Differential dynamical systems:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Philadelphia, PA
SIAM
2007
|
| Schriftenreihe: | Mathematical modeling and computation
14 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Includes bibliographical references and index |
| Beschreibung: | XXII, 412 S. Ill., graph. Darst. |
| ISBN: | 9780898716351 |
Internformat
MARC
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| 020 | |a 9780898716351 |9 978-0-898716-35-1 | ||
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| 100 | 1 | |a Meiss, James D. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Differential dynamical systems |c James D. Meiss |
| 264 | 1 | |a Philadelphia, PA |b SIAM |c 2007 | |
| 300 | |a XXII, 412 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Mathematical modeling and computation |v 14 | |
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Mathematisches Modell | |
| 650 | 4 | |a Differentiable dynamical systems |x Mathematical models | |
| 650 | 0 | 7 | |a Chaotisches System |0 (DE-588)4316104-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Lineare gewöhnliche Differentialgleichung |0 (DE-588)4353441-7 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Verzweigung |g Mathematik |0 (DE-588)4078889-1 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Dynamisches System |0 (DE-588)4013396-5 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Hamiltonsches System |0 (DE-588)4139943-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Differenzierbares dynamisches System |0 (DE-588)4137931-7 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Differenzierbares dynamisches System |0 (DE-588)4137931-7 |D s |
| 689 | 0 | |5 DE-604 | |
| 689 | 1 | 0 | |a Lineare gewöhnliche Differentialgleichung |0 (DE-588)4353441-7 |D s |
| 689 | 1 | 1 | |a Dynamisches System |0 (DE-588)4013396-5 |D s |
| 689 | 1 | 2 | |a Chaotisches System |0 (DE-588)4316104-2 |D s |
| 689 | 1 | 3 | |a Hamiltonsches System |0 (DE-588)4139943-2 |D s |
| 689 | 1 | 4 | |a Verzweigung |g Mathematik |0 (DE-588)4078889-1 |D s |
| 689 | 1 | |8 1\p |5 DE-604 | |
| 830 | 0 | |a Mathematical modeling and computation |v 14 |w (DE-604)BV035421440 |9 14 | |
| 856 | 4 | 2 | |m Digitalisierung UB Bayreuth |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016146804&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-016146804 | ||
| 883 | 1 | |8 1\p |a cgwrk |d 20201028 |q DE-101 |u https://d-nb.info/provenance/plan#cgwrk | |
Datensatz im Suchindex
| _version_ | 1804137180084305920 |
|---|---|
| adam_text | Contents
List of Figures
xi
Preface
xvii
Acknowledgments
xxi
1
Introduction
1
1.1
Modeling
................................. 1
1.2
What Are Differential Equations?
.................... 2
1.3
One-Dimensional Dynamics
....................... 5
1
.4
Examples
................................. 8
Population Dynamics
.......................... 8
Mechanical Systems
........................... 10
Oscillating Circuits
...........................
II
Fluid Mixing
............................... 13
1.5
Two-Dimensional Dynamics
...................... 14
Nullclines
................................ 14
Phase Curves
............................... 17
1.6
The
Lorenz
Model
............................ 19
1.7
Quadratic ODEs: The Simplest Chaotic Systems
............ 21
1.8
Exercises
................................. 23
2
Linear Systems
29
2.1
Matrix ODEs
............................... 29
Eigenvalues and Eigenvectors
...................... 30
Diagonalization
............................. 33
2.2
Two-Dimensional Linear Systems
.................... 35
2.3
Exponentials of Operators
........................ 40
2.4
Fundamental Solution Theorem
..................... 45
2.5
Complex Eigenvalues
.......................... 48
2.6
Multiple Eigenvalues
.......................... 50
Semisimple-Nilpotent Decomposition
.................. 51
The Exponential
............................. 53
Alternative Methods
........................... 56
VII
v¡¡¡
Contents
2.7
Linear
Stability
.............................57
2.8
Nonautonomous Linear Systems
and Floquet Theory
..........61
2.9
Exercises.................................
67
3
Existence and Uniqueness
73
3.1
Set and Topological Preliminaries
.................... 73
Convergence
............................... 75
Uniform Convergence
.......................... 75
3.2
Function Space Preliminaries
...................... 76
Metric Spaces
.............................. 77
Contraction Maps
............................ 80
Lipschitz Functions
........................... 81
3.3
Existence and Uniqueness Theorem
................... 84
3.4
Dependence on Initial Conditions and Parameters
........... 92
3.5
Maximal Interval of Existence
...................... 98
3.6
Exercises
................................. 101
4
Dynamical Systems
105
4.1
Definitions
................................105
4.2
Flows
...................................107
4.3
Global Existence of Solutions
......................109
4.4
Linearization
...............................112
4.5
Stability
.................................116
4.6
Lyapunov Functions
...........................123
4.7
Topological Conjugacy and Equivalence
................130
4.8
Hartman-Grobman Theorem
......................138
4.9
Omega-Limit Sets
............................143
4.10
Attractors and Basins
..........................148
4.11
Stability of Periodic Orbits
.......................152
4.12
PoincareMaps
..............................154
4.13
Exercises
.................................159
5
Invariant Manifolds
165
5.1
Stable and Unstable Sets
.........................165
5.2
Heteroclinic Orbits
..........................167
5.3
Stable Manifolds
.............................170
5.4
Local Stable Manifold Theorem
.....................173
5.5
Global Stable Manifolds
.........................181
5.6
Center Manifolds
............................186
5.7
Exercises
.................... 192
6
The Phase Plane
197
6.1
Nonhyperbolic Equilibria in the Plane
................. 197
6.2
Two Zero Eigenvalues and Nonhyperbolic Nodes
........... 198
6.3
Imaginary Eigenvalues: Topological Centers
.............. 203
6.4
Symmetries and Reversors
....................... 211
Contents
¡χ
6.5 Index
Theory
...............................214
Higher
Dimensions: The Degree
....................217
6.6
Poincaré-Bendixson
Theorem
......................219
6.7
Liénard
Systems
.............................224
6.8
Behavior at Infinity: The Poincare Sphere
...............229
6.9
Exercises
.................................238
7
Chaotic Dynamics
243
7.1
Chaos
..................................243
7.2
Lyapunov Exponents
..........................248
Definition
................................250
Properties of Lyapunov Exponents
...................252
Computing Exponents
..........................255
7.3
Strange Attractors
............................259
Hausdorff Dimension
..........................260
Strange, Nonchaotic Attractors
.....................262
7.4
Exercises
.................................265
8
Bifurcation Theory
267
8.1
Bifurcations of Equilibria
........................267
8.2
Preservation of Equilibria
........................271
8.3
Unfolding Vector Fields
.........................273
Unfolding Two-Dimensional Linear Flows
...............275
8.4
Saddle-Node Bifurcation in One Dimension
..............278
8.5
Normal Forms
..............................281
Homological Operator
..........................282
Matrix Representation
..........................285
Higher-Order Normal Forms
......................287
8.6
Saddle-Node Bifurcation in
Ж
.....................290
Transversality
..............................291
Center Manifold Methods
........................293
8.7
Degenerate Saddle-Node Bifurcations
.................295
8.8
Andronov-Hopf Bifurcation
.......................296
8.9
The Cusp Bifurcation
..........................301
8.10
Takens-Bogdanov Bifurcation
.....................304
8.11
Homoclinic Bifurcations
.........................306
Fragility of Heteroclinic Orbits
.....................306
Generic Homoclinic Bifurcations in K2
.................309
8.12
Melnikov s Method
...........................311
8.13
Melnikov s Method for Nonautonomous Perturbations
.........314
8.14
Shilnikov Bifurcation
..........................322
8.15
Exercises
.................................325
9
Hamiltonian Dynamics
333
9.1
Conservative Dynamics
.........................333
9.2
Volume-Preserving Flows
........................335
x
Contents
9.3 Hamiltonian Systems..........................336
9.4
Poisson
Dynamics............................340
9.5 The
Action
Principle...........................
343
9.6
Poincaré
Invariant............................346
9.7 Lagrangian Systems ...........................348
Coordinate Independence of the Action
.................350
Symmetries and Invariants
.......................354
9.8
The Calculus of Variations
........................356
9.9
Equivalence of Hamiltonian and Lagrangian Mechanics
........358
9.10
Linearized Hamiltonian Systems
....................360
Eigenvalues of Hamiltonian Matrices
..................362
9.11
Krein Collisions
.............................365
9.12
Integrability
...............................368
9.13
Nearly
Integrable
Dynamics
.......................369
Invariant Tori
..............................370
KAM
Theory
..............................371
9.14
Onset of Chaos in Two Degrees of Freedom
..............373
9.15
Resonances: Single Wave Model
....................378
9.16
Resonances: Multiple Waves
......................381
9.17
Resonance Overlap and Chaos
.....................382
9.18
Exercises
.................................386
Appendix Mathematical Software
393
A.I Vector Fields
...............................393
A.2 Matrix Exponentials
...........................394
A.3 Lyapunov Exponents
..........................395
A.4 Bifurcation Diagrams
..........................396
A.5
Poincaré
Maps
..............................397
Bibliography
399
Index
407
|
| adam_txt |
Contents
List of Figures
xi
Preface
xvii
Acknowledgments
xxi
1
Introduction
1
1.1
Modeling
. 1
1.2
What Are Differential Equations?
. 2
1.3
One-Dimensional Dynamics
. 5
1
.4
Examples
. 8
Population Dynamics
. 8
Mechanical Systems
. 10
Oscillating Circuits
.
II
Fluid Mixing
. 13
1.5
Two-Dimensional Dynamics
. 14
Nullclines
. 14
Phase Curves
. 17
1.6
The
Lorenz
Model
. 19
1.7
Quadratic ODEs: The Simplest Chaotic Systems
. 21
1.8
Exercises
. 23
2
Linear Systems
29
2.1
Matrix ODEs
. 29
Eigenvalues and Eigenvectors
. 30
Diagonalization
. 33
2.2
Two-Dimensional Linear Systems
. 35
2.3
Exponentials of Operators
. 40
2.4
Fundamental Solution Theorem
. 45
2.5
Complex Eigenvalues
. 48
2.6
Multiple Eigenvalues
. 50
Semisimple-Nilpotent Decomposition
. 51
The Exponential
. 53
Alternative Methods
. 56
VII
v¡¡¡
Contents
2.7
Linear
Stability
.57
2.8
Nonautonomous Linear Systems
and Floquet Theory
.61
2.9
Exercises.
67
3
Existence and Uniqueness
73
3.1
Set and Topological Preliminaries
. 73
Convergence
. 75
Uniform Convergence
. 75
3.2
Function Space Preliminaries
. 76
Metric Spaces
. 77
Contraction Maps
. 80
Lipschitz Functions
. 81
3.3
Existence and Uniqueness Theorem
. 84
3.4
Dependence on Initial Conditions and Parameters
. 92
3.5
Maximal Interval of Existence
. 98
3.6
Exercises
. 101
4
Dynamical Systems
105
4.1
Definitions
.105
4.2
Flows
.107
4.3
Global Existence of Solutions
.109
4.4
Linearization
.112
4.5
Stability
.116
4.6
Lyapunov Functions
.123
4.7
Topological Conjugacy and Equivalence
.130
4.8
Hartman-Grobman Theorem
.138
4.9
Omega-Limit Sets
.143
4.10
Attractors and Basins
.148
4.11
Stability of Periodic Orbits
.152
4.12
PoincareMaps
.154
4.13
Exercises
.159
5
Invariant Manifolds
165
5.1
Stable and Unstable Sets
.165
5.2
Heteroclinic Orbits
.167
5.3
Stable Manifolds
.170
5.4
Local Stable Manifold Theorem
.173
5.5
Global Stable Manifolds
.181
5.6
Center Manifolds
.186
5.7
Exercises
. 192
6
The Phase Plane
197
6.1
Nonhyperbolic Equilibria in the Plane
. 197
6.2
Two Zero Eigenvalues and Nonhyperbolic Nodes
. 198
6.3
Imaginary Eigenvalues: Topological Centers
. 203
6.4
Symmetries and Reversors
. 211
Contents
¡χ
6.5 Index
Theory
.214
Higher
Dimensions: The Degree
.217
6.6
Poincaré-Bendixson
Theorem
.219
6.7
Liénard
Systems
.224
6.8
Behavior at Infinity: The Poincare Sphere
.229
6.9
Exercises
.238
7
Chaotic Dynamics
243
7.1
Chaos
.243
7.2
Lyapunov Exponents
.248
Definition
.250
Properties of Lyapunov Exponents
.252
Computing Exponents
.255
7.3
Strange Attractors
.259
Hausdorff Dimension
.260
Strange, Nonchaotic Attractors
.262
7.4
Exercises
.265
8
Bifurcation Theory
267
8.1
Bifurcations of Equilibria
.267
8.2
Preservation of Equilibria
.271
8.3
Unfolding Vector Fields
.273
Unfolding Two-Dimensional Linear Flows
.275
8.4
Saddle-Node Bifurcation in One Dimension
.278
8.5
Normal Forms
.281
Homological Operator
.282
Matrix Representation
.285
Higher-Order Normal Forms
.287
8.6
Saddle-Node Bifurcation in
Ж"
.290
Transversality
.291
Center Manifold Methods
.293
8.7
Degenerate Saddle-Node Bifurcations
.295
8.8
Andronov-Hopf Bifurcation
.296
8.9
The Cusp Bifurcation
.301
8.10
Takens-Bogdanov Bifurcation
.304
8.11
Homoclinic Bifurcations
.306
Fragility of Heteroclinic Orbits
.306
Generic Homoclinic Bifurcations in K2
.309
8.12
Melnikov's Method
.311
8.13
Melnikov's Method for Nonautonomous Perturbations
.314
8.14
Shilnikov Bifurcation
.322
8.15
Exercises
.325
9
Hamiltonian Dynamics
333
9.1
Conservative Dynamics
.333
9.2
Volume-Preserving Flows
.335
x
Contents
9.3 Hamiltonian Systems.336
9.4
Poisson
Dynamics.340
9.5 The
Action
Principle.
343
9.6
Poincaré
Invariant.346
9.7 Lagrangian Systems .348
Coordinate Independence of the Action
.350
Symmetries and Invariants
.354
9.8
The Calculus of Variations
.356
9.9
Equivalence of Hamiltonian and Lagrangian Mechanics
.358
9.10
Linearized Hamiltonian Systems
.360
Eigenvalues of Hamiltonian Matrices
.362
9.11
Krein Collisions
.365
9.12
Integrability
.368
9.13
Nearly
Integrable
Dynamics
.369
Invariant Tori
.370
KAM
Theory
.371
9.14
Onset of Chaos in Two Degrees of Freedom
.373
9.15
Resonances: Single Wave Model
.378
9.16
Resonances: Multiple Waves
.381
9.17
Resonance Overlap and Chaos
.382
9.18
Exercises
.386
Appendix Mathematical Software
393
A.I Vector Fields
.393
A.2 Matrix Exponentials
.394
A.3 Lyapunov Exponents
.395
A.4 Bifurcation Diagrams
.396
A.5
Poincaré
Maps
.397
Bibliography
399
Index
407 |
| any_adam_object | 1 |
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| author | Meiss, James D. |
| author_facet | Meiss, James D. |
| author_role | aut |
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| dewey-ones | 515 - Analysis |
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| dewey-search | 515/.39 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
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| id | DE-604.BV022942175 |
| illustrated | Illustrated |
| index_date | 2024-07-02T18:58:27Z |
| indexdate | 2024-07-09T21:08:11Z |
| institution | BVB |
| isbn | 9780898716351 |
| language | English |
| lccn | 2007061747 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016146804 |
| oclc_num | 263970040 |
| open_access_boolean | |
| owner | DE-20 DE-29T DE-703 DE-11 DE-83 DE-384 |
| owner_facet | DE-20 DE-29T DE-703 DE-11 DE-83 DE-384 |
| physical | XXII, 412 S. Ill., graph. Darst. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | SIAM |
| record_format | marc |
| series | Mathematical modeling and computation |
| series2 | Mathematical modeling and computation |
| spelling | Meiss, James D. Verfasser aut Differential dynamical systems James D. Meiss Philadelphia, PA SIAM 2007 XXII, 412 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Mathematical modeling and computation 14 Includes bibliographical references and index Mathematisches Modell Differentiable dynamical systems Mathematical models Chaotisches System (DE-588)4316104-2 gnd rswk-swf Lineare gewöhnliche Differentialgleichung (DE-588)4353441-7 gnd rswk-swf Verzweigung Mathematik (DE-588)4078889-1 gnd rswk-swf Dynamisches System (DE-588)4013396-5 gnd rswk-swf Hamiltonsches System (DE-588)4139943-2 gnd rswk-swf Differenzierbares dynamisches System (DE-588)4137931-7 gnd rswk-swf Differenzierbares dynamisches System (DE-588)4137931-7 s DE-604 Lineare gewöhnliche Differentialgleichung (DE-588)4353441-7 s Dynamisches System (DE-588)4013396-5 s Chaotisches System (DE-588)4316104-2 s Hamiltonsches System (DE-588)4139943-2 s Verzweigung Mathematik (DE-588)4078889-1 s 1\p DE-604 Mathematical modeling and computation 14 (DE-604)BV035421440 14 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016146804&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Meiss, James D. Differential dynamical systems Mathematical modeling and computation Mathematisches Modell Differentiable dynamical systems Mathematical models Chaotisches System (DE-588)4316104-2 gnd Lineare gewöhnliche Differentialgleichung (DE-588)4353441-7 gnd Verzweigung Mathematik (DE-588)4078889-1 gnd Dynamisches System (DE-588)4013396-5 gnd Hamiltonsches System (DE-588)4139943-2 gnd Differenzierbares dynamisches System (DE-588)4137931-7 gnd |
| subject_GND | (DE-588)4316104-2 (DE-588)4353441-7 (DE-588)4078889-1 (DE-588)4013396-5 (DE-588)4139943-2 (DE-588)4137931-7 |
| title | Differential dynamical systems |
| title_auth | Differential dynamical systems |
| title_exact_search | Differential dynamical systems |
| title_exact_search_txtP | Differential dynamical systems |
| title_full | Differential dynamical systems James D. Meiss |
| title_fullStr | Differential dynamical systems James D. Meiss |
| title_full_unstemmed | Differential dynamical systems James D. Meiss |
| title_short | Differential dynamical systems |
| title_sort | differential dynamical systems |
| topic | Mathematisches Modell Differentiable dynamical systems Mathematical models Chaotisches System (DE-588)4316104-2 gnd Lineare gewöhnliche Differentialgleichung (DE-588)4353441-7 gnd Verzweigung Mathematik (DE-588)4078889-1 gnd Dynamisches System (DE-588)4013396-5 gnd Hamiltonsches System (DE-588)4139943-2 gnd Differenzierbares dynamisches System (DE-588)4137931-7 gnd |
| topic_facet | Mathematisches Modell Differentiable dynamical systems Mathematical models Chaotisches System Lineare gewöhnliche Differentialgleichung Verzweigung Mathematik Dynamisches System Hamiltonsches System Differenzierbares dynamisches System |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016146804&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV035421440 |
| work_keys_str_mv | AT meissjamesd differentialdynamicalsystems |