| Titre : | Ordinary Differential Equations with Applications |
| Auteurs : | Carmen Chicone, Auteur |
| Type de document : | Monographie imprimée |
| Editeur : | Springer, 2024 |
| ISBN/ISSN/EAN : | 978-3-031-51651-1 |
| Format : | p729 / 24cm |
| Langues: | Anglais |
| Résumé : |
This book, developed during 20 years of the author teaching differential equations courses at his home university, is designed to serve as a text for a graduate level course focused on the central theory of the subject with attention paid to applications and connections to other advanced topics in mathematics. Core theory includes local existence and uniqueness, the phase plane, Poincaré-Bendixson theory, Lyapunov and linearized stability, linear systems, Floquet theory, the Grobman–Hartman theorem, persistence of rest points and periodic orbits, the stable and center manifold theorems, and bifurcation theory. This edition includes expanded treatment of deterministic chaos, perturbation theory for periodic solutions, boundary value problems, optimization, and a wide range of their applications. In addition, it contains a formulation and new proof of a theorem on instability of rest points in the presence of an eigenvalue with positive real part, and new proofs of differential inequalities and Lyapunov’s center theorem. New sections present discussions of global bifurcation, the Crandall–Rabinowitz theorem, and Alekseev’s formula. Of particular note is a new chapter on basic control theory, a discussion of optimal control, and a proof of a useful special case of the maximum principle. A key feature of earlier editions, a wide selection of original exercises, is respected in this edition with the inclusion of a wealth of new exercises. Reviews of the first edition: “As an applied mathematics text on linear and nonlinear equations, the book by Chicone is written with stimulating enthusiasm. It will certainly appeal to many students and researchers.”―F. Verhulst, SIAM Review “The author writes lucidly and in an engaging conversational style. His book is wide-ranging in its subject matter, thorough in its presentation, and written at a generally high level of generality, detail, and rigor.”―D. S. Shafer, Mathematical Reviews |
| Sommaire : |
Series Preface Preface Preface to the Second Edition Preface to the Third Edition Acknowledgments Contents 1 Introduction to Ordinary Differential Equations 1.1 Existence and Uniqueness 1.2 Types of Differential Equations 1.3 Geometric Interpretation of Autonomous Systems 1.4 Flows 1.5 Stability and Linearization 1.6 Stability and the Direct Method of Lyapunov 1.7 Manifolds 1.7.1 Introduction to Invariant Manifolds 1.7.2 Smooth Manifolds 1.7.3 Tangent Spaces 1.7.4 Change of Coordinates 1.7.5 Reparametrization of Time 1.7.6 Polar Coordinates 1.8 Periodic Solutions 1.8.1 The Poincaré Map 1.8.2 Limit Sets and Poincaré–Bendixson Theory 1.9 Regular and Singular Perturbation 1.10 Review of Calculus 1.10.1 The Mean Value Theorem 1.10.2 Integration in Banach Spaces 1.11 Contraction 1.11.1 The Contraction Mapping Theorem 1.11.2 Uniform Contraction 1.11.3 Fiber Contraction 1.11.4 The Implicit Function Theorem 1.12 Existence, Uniqueness, and Extension 2 Homogeneous Linear Systems 2.1 Gronwall's Inequality 2.2 Existence Theory 2.3 Principle of Superposition 2.4 Linear Equations with Constant Coefficients 2.5 The Matrix Exponential 2.6 Lie–Trotter and Baker–Campbell–Hausdorff formulas 3 Stability of Linear Systems 4 Stability of Nonlinear Systems 4.1 Variation of Parameters and Solution of Inhomogeneous … 4.2 Alekseev-Gröbner formula 4.3 Stability of Nonlinear Systems 4.4 An Instability Criterion 5 Floquet Theory 5.1 Lyapunov Exponents 5.2 Hill's Equation 5.3 Periodic Orbits of Linear Systems 5.4 Stability of Periodic Orbits 6 Applications 6.1 Origins of ODE: Calculus of Variations 6.2 Origins of ODE: Classical Physics 6.2.1 Motion of a Charged Particle 6.2.2 Motion of a Binary System 6.2.3 Perturbed Kepler Motion and Delaunay Elements 6.2.4 Satellite Orbiting an Oblate Planet 6.2.5 The Diamagnetic Kepler Problem 6.3 Coupled Pendula: Normal Modes and Beats 6.4 The Fermi-Ulam-Pasta Oscillator 6.5 The Inverted Pendulum 6.6 Origins of ODE: Partial Differential Equations 6.6.1 Infinite-Dimensional ODE 6.6.2 Galërkin Approximation 6.6.3 Traveling Waves 6.6.4 First Order PDE 6.7 Control 6.7.1 Controllability of Time-Invariant Linear Systems 6.7.2 Optimal Control 6.7.3 Quadratic Regulator 6.7.4 Optimal Control Example 6.7.5 Parameter Estimation and the Adjoint Method 7 Hyperbolic Theory 7.1 Invariant Manifolds 7.2 Applications of Invariant Manifolds 7.3 The Hartman–Grobman Theorem 7.3.1 Diffeomorphisms 7.3.2 Differential Equations 7.3.3 Linearization via the Lie Derivative 8 Continuation of Periodic Solutions 8.1 A Classic Example: van der Pol's Oscillator 8.1.1 Continuation Theory and Applied Mathematics 8.2 Autonomous Perturbations 8.2.1 Poincaré's Method of Continuation 8.2.2 Continuation of Periodic Orbits of Planar Systems 8.2.3 Diliberto's Theorem 8.2.4 Preparation Theorem and Persistence of Nonhyperbolic Periodic Orbits 8.2.5 Continuation from an Annulus of Period Orbits 8.2.6 Periodic Orbits of Multidimensional Systems with First Integrals 8.3 Nonautonomous Perturbations 8.3.1 Rest Points 8.3.2 Isochronous Period Annulus 8.3.3 The Forced van der Pol Oscillator 8.3.4 Regular Period Annulus and Lyapunov–Schmidt Reduction 8.3.5 Limit Cycles–Entrainment–Resonance Zones 8.3.6 Lindstedt Series and the Perihelion of Mercury 8.3.7 Entrainment Domains for van der Pol's Oscillator 8.4 Forced Oscillators 9 Homoclinic Orbits, Melnikov's Method, and Chaos 9.1 Autonomous Perturbations: Separatrix Splitting 9.2 Periodic Perturbations: Transverse Homoclinic Points 9.3 Origins of ODE: Fluid Dynamics 9.3.1 The Equations of Fluid Motion 9.3.2 ABC Flows 9.3.3 Chaotic ABC Flows 10 Averaging 10.1 The Averaging Principle 10.2 Averaging at Resonance 10.3 Action-Angle Variables 11 Bifurcation 11.1 One-Dimensional State Space 11.1.1 The Saddle-Node Bifurcation 11.1.2 A Normal Form 11.1.3 Bifurcation in Applied Mathematics 11.1.4 Families, Transversality, and Jets 11.2 Saddle-Node Bifurcation via Lyapunov-Schmidt Reduction 11.3 Poincaré-Andronov-Hopf Bifurcation 11.3.1 Multiple Hopf Bifurcation 11.4 Dynamic Bifurcation 11.5 Global Continuation and the Crandall-Rabinowitz Theorem 11.5.1 Finite-Dimensional Approximation 11.5.2 Continuation 11.5.3 Bifurcation 11.5.4 Summary References Index |
Disponibilité (2)
| Cote | Support | Localisation | Statut |
|---|---|---|---|
| MAT/2014 | Livre | bibliothèque sciences exactes | Consultable |
| MAT/2014 | Livre | bibliothèque sciences exactes | Empruntable |
