The Z2 Equivariant Reduction by a Symmetry Adapted Double
Bordering 266
8.2.4 Computation of a Corank 2 Point 268
8.2.5 Analysis and Computation of the Singularity Properties of a
Corank 2 Point 269
8.2.6 The Z2 Equivariant Classification of Corank 2 Points with
Distinguished Bifurcation Parameter 272
8.3 Rank Drop on a Curve of Singular Points 275
8.3.1 Corank 1 Singularities in Two State Variables 275
8.3.2 The Case of a Symmetry Adapted Bordering 277
8.3.3 Numerical Example: A Corank 2 Point on a Curve of Cusps .... 278
8.4 Other Symmetry Groups 280
8.4.1 Symmetry Adapted Bases 280
8.4.2 The Equivariant Branching Lemma 283
8.4.3 Example: A System with D4 Symmetry 286
8.4.4 Numerical Implementation 290
8.5 Notes and Further Reading 292
8.6 Exercises 292
9 Bifurcations with Degeneracies in the Nonlinear Terms 295
9.1 Principles of Center Manifold Theory 296
9.1.1 The Homological Equation for Dynamics in the Center Manifold . 297
9.1.2 Normal Form Results 298
9.1.3 General Remarks on the Computation 301
9.2 Computation of CPs 301
9.2.1 The Manifold 302
9.2.2 A Minimally Extended Denning System 303
9.2.3 A Large Denning System 304
9.3 Computation of GH Points 306
9.3.1 The Manifold 306
9.3.2 A Minimally Extended Defining System 307
9.3.3 A Large Defining System 308
9.4 Examples 311
9.4.1 A Turning Point of Periodic Orbits in the Hodgkin Huxley Model 311
9.4.2 Bifurcations with High Codimension in the LP Neuron Model . . . 314
9.4.3 Dynamics of Corruption in Democratic Societies 315
9.5 Notes and Further Reading 320
9.6 Exercises 320
xii Contents
10 An Introduction to Large Dynamical Systems 323
10.1 A Class of One Dimensional PDEs 324
10.1.1 Space Discretization 325
10.1.2 Integration by Crank Nicolson 328
10.1.3 B stability and the Implicit Midpoint Rule 332
10.1.4 Numerical Continuation 332
10.1.5 Solution of Linear Systems 334
10.1.6 Example: The Nonadiabatic Tubular Reactor 335
10.2 Bifurcations: Reduction to a Low Dimensional State Space 336
10.3 Notes and Further Reading 339
10.4 Exercises 340
Bibliography 343
Index 359
|
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| id | DE-604.BV013365694 |
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| indexdate | 2024-07-09T18:44:34Z |
| institution | BVB |
| isbn | 0898714427 |
| language | English |
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| spelling | Govaerts, Willy J. Verfasser aut Numerical methods for bifurcations of dynamical equilibria Willy J. F. Govaerts Philadelphia, PA SIAM 2000 XXII, 362 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Differentialgleichung - Numerisches Verfahren Differenzierbares dynamisches System Verzweigung <Mathematik> Bifurcation theory Differentiable dynamical systems Differential equations Numerical solutions Verzweigung Mathematik (DE-588)4078889-1 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Dynamisches System (DE-588)4013396-5 gnd rswk-swf Verzweigung Mathematik (DE-588)4078889-1 s Dynamisches System (DE-588)4013396-5 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009116874&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Govaerts, Willy J. Numerical methods for bifurcations of dynamical equilibria Differentialgleichung - Numerisches Verfahren Differenzierbares dynamisches System Verzweigung <Mathematik> Bifurcation theory Differentiable dynamical systems Differential equations Numerical solutions Verzweigung Mathematik (DE-588)4078889-1 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Dynamisches System (DE-588)4013396-5 gnd |
| subject_GND | (DE-588)4078889-1 (DE-588)4128130-5 (DE-588)4013396-5 |
| title | Numerical methods for bifurcations of dynamical equilibria |
| title_auth | Numerical methods for bifurcations of dynamical equilibria |
| title_exact_search | Numerical methods for bifurcations of dynamical equilibria |
| title_full | Numerical methods for bifurcations of dynamical equilibria Willy J. F. Govaerts |
| title_fullStr | Numerical methods for bifurcations of dynamical equilibria Willy J. F. Govaerts |
| title_full_unstemmed | Numerical methods for bifurcations of dynamical equilibria Willy J. F. Govaerts |
| title_short | Numerical methods for bifurcations of dynamical equilibria |
| title_sort | numerical methods for bifurcations of dynamical equilibria |
| topic | Differentialgleichung - Numerisches Verfahren Differenzierbares dynamisches System Verzweigung <Mathematik> Bifurcation theory Differentiable dynamical systems Differential equations Numerical solutions Verzweigung Mathematik (DE-588)4078889-1 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Dynamisches System (DE-588)4013396-5 gnd |
| topic_facet | Differentialgleichung - Numerisches Verfahren Differenzierbares dynamisches System Verzweigung <Mathematik> Bifurcation theory Differentiable dynamical systems Differential equations Numerical solutions Verzweigung Mathematik Numerisches Verfahren Dynamisches System |
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