Ordinary differential equations

Cover......Page 1
Ordinary Differential Equations......Page 2
Copyright......Page 3
Preface to the Third Edition......Page 4
Preface to the First Edition......Page 7
Frequently used notation......Page 8
Contents......Page 9
1 . Examples of Evolutionary Processes......Page 16
2 . Phase Spaces......Page 17
3 . The Integral Curves of a Direction Field......Page 19
4 . A Differential Equation and its Solutions......Page 20
5 . The Evolutionary Equation with a One-dimensional Phase Space......Page 22
6 . Example: The Equation of Normal Reproduction......Page 24
7 . Example: The Explosion Equation......Page 26
8 . Example: The Logistic Curve......Page 27
9 . Example: Harvest Quotas......Page 28
10 . Example: Harvesting with a Relative Quota.......Page 29
11 . Equations with a Multidimensional Phase Space......Page 30
12 . Example: The Differential Equation of a Predator-Prey System......Page 31
13 . Example: A Free Particle on a Line......Page 34
15 . Example: Small Oscillations......Page 35
16 . Example: The Mathematical Pendulum......Page 36
18 . Example: Small Oscillations of a Spherical Pendulum......Page 37
2 . A Counterexample......Page 39
3 . Proof of Uniqueness......Page 40
5 . Examples of Direct Products......Page 42
6 . Equations with Separable Variables......Page 44
7 . An Example: The Lotka-Volterra Model......Page 46
1 . Homogeneous Linear Equations......Page 51
2 . First-order Homogeneous Linear Equations with Periodic Coefficients......Page 52
3 . Inhomogeneous Linear Equations......Page 54
4 . The Influence Function and delta-shaped Inhomogeneities......Page 10
5 . Inhomogeneous Linear Equations with Periodic Coefficients......Page 59
1 . The Action of a Group on a Set......Page 60
2 . One-parameter Transformation Groups......Page 63
3 . One-parameter Diffeomorphism Groups......Page 64
4 . The Phase Velocity Vector Field......Page 66
1 . The Action of Smooth Mappings on Vectors......Page 69
2 . The Action of Diffeomorphisms on Vector Fields......Page 73
3 . Change of Variables in an Equation......Page 75
4 . The Action of a Diffeomorphism on a Direction Field......Page 76
5 . The Action of a Diffeoinorphisin on a Phase Flow......Page 78
1 . Symmetry Groups......Page 79
2 . Application of a One-parameter Symmetry Group to Integrate an Equation......Page 80
3 . Homogeneous Equations......Page 82
4 . Quasi-homogeneous Equations......Page 85
5 . Similarity and Dimensional Considerations......Page 87
6 . Methods of Integrating Differential Equations......Page 89
1 . Rectification of a Direction Field......Page 92
2 . Existence and Uniqueness Theorems......Page 95
3 . Theorems on Continuous and Differentiable Dependence of the Solutions on the Initial Condition......Page 96
4 . Transformation over the Time Interval from t_0 zero to t......Page 99
5 . Theorems on Continuous and Differentiable Dependence on a Parameter......Page 100
6 . Extension Theorems......Page 103
7 . Rectification of a Vector Field......Page 106
1 . The Equivalence of an Equation of Order n and a System of n First-order Equations......Page 107
2 . Existence and Uniqueness Theorems......Page 110
3 . Differentiability and Extension Theorems......Page 111
4 . Systems of Equations......Page 112
5 . Remarks on Terminology......Page 115
9 . The Phase Curves of an Autonomous System......Page 119
2 . Translation over Time......Page 120
3 . Closed Phase Curves......Page 122
1 . The Derivative in the Direction of a Vector......Page 124
2 . The Derivative in the Direction of a Vector Field......Page 125
3 . Properties of the Directional Derivative......Page 126
4 . The Lie Algebra of Vector Fields......Page 127
5 . First Integrals......Page 128
6 . Local First Integrals......Page 129
7 . Time-Dependent First Integrals......Page 130
1 . The Homogeneous Linear Equation......Page 132
2 . The Cauchy Problem......Page 133
3 . The Inhomogeneous Linear Equation......Page 134
4 . The Quasi-linear Equation......Page 135
5 . The Characteristics of a Quasi-linear Equation......Page 136
6 . Integration of a Quasi-linear Equation......Page 138
7 . The First-order Nonlinear Partial Differential Equation......Page 139
1 . Definitions......Page 141
2 . The Law of Conservation of Energy......Page 142
3 . The Level Lines of the Energy......Page 143
4 . The Level Lines of the Energy Near a Singular Point......Page 145
5 . Extension of the Solutions of Newton\'s Equation......Page 147
6 . Noncritical Level Lines of the Energy......Page 148
7 . Proof of the Theorem of Sect . 6......Page 149
8 . Critical Level Lines......Page 150
9 . An Example......Page 151
10 . Small Perturbations of a Conservative System......Page 152
1 . Example: Linearization......Page 155
2 . Example: One-parameter Groups of Linear Transformations of R^n......Page 156
3 . The Linear Equation......Page 157
1 . The Norm of an Operator......Page 158
3 . Proof of Completeness......Page 159
4 . Series......Page 160
5 . Definition of the Exponential e^A......Page 161
6 . An Example......Page 162
8 . The Exponential of a Nilpotent Operator......Page 163
9 . Quasi-polynomials......Page 164
15 . Properties of the Exponential......Page 165
1 . The Group Property......Page 166
2 . The Fundamental Theorem of the Theory of Linear Equations with Constant Coefficients......Page 167
4 . A Second Definition of the Exponential......Page 168
5 . An Example: Euler’s Formula for e^z......Page 169
6 . Euler’s Broken Lines......Page 170
1 . The Deteminant of an Operator......Page 172
2 . The Trace of an Operator......Page 173
4 . The Determinant of the 0perator e^A......Page 174
1 . The Diagonalizable Operator......Page 176
2 . An Example......Page 177
3 . The Discrete Case......Page 178
2 . Complexification......Page 180
3 . The Complex Conjugate......Page 181
4 . The Exponential, Determinant, and Trace of a Complex Operator......Page 182
5 . The Derivative of a Curve with Complex Values......Page 183
2 . The Fundamental Theorem......Page 184
4 . Example: A Linear Equation whose Phase Space is a Complex Line......Page 185
1 . The Complexified Equation......Page 188
2 . The Invariant Subspaces of a Real Operator......Page 190
3 . The Linear Equation on the Plane......Page 192
4 . The Classification of Singular Points in the Plane......Page 193
5 . Example: The Pendulum with Friction......Page 194
6 . The General Solution of a Linear Equation in the Case when the Characteristic Equation Has Only Simple Roots......Page 196
1 . Example: Singular Points in Three-dimensional Space......Page 198
2 . Linear, Differentiable, and Topological Equivalence......Page 200
3 . The Linear Classification......Page 201
1 . Theorem......Page 202
2 . Reduction to the Case m_- = 0......Page 203
3 . The Lyapunov Function......Page 204
4 . Construction of the Lyapunov Function......Page 205
5 . An Estimate of the Derivative......Page 207
6 . Construction of the Homeomorphism h......Page 209
7 . Proof of Lemma 3......Page 13
8 . Proof of the Topological Classification Theorem......Page 211
1 . Lyapunov Stability......Page 213
3 . A Theorem on Stability in First Approximation......Page 214
4 . Proof of the Theorem......Page 215
2 . An Example......Page 218
3 . The Phase Curves of Eq . (4) on the Torus......Page 220
5 . The Multidimensional Case......Page 222
6 . The Uniform Distribution......Page 223
1 . The Computation of e^{At}, where A is a Jordan Block......Page 224
2 . Applications......Page 226
3 . Applications to Systems of Equations of Order Higher than the First......Page 227
5 . On Recursive Sequences......Page 228
6 . Small Oscillations......Page 230
1 . A Linear Function Space......Page 232
2 . The Vector Space of Solutions of a Linear Equation......Page 233
3 . Translastion-invariance......Page 234
4 . Historical Remark......Page 235
5 . Inhomogeneous Equations......Page 236
6 . The Method of Complex Amplitudes......Page 238
7 . Application to the Calculation of Weakly Nonlinear Oscillations......Page 243
1 . Definition......Page 244
2 . The Existence of Solutions......Page 245
3 . The Vector Space of Solutions......Page 247
4 . The Wronskian Determinant......Page 248
5 . The Case of a Single Equation......Page 249
6 . Liouville\'s Theorem......Page 251
7 . Sturm\'s Theorems on the Zeros of Solutions of Second-order Equations......Page 254
1 . The Mapping over a Period......Page 259
2 . Stability Conditions......Page 261
3 . Strongly Stable Systems......Page 262
4 . Computations......Page 265
2 . The General Case......Page 267
3 . Computations......Page 268
1 . Definition......Page 270
2 . The Contraction Mapping Theorem......Page 271
1 . The Successive Approximations of Picard......Page 272
2 . Preliminary Estimates......Page 274
4 . Differentiability and the Lipschitz Condition......Page 275
5 . The Quantities C , L , a\', b\'......Page 276
6 . The Metric Space M......Page 277
7 . The Contraction Mapping A: M --> M......Page 278
8 . The Existence and Uniqueness Theorem......Page 279
9 . Other Applications of Contraction Mappings......Page 280
1 . The Equation of Variations......Page 282
2 . The Differentiability Theorem......Page 283
4 . Derivatives in x and t......Page 284
5 . The Rectification Theorem......Page 285
6 . The Last Derivative......Page 288
2 . Definitions......Page 291
3 . Examples of Atlases......Page 294
5 . Connectedness and Dimension......Page 296
6 . Differentiable Mappings......Page 297
8 . Submanifolds......Page 299
9 . An Example......Page 300
1 . The Tangent Space......Page 301
2 . The Tangent Bundle......Page 302
3 . A Remark on Parallelizability......Page 304
4 . The Tangent Mapping......Page 305
5 . Vector Fields......Page 306
1 . Theorem......Page 307
2 . Construction of the Diffeomorphisms g^t for Small t......Page 308
3 . The Construction of g^t for any t......Page 15
4 . A Remark......Page 310
1 . The Index of a Curve......Page 312
3 . Examples......Page 313
4 . The Index of a Singular Point of a Vector Field......Page 315
5 . The Theorem on the Sum of the Indices......Page 316
6 . The Sum of the Indices of the Singular Points on a Sphere......Page 318
7 . Justification......Page 320
8 . The Multidimensional Case......Page 321
Examination Topics......Page 326
Sample Examination Problems......Page 327
Supplementary Problems......Page 329
Subject Index......Page 334