Differential Equations: A Dynamical Systems Approach to Theory and Practice (Graduate Studies in Mathematics, 212)

This graduate-level introduction to ordinary differential equations combines both qualitative and numerical analysis of solutions, in line with Poincaré's vision for the field over a century ago. Taking into account the remarkable development of dynamical systems since then, the authors present the core topics that every young mathematician of our time—pure and applied alike—ought to learn. The book features a dynamical perspective that drives the motivating questions, the style of exposition, and the arguments and proof techniques. The text is organized in six cycles. The first cycle deals with the foundational questions of existence and uniqueness of solutions. The second introduces the basic tools, both theoretical and practical, for treating concrete problems. The third cycle presents autonomous and non-autonomous linear theory. Lyapunov stability theory forms the fourth cycle. The fifth one deals with the local theory, including the Grobman–Hartman theorem and the stable manifold theorem. The last cycle discusses global issues in the broader setting of differential equations on manifolds, culminating in the Poincaré–Hopf index theorem. The book is appropriate for use in a course or for self-study. The reader is assumed to have a basic knowledge of general topology, linear algebra, and analysis at the undergraduate level. Each chapter ends with a computational experiment, a diverse list of exercises, and detailed historical, biographical, and bibliographic notes seeking to help the reader form a clearer view of how the ideas in this field unfolded over time. Cover 1 Title page 4 Preface 10 Chapter 1. Introduction 16 1.1. Differential equations and their solutions 17 1.2. Qualitative theory of differential equations 20 1.3. Numerical analysis of differential equations 27 1.4. Experiment: population dynamics 29 1.5. Exercises 32 1.6. Notes 37 Chapter 2. Local solutions 42 2.1. Existence and uniqueness theorem (Picard’s theorem) 43 2.2. Existence theorem (Peano’s theorem) 54 2.3. Theorem of continuous dependence 59 2.4. Theorem of differentiable dependence 63 2.5. Generalizations 70 2.6. Experiment: Picard’s method 77 2.7. Exercises 81 2.8. Notes 87 Chapter 3. Maximal solutions 94 3.1. Existence and uniqueness 95 3.2. Boundary behavior 99 3.3. Globally Lipschitz equations 102 3.4. Theorem of continuous dependence (global) 105 3.5. Theorem of differentiable dependence (global) 108 3.6. Experiment: continuation of solutions 110 3.7. Exercises 113 3.8. Notes 119 Chapter 4. Numerical integration 124 4.1. Euler method 126 4.2. Runge–Kutta methods 133 4.3. Convergence of one-step methods 139 4.4. Adams methods 144 4.5. Convergence of multistep methods 148 4.6. Stiffness 153 4.7. Experiment: level curves 161 4.8. Exercises 164 4.9. Notes 168 Chapter 5. Autonomous equations 174 5.1. Flow of an autonomous equation 176 5.2. Tubular flow theorem 181 5.3. Poincaré maps 183 5.4. Conjugacy and equivalence of flows 189 5.5. Poincaré recurrence theorem 192 5.6. Experiment: electrical circuits 196 5.7. Exercises 200 5.8. Notes 204 Chapter 6. Autonomous linear equations 210 6.1. Exponential of a linear map 211 6.2. Calculation of the exponential 214 6.3. Two-dimensional case 220 6.4. Differentiable conjugacy of linear flows 226 6.5. Topological classification of hyperbolic flows 227 6.6. Experiment: aerodynamic instability 235 6.7. Exercises 240 6.8. Notes 244 Chapter 7. Nonautonomous linear equations 248 7.1. Solution space of the homogeneous equation 249 7.2. Fundamental solutions of the homogeneous equation 251 7.3. Liouville–Ostrogradskiĭ formula 252 7.4. Solution space of the nonhomogeneous equation 256 7.5. Floquet’s theorem 258 7.6. Experiment: resonance 262 7.7. Exercises 265 7.8. Notes 270 Chapter 8. Lyapunov stability 276 8.1. Autonomous equations: linear stability 280 8.2. Autonomous equations: Lyapunov functions 286 8.3. Lyapunov analysis of nonautonomous equations 294 8.4. Linear stability and Lyapunov exponents 303 8.5. Experiment: largest Lyapunov exponent 315 8.6. Exercises 319 8.7. Notes 324 Chapter 9. Grobman–Hartman theorem 330 9.1. Hyperbolic stationary points 331 9.2. Grobman–Hartman theorem for flows 335 9.3. Proof of the Grobman–Hartman theorem 337 9.4. Grobman–Hartman theorem for diffeomorphisms 347 9.5. Differentiable conjugacy 349 9.6. Experiment: shooting method 355 9.7. Exercises 358 9.8. Notes 362 Chapter 10. Stable manifold theorem 366 10.1. Local stable and unstable manifolds 367 10.2. Stable manifold theorem 369 10.3. Proof of the stable manifold theorem 370 10.4. Hyperbolic periodic trajectories 385 10.5. Experiment: planetary systems 386 10.6. Exercises 389 10.7. Notes 393 Chapter 11. Vector fields on surfaces 400 11.1. α- and ω-limit sets 401 11.2. Poincaré–Bendixson theorem 403 11.3. Limit sets of flows on surfaces 411 11.4. Mayer’s theorem on conservative flows 415 11.5. Comments on structural stability 432 11.6. Experiment: Lorenz attractor 434 11.7. Exercises 440 11.8. Notes 445 Chapter 12. Poincaré–Hopf theorem 450 12.1. Index of a stationary point 451 12.2. Euler characteristic 460 12.3. Indices and curvature 467 12.4. Proof of the theorem 469 12.5. Comments on Mayer’s theorem 471 12.6. Experiment: oxygen–ozone cycle 473 12.7. Exercises 475 12.8. Notes 480 Appendix A. Metric spaces and differentiable manifolds 484 A.1. Metric spaces and sequences 484 A.2. Continuous maps 488 A.3. Differentiable manifolds 490 A.4. Tangent space and derivative map 493 A.5. Cotangent space and differential forms 495 A.6. Transversality 498 A.7. Riemannian manifolds 499 A.8. Euler characteristic 500 A.9. Curvature and connection forms 503 A.10. Notes 505 Bibliography 510 Index 532 Back Cover 553 Taking into account the remarkable development of dynamical systems, the authors present the core topics that every young mathematician of our time - pure and applied alike - ought to learn. The book features a dynamical perspective that drives the motivating questions, the style of exposition, and the arguments and proof techniques.