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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