Ordinary Differential Equations
by Arnold, V.I.; Silverman, Richard A.Format: Paperback
Pub. Date: 1978-07-15
Publisher(s): The MIT Press
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Summary
Few books on Ordinary Differential Equations (ODEs) have the elegant geometric insight of this one, which puts emphasis on the qualitative and geometric properties of ODEs and their solutions, rather than on routine presentation of algorithms.
Table of Contents
| Preface | p. vii |
| Frequently Used Notation | p. ix |
| Basic Concepts | p. 1 |
| Phase Spaces and Phase Flows | p. 1 |
| Vector Fields on the Line | p. 11 |
| Phase Flows on the Line | p. 19 |
| Vector Fields and Phase Flows in the Plane | p. 24 |
| Nonautonomous Equations | p. 28 |
| The Tangent Space | p. 33 |
| Basic Theorems | p. 48 |
| The Vector Field near a Nonsingular Point | p. 48 |
| Applications to the Nonautonomous Case | p. 56 |
| Applications to Equations of Higher Order | p. 59 |
| Phase Curves of Autonomous Systems | p. 68 |
| The Directional Derivative. First Integrals | p. 72 |
| Conservative Systems with One Degree of Freedom | p. 79 |
| Linear Systems | p. 95 |
| Linear Problems | p. 95 |
| The Exponential of an Operator | p. 97 |
| Properties of the Exponential | p. 104 |
| The Determinant of the Exponential | p. 111 |
| The Case of Distinct Real Eigenvalues | p. 115 |
| Complexification and Decomplexification | p. 119 |
| Linear Equations with a Complex Phase Space | p. 124 |
| Complexification of a Real Linear Equation | p. 129 |
| Classification of Singular Points of Linear Systems | p. 139 |
| Topological Classification of Singular Points | p. 143 |
| Stability of Equilibrium Positions | p. 154 |
| The Case of Purely Imaginary Eigenvalues | p. 160 |
| The Case of Multiple Eigenvalues | p. 167 |
| More on Quasi-Polynomials | p. 176 |
| Nonautonomous Linear Equations | p. 188 |
| Linear Equations with Periodic Coefficients | p. 199 |
| Variation of Constants | p. 208 |
| Proofs of the Basic Theorems | p. 211 |
| Contraction Mappings | p. 211 |
| The Existence, Uniqueness, and Continuity Theorems | p. 213 |
| The Differentiable Manifolds | p. 233 |
| The Tangent Bundle. Vector Fields on a Manifold | p. 243 |
| The Phase Flow Determined by a Vector Field | p. 250 |
| The Index of a Singular Point of a Vector Field | p. 254 |
| Sample Examination Problems | p. 269 |
| Bibliography | p. 273 |
| Index | p. 275 |
| Table of Contents provided by Publisher. All Rights Reserved. |
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