Table of contents for Ordinary differential equations with applications / Carmen Chicone.


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1 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 . . . . . . . . . . . . ....  60
1.8.5  Polar Coordinates ... . . . . .  .  .  .  .  ......  65
1.9  Periodic Solutions  . ......................           82
1.9.1  The Poincar6 Map . . . . . . . . . .  . . . .....  82
1.9.2  Limit Sets and Poincar6-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
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-Ulam-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  Gal6rkin 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
5 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
6 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
7 Averaging                                               511
7.1 The Averaging Principle . . . . . . . .  .  .  .  .....  . 511
7.2 Averaging at Resonance . ................... ..522
7.3 Action-Angle Variables ... . . . . . .  .  .   . . . ..539
8 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 Lyapunov-Schmidt Reduction . 559
8.3 Poincar6-Andronov-Hopf Bifurcation  . . . . . . . . . ... 565
8.3.1  Multiple Hopf Bifurcation . . . . . . . . . . . ....  576
8.4 Dynamic Bifurcation .... . . . . . . .  .   .  .  .  .   . 595



Library of Congress subject headings for this publication: Differential equations