Ordinary differential equations with applications:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY
Springer
2006
|
| Edition: | 2. ed. |
| Series: | Texts in Applied Mathematics
34 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XIX, 636 S. graph. Darst. |
| ISBN: | 0387307699 9780387307695 |
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Record in the Search Index
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| adam_text | Contents
Introduction
to Ordinary Differential Equations
1
1.1
Existence and Uniqueness
................... 1
1.2
Types of Differential Equations
................ 6
1.3
Geometric Interpretation of Autonomous Systems
...... 8
1.4
Flows
.............................. 14
1.5
Reparametrization of Time
.................. 16
1.6
Stability and Linearization
................... 20
1.7
Stability and the Direct Method of Lyapunov
........ 28
1.8
Manifolds
............................ 33
1.8.1
Introduction to Invariant Manifolds
......... 34
1.8.2
Smooth Manifolds
................... 43
1.8.3
Tangent Spaces
..................... 52
1.8.4
Change of Coordinates
.................
GO
1.8.5
Polar Coordinates
................... 65
1.9
Periodic Solutions
....................... 82
1.9.1
The
Poincaré Map
................... 82
1.9.2
Limit Sets and
Poincaré
Bendixson Theory
..... 91
1.10
Review of Calculus
....................... 108
1.10.1
The Mean Value Theorem
............... 113
1.10.2
Integration in Banach Spaces
............. 115
1.11
Contraction
........................... 121
1.11.1
The Contraction Mapping Theorem
......... 121
xviii
Contents
1.11.2
Uniform Contraction
.................. 123
1.11.3
Fiber Contraction
................... 127
1.11.4
The Implicit Function Theorem
............ 134
1.12
Existence, Uniqueness, and Extension
............ 135
2
Linear Systems and Stability of Nonlinear Systems
145
2.1
Homogeneous Linear Differential Equations
......... 146
2.1.1
Gronwall s Inequality
................. 146
2.1.2
Homogeneous Linear Systems: General Theory
. . . 148
2.1.3
Principle of Superposition
............... 149
2.1.4
Linear Equations with Constant Coefficients
..... 154
2.2
Stability of Linear Systems
.................. 174
2.3
Stability of Nonlinear Systems
................ 179
2.4
Floquet Theory
......................... 187
2.4.1
Lyapunov Exponents
.................. 202
2.4.2
Hill s Equation
..................... 206
2.4.3
Periodic Orbits of Linear Systems
.......... 210
2.4.4
Stability of Periodic Orbits
.............. 212
3
Applications
225
3.1
Origins of ODE: The Euler-Lagrange Equation
....... 225
3.2
Origins of ODE: Classical Physics
............... 236
3.2.1
Motion of a Charged Particle
............. 239
3.2.2
Motion of a Binary System
.............. 240
3.2.3
Perturbed Kepler Motion and Delaunay Elements
. 249
3.2.4
Satellite Orbiting an Oblate Planet
.......... 257
3.2.5
The Diamagnetic Kepler Problem
.......... 263
3.3
Coupled
Pendula:
Normal Modes and Beats
......... 269
3.4
The Fermi-
Ulani-Pasta
Oscillator
.............. 273
3.5
The Inverted Pendulum
.................... 278
3.6
Origins of ODE: Partial Differential Equations
....... 284
3.6.1
Infinite Dimensional ODE
............... 286
3.6.2
Galër
kin Approximation
................ 299
3.6.3
Traveling Waves
.................... 312
3.6.4
First Order PDE
.................... 316
4
Hyperbolic Theory
323
4.1
Invariant Manifolds
....................... 323
4.2
Applications of Invariant Manifolds
.............. 345
4.3
The
Hartman Grobman
Theorem
............... 347
4.3.1
Diffeomorphisms
.................... 348
4.3.2
Differential Equations
................. 354
4.3.3
Linearization via the Lie Derivative
......... 358
Contents xix
Continuation
of Periodic Solutions
367
5.1
A Classic Example: van
der
Pol s Oscillator
......... 368
5.1.1
Continuation
Theory and Applied Mathematics
. . . 374
5.2
Autonomous Perturbations
.................. 376
5.3
Nonautonomous Perturbations
................ 390
5.3.1
Rest Points
....................... 393
5.3.2
Isochronous Period Annulus
.............. 394
5.3.3
The Forced van
der
Pol Oscillator
.......... 398
5.3.4
Regular Period Annulus
................ 406
5.3.5
Limit Cycles-Entrainment-Resonance Zones
.... 417
5.3.6
Lindstedt
Series and the Perihelion of Mercury
. . . 425
5.3.7
Entrainment Domains for van
der
Pol s Oscillator
. 434
5.3.8
Periodic Orbits of Multidimensional Systems with
First Integrals
..................... 436
5.4
Forced Oscillators
........................ 442
Homoclinic Orbits, Melnikov s Method, and Chaos
449
6.1
Autonomous Perturbations: Separatrix Splitting
...... 454
6.2
Periodic Perturbations: Transverse Homoclinic Points
. . . 465
6.3
Origins of ODE: Fluid Dynamics
............... 479
6.3.1
The Equations of Fluid Motion
............ 480
6.3.2
ABC Flows
....................... 490
6.3.3
Chaotic ABC Flows
.................. 493
Averaging
511
7.1
The Averaging Principle
.................... 511
7.2
Averaging at Resonance
.................... 522
7.3
Action-Angle Variables
..................... 539
Local Bifurcation
545
8.1
One-Dimensional State Space
................. 546
8.1.1
The Saddle-Node Bifurcation
............. 546
8.1.2
A Normal Form
..................... 548
8.1.3
Bifurcation in Applied Mathematics
......... 549
8.1.4
Families. Transversality. and Jets
........... 551
8.2
Saddle-Node Bifurcation by Lyapimov Schmidt Reduction
. 559
8.3
Poiiicarc-Aiidronov-Hopf Bifurcation
............ 565
8.3.1
.Multiple
Hopf
Bifurcation
............... 576
8.4
Dynamic Bifurcation
...................... 595
References
603
|
| adam_txt |
Contents
Introduction
to Ordinary Differential Equations
1
1.1
Existence and Uniqueness
. 1
1.2
Types of Differential Equations
. 6
1.3
Geometric Interpretation of Autonomous Systems
. 8
1.4
Flows
. 14
1.5
Reparametrization of Time
. 16
1.6
Stability and Linearization
. 20
1.7
Stability and the Direct Method of Lyapunov
. 28
1.8
Manifolds
. 33
1.8.1
Introduction to Invariant Manifolds
. 34
1.8.2
Smooth Manifolds
. 43
1.8.3
Tangent Spaces
. 52
1.8.4
Change of Coordinates
.
GO
1.8.5
Polar Coordinates
. 65
1.9
Periodic Solutions
. 82
1.9.1
The
Poincaré Map
. 82
1.9.2
Limit Sets and
Poincaré
Bendixson Theory
. 91
1.10
Review of Calculus
. 108
1.10.1
The Mean Value Theorem
. 113
1.10.2
Integration in Banach Spaces
. 115
1.11
Contraction
. 121
1.11.1
The Contraction Mapping Theorem
. 121
xviii
Contents
1.11.2
Uniform Contraction
. 123
1.11.3
Fiber Contraction
. 127
1.11.4
The Implicit Function Theorem
. 134
1.12
Existence, Uniqueness, and Extension
. 135
2
Linear Systems and Stability of Nonlinear Systems
145
2.1
Homogeneous Linear Differential Equations
. 146
2.1.1
Gronwall's Inequality
. 146
2.1.2
Homogeneous Linear Systems: General Theory
. . . 148
2.1.3
Principle of Superposition
. 149
2.1.4
Linear Equations with Constant Coefficients
. 154
2.2
Stability of Linear Systems
. 174
2.3
Stability of Nonlinear Systems
. 179
2.4
Floquet Theory
. 187
2.4.1
Lyapunov Exponents
. 202
2.4.2
Hill's Equation
. 206
2.4.3
Periodic Orbits of Linear Systems
. 210
2.4.4
Stability of Periodic Orbits
. 212
3
Applications
225
3.1
Origins of ODE: The Euler-Lagrange Equation
. 225
3.2
Origins of ODE: Classical Physics
. 236
3.2.1
Motion of a Charged Particle
. 239
3.2.2
Motion of a Binary System
. 240
3.2.3
Perturbed Kepler Motion and Delaunay Elements
. 249
3.2.4
Satellite Orbiting an Oblate Planet
. 257
3.2.5
The Diamagnetic Kepler Problem
. 263
3.3
Coupled
Pendula:
Normal Modes and Beats
. 269
3.4
The Fermi-
Ulani-Pasta
Oscillator
. 273
3.5
The Inverted Pendulum
. 278
3.6
Origins of ODE: Partial Differential Equations
. 284
3.6.1
Infinite Dimensional ODE
. 286
3.6.2
Galër
kin Approximation
. 299
3.6.3
Traveling Waves
. 312
3.6.4
First Order PDE
. 316
4
Hyperbolic Theory
323
4.1
Invariant Manifolds
. 323
4.2
Applications of Invariant Manifolds
. 345
4.3
The
Hartman Grobman
Theorem
. 347
4.3.1
Diffeomorphisms
. 348
4.3.2
Differential Equations
. 354
4.3.3
Linearization via the Lie Derivative
. 358
Contents xix
Continuation
of Periodic Solutions
367
5.1
A Classic Example: van
der
Pol's Oscillator
. 368
5.1.1
Continuation
Theory and Applied Mathematics
. . . 374
5.2
Autonomous Perturbations
. 376
5.3
Nonautonomous Perturbations
. 390
5.3.1
Rest Points
. 393
5.3.2
Isochronous Period Annulus
. 394
5.3.3
The Forced van
der
Pol Oscillator
. 398
5.3.4
Regular Period Annulus
. 406
5.3.5
Limit Cycles-Entrainment-Resonance Zones
. 417
5.3.6
Lindstedt
Series and the Perihelion of Mercury
. . . 425
5.3.7
Entrainment Domains for van
der
Pol's Oscillator
. 434
5.3.8
Periodic Orbits of Multidimensional Systems with
First Integrals
. 436
5.4
Forced Oscillators
. 442
Homoclinic Orbits, Melnikov's Method, and Chaos
449
6.1
Autonomous Perturbations: Separatrix Splitting
. 454
6.2
Periodic Perturbations: Transverse Homoclinic Points
. . . 465
6.3
Origins of ODE: Fluid Dynamics
. 479
6.3.1
The Equations of Fluid Motion
. 480
6.3.2
ABC Flows
. 490
6.3.3
Chaotic ABC Flows
. 493
Averaging
511
7.1
The Averaging Principle
. 511
7.2
Averaging at Resonance
. 522
7.3
Action-Angle Variables
. 539
Local Bifurcation
545
8.1
One-Dimensional State Space
. 546
8.1.1
The Saddle-Node Bifurcation
. 546
8.1.2
A Normal Form
. 548
8.1.3
Bifurcation in Applied Mathematics
. 549
8.1.4
Families. Transversality. and Jets
. 551
8.2
Saddle-Node Bifurcation by Lyapimov Schmidt Reduction
. 559
8.3
Poiiicarc-Aiidronov-Hopf Bifurcation
. 565
8.3.1
.Multiple
Hopf
Bifurcation
. 576
8.4
Dynamic Bifurcation
. 595
References
603 |
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| discipline | Mathematik |
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| illustrated | Illustrated |
| index_date | 2024-07-02T14:22:39Z |
| indexdate | 2024-07-09T20:37:43Z |
| institution | BVB |
| isbn | 0387307699 9780387307695 |
| language | English |
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| physical | XIX, 636 S. graph. Darst. |
| publishDate | 2006 |
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| publisher | Springer |
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| series | Texts in Applied Mathematics |
| series2 | Texts in Applied Mathematics |
| spelling | Chicone, Carmen 1946- Verfasser (DE-588)1056093501 aut Ordinary differential equations with applications Carmen Chicone 2. ed. New York, NY Springer 2006 XIX, 636 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Texts in Applied Mathematics 34 Differential equations Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd rswk-swf Gewöhnliche Differentialgleichung (DE-588)4020929-5 s DE-604 Erscheint auch als Online-Ausgabe 0-387-22623-0 Texts in Applied Mathematics 34 (DE-604)BV002476038 34 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=2707425&prov=M&dok_var=1&dok_ext=htm Inhaltstext Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014737993&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Chicone, Carmen 1946- Ordinary differential equations with applications Texts in Applied Mathematics Differential equations Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| subject_GND | (DE-588)4020929-5 |
| title | Ordinary differential equations with applications |
| title_auth | Ordinary differential equations with applications |
| title_exact_search | Ordinary differential equations with applications |
| title_exact_search_txtP | Ordinary differential equations with applications |
| title_full | Ordinary differential equations with applications Carmen Chicone |
| title_fullStr | Ordinary differential equations with applications Carmen Chicone |
| title_full_unstemmed | Ordinary differential equations with applications Carmen Chicone |
| title_short | Ordinary differential equations with applications |
| title_sort | ordinary differential equations with applications |
| topic | Differential equations Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| topic_facet | Differential equations Gewöhnliche Differentialgleichung |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=2707425&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014737993&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV002476038 |
| work_keys_str_mv | AT chiconecarmen ordinarydifferentialequationswithapplications |