Ordinary Differential Equations

by Arnold, V.I.; Silverman, Richard A.

• ISBN13: 9780262510189
• ISBN10: 0262510189
• Format: Paperback
• Copyright: 1978-07-15
• Publisher: The MIT Press

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
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