Differential Equations - A Dynamical Systems Approach to Theory and Practice

Description

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 Title page Preface Chapter 1. Introduction 1.1. Differential equations and their solutions 1.2. Qualitative theory of differential equations 1.3. Numerical analysis of differential equations 1.4. Experiment: population dynamics 1.5. Exercises 1.6. Notes Chapter 2. Local solutions 2.1. Existence and uniqueness theorem (Picard’s theorem) 2.2. Existence theorem (Peano’s theorem) 2.3. Theorem of continuous dependence 2.4. Theorem of differentiable dependence 2.5. Generalizations 2.6. Experiment: Picard’s method 2.7. Exercises 2.8. Notes Chapter 3. Maximal solutions 3.1. Existence and uniqueness 3.2. Boundary behavior 3.3. Globally Lipschitz equations 3.4. Theorem of continuous dependence (global) 3.5. Theorem of differentiable dependence (global) 3.6. Experiment: continuation of solutions 3.7. Exercises 3.8. Notes Chapter 4. Numerical integration 4.1. Euler method 4.2. Runge–Kutta methods 4.3. Convergence of one-step methods 4.4. Adams methods 4.5. Convergence of multistep methods 4.6. Stiffness 4.7. Experiment: level curves 4.8. Exercises 4.9. Notes Chapter 5. Autonomous equations 5.1. Flow of an autonomous equation 5.2. Tubular flow theorem 5.3. Poincaré maps 5.4. Conjugacy and equivalence of flows 5.5. Poincaré recurrence theorem 5.6. Experiment: electrical circuits 5.7. Exercises 5.8. Notes Chapter 6. Autonomous linear equations 6.1. Exponential of a linear map 6.2. Calculation of the exponential 6.3. Two-dimensional case 6.4. Differentiable conjugacy of linear flows 6.5. Topological classification of hyperbolic flows 6.6. Experiment: aerodynamic instability 6.7. Exercises 6.8. Notes Chapter 7. Nonautonomous linear equations 7.1. Solution space of the homogeneous equation 7.2. Fundamental solutions of the homogeneous equation 7.3. Liouville–Ostrogradskiĭ formula 7.4. Solution space of the nonhomogeneous equation 7.5. Floquet’s theorem 7.6. Experiment: resonance 7.7. Exercises 7.8. Notes Chapter 8. Lyapunov stability 8.1. Autonomous equations: linear stability 8.2. Autonomous equations: Lyapunov functions 8.3. Lyapunov analysis of nonautonomous equations 8.4. Linear stability and Lyapunov exponents 8.5. Experiment: largest Lyapunov exponent 8.6. Exercises 8.7. Notes Chapter 9. Grobman–Hartman theorem 9.1. Hyperbolic stationary points 9.2. Grobman–Hartman theorem for flows 9.3. Proof of the Grobman–Hartman theorem 9.4. Grobman–Hartman theorem for diffeomorphisms 9.5. Differentiable conjugacy 9.6. Experiment: shooting method 9.7. Exercises 9.8. Notes Chapter 10. Stable manifold theorem 10.1. Local stable and unstable manifolds 10.2. Stable manifold theorem 10.3. Proof of the stable manifold theorem 10.4. Hyperbolic periodic trajectories 10.5. Experiment: planetary systems 10.6. Exercises 10.7. Notes Chapter 11. Vector fields on surfaces 11.1. 𝛼- and 𝜔-limit sets 11.2. Poincaré–Bendixson theorem 11.3. Limit sets of flows on surfaces 11.4. Mayer’s theorem on conservative flows 11.5. Comments on structural stability 11.6. Experiment: Lorenz attractor 11.7. Exercises 11.8. Notes Chapter 12. Poincaré–Hopf theorem 12.1. Index of a stationary point 12.2. Euler characteristic 12.3. Indices and curvature 12.4. Proof of the theorem 12.5. Comments on Mayer’s theorem 12.6. Experiment: oxygen–ozone cycle 12.7. Exercises 12.8. Notes Appendix A. Metric spaces and differentiable manifolds A.1. Metric spaces and sequences A.2. Continuous maps A.3. Differentiable manifolds A.4. Tangent space and derivative map A.5. Cotangent space and differential forms A.6. Transversality A.7. Riemannian manifolds A.8. Euler characteristic A.9. Curvature and connection forms A.10. Notes Bibliography Index Back Cover