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Chapter 1.Basic Concepts 13

1.Phase Spaces 13

1.Examples of Evolutionary Processes 13

2.Phase Spaces 14

3.The Integral Curves of a Direction Field 16

4.A Differential Equation and its Solutions 17

5.The Evolutionary Equation with a One-dimensional Phase Space 19

6.Example： The Equation of Normal Reproduction 21

7.Example： The Explosion Equation 23

8.Example： The Logistic Curve 24

9.Example： Harvest Quotas 25

10.Example： Harvesting with a Relative Quota 26

11.Equations with a Multidimensional Phase Space 27

12.Example： The Differential Equation of a Predator-Prey System 28

13.Example： A Free Particle on a Line 31

14.Example： Free Fall 32

15.Example： Small Oscillations 32

16.Example： The Mathematical Pendulum 33

17.Example： The Inverted Pendulum 34

18.Example： Small Oscillations of a Spherical Pendulum 34

2.Vector Fields on the Line 36

1.Existence and Uniqueness of Solutions 36

2.A Counterexample 36

3.Proof of Uniqueness 37

4.Direct Products 39

5.Examples of Direct Products 39

6.Equations with Separable Variables 41

7.An Example： The Lotka-Volterra Model 43

3.Linear Equations 48

1.Homogeneous Linear Equations 48

2.First-order Homogeneous Linear Equations with Periodic Coefficients 49

3.Inhomogeneous Linear Equations 51

4.The Influence Function and δ-shaped Inhomogeneities 53

5.Inhomogeneous Linear Equations with Periodic Coefficients 56

4.Phase Flows 57

1.The Action of a Group on a Set 57

2.One-parameter Transformation Groups 59

3.One-parameter Diffeomorphism Groups 61

4.The Phase Velocity Vector Field 63

5.The Action of Diffeomorphisms on Vector Fields and Direction Fields 66

1.The Action of Smooth Mappings on Vectors 66

2.The Action of Diffeomorphisms on Vector Fields 70

3.Change of Variables in an Equation 72

4.The Action of a Diffeomorphism on a Direction Field 73

5.The Action of a Diffeomorphisms on a Phase Flow 75

6.Symmetries 76

1.Symmetry Groups 76

2.Application of a One-parameter Symmetry Group to Integrate an Equation 77

3.Homogeneous Equations 79

4.Quasi-homogeneous Equations 82

5.Similarity and Dimensional Considerations 84

6.Methods of Integrating Differential Equations 86

Chapter 2.Basic Theorems 89

7.Rectification Theorems 89

1.Rectification of a Direction Field 89

2.Existence and Uniqueness Theorems 92

3.Theorems on Continuous and Differentiable Dependence of theSolutions on the Initial Condition 93

4.Transformation over the Time Interval from to to t 96

5.Theorems on Continuous and Differentiable Dependence on a Parameter 97

6.Extension Theorems 100

7.Rectification of a Vector Field 103

8.Applications to Equations of Higher Order than First 104

1.The Equivalence of an Equation of Order n and a System of n First-order Equations 104

2.Existence and Uniqueness Theorems 107

3.Differentiability and Extension Theorems 108

4.Systems of Equations 109

5.Remarks on Terminology 112

9.The Phase Curves of an Autonomous System 116

1.Autonomous Systems 117

2.Translation over Time 117

3.Closed Phase Curves 119

10.The Derivative in the Direction of a Vector Field and First Integrals 121

1.The Derivative in the Direction of a Vector 121

2.The Derivative in the Direction of a Vector Field 122

3.Properties of the Directional Derivative 123

4.The Lie Algebra of Vector Fields 124

5.First Integrals 125

6.Local First Integrals 126

7.Time-Dependent First Integrals 127

11.First-order Linear and Quasi-linear Partial Differential Equations 129

1.The Homogeneous Linear Equation 129

2.The Cauchy Problem 130

3.The Inhomogeneous Linear Equation 131

4.The Quasi-linear Equation 132

5.The Characteristics of a Quasi-linear Equation 133

6.Integration of a Quasi-linear Equation 135

7.The First-order Nonlinear Partial Differential Equation 136

12.The Conservative System with one Degree of Freedom 138

1.Definitions 138

2.The Law of Conservation of Energy 139

3.The Level Lines of the Energy 140

4.The Level Lines of the Energy Near a Singular Point 142

5.Extension of the Solutions of Newton’s Equation 144

6.Noncritical Level Lines of the Energy 145

7.Proof of the Theorem of Sect.6 146

8.Critical Level Lines 147

9.An Example 148

10.Small Perturbations of a Conservative System 149

Chapter 3.Linear Systems 152

13.Linear Problems 152

1.Example： Linearization 152

2.Example： One-parameter Groups of Linear Transformations of Rn 153

3.The Linear Equation 154

14.The Exponential Function 155

1.The Norm of an Operator 155

2.The Metric Space of Operators 156

3.Proof of Completeness 156

4.Series 157

5.Definition of the Exponential eA 158

6.An Example 159

7.The Exponential of a Diagonal Operator 160

8.The Exponential of a Nilpotent Operator 160

9.Quasi-polynomials 161

15.Properties of the Exponential 162

1.The Group Property 163

2.The Fundamental Theorem of the Theory of Linear Equations with Constant Coefficients 164

3.The General Form of One-parameter Groups of Linear Transformations of the Space Rn 165

4.A Second Definition of the Exponential 165

5.An Example： Euler’s Formula for e 166

6.Euler’s Broken Lines 167

16.The Determinant of an Exponential 169

1.The Determinant of an Operator 169

2.The Trace of an Operator 170

3.The Connection Between the Determinant and the Trace 171

4.The Determinant of the Operator eA 171

17.Practical Computation of the Matrix of an Exponential -The Case when the Eigenvalues are Real and Distinct 173

1.The Diagonalizable Operator 173

2.An Example 174

3.The Discrete Case 175

18.Complexification and Realification 177

1.Realification 177

2.Complexification 177

3.The Complex Conjugate 178

4.The Exponential， Determinant， and Trace of a Complex Operator 179

5.The Derivative of a Curve with Complex Values 180

19.The Linear Equation with a Complex Phase Space 181

1.Definitions 181

2.The Fundamental Theorem 181

3.The Diagonalizable Case 182

4.Example： A Linear Equation whose Phase Space is a Complex Line 182

5.Corollary 185

20.The Complexification of a Real Linear Equation 185

1.The Complexified Equation 185

2.The Invariant Subspaces of a Real Operator 187

3.The Linear Equation on the Plane 189

4.The Classification of Singular Points in the Plane 190

5.Example： The Pendulum with Friction 191

6.The General Solution of a Linear Equation in the Case when the Characteristic Equation Has Only Simple Roots 193

21.The Classification of Singular Points of Linear Systems 195

1.Example：Singular Points in Three-dimensional Space 195

2.Linear，Differentiable， and Topological Equivalence 197

3.The Linear Classification 198

4.The Differentiable Classification 199

22.The Topological Classification of Singular Points 199

1.Theorem 199

2.Reduction to the Case m _ = 0 200

3.The Lyapunov Function 201

4.Construction of the Lyapunov Function 202

5.An Estimate of the Derivative 204

6.Construction of the Homeomorphism h 206

7.Proof of Lemma 3 207

8.Proof of the Topological Classification Theorem 208

23.Stability of Equilibrium Positions 210

1.Lyapunov Stability 210

2.Asymptotic Stability 211

3.A Theorem on Stability in First Approximation 211

4.Proof of the Theorem 212

24.The Case of Purely Imaginary Eigenvalues 215

1.The Topological Classification 215

2.An Example 215

3.The Phase Curves of Eq.（4） on the Torus 217

4.Corollaries 219

5.The Multidimensional Case 219

6.The Uniform Distribution 220

25.The Case of Multiple Eigenvalues 221

1.The Computation of eat， where A is a Jordan Block 221

2.Applications 223

3.Applications to Systems of Equations of Order Higher than the First 224

4.The Case of a Single nth-order Equation 225

5.On Recursive Sequences 226

6.Small Oscillations 227

26.Quasi-polynomials 229

1.A Linear Function Space 229

2.The Vector Space of Solutions of a Linear Equation 230

3.Translation-invariance 231

4.Historical Remark 232

5.Inhomogeneous Equations 233

6.The Method of Complex Amplitudes 235

7.Application to the Calculation of Weakly Nonlinear Oscillations 240

27.Nonautonomous Linear Equations 241

1.Definition 241

2.The Existence of Solutions 242

3.The Vector Space of Solutions 244

4.The Wronskian Determinant 245

5.The Case of a Single Equation 246

6.Liouville’s Theorem 248

7.Sturm’s Theorems on the Zeros of Solutions of Second-order Equations 251

28.Linear Equations with Periodic Coefficients 256

1.The Mapping over a Period 256

2.Stability Conditions 258

3.Strongly Stable Systems 259

4.Computations 262

29.Variation of Constants 264

1.The Simplest Case 264

2.The General Case 264

3.Computations 265

Chapter 4.Proofs of the Main Theorems 267

30.Contraction Mappings 267

1.Definition 267

2.The Contraction Mapping Theorem 268

3.Remark 269

31.Proof of the Theorems on Existence and Continuous Dependence on the Initial Conditions 269

1.The Successive Approximations of Picard 269

2.Preliminary Estimates 271

3.The Lipschitz Condition 272

4.Differentiability and the Lipschitz Condition 272

5.The Quantities C， L， a’， b 273

6.The Metric Space M 274

7.The Contraction Mapping A ：M→M 275

8.The Existence and Uniqueness Theorem 276

9.Other Applications of Contraction Mappings 277

32.The Theorem on Differentiability 279

1.The Equation of Variations 279

2.The Differentiability Theorem 280

3.Higher Derivatives with Respect to x 281

4.Derivatives in x and t 281

5.The Rectification Theorem 282

6.The Last Derivative 285

Chapter 5.Differential Equations on Manifolds 288

33.Differentiable Manifolds 288

1.Examples of Manifolds 288

2.Definitions 288

3.Examples of Atlases 291

4.Compactness 293

5.Connectedness and Dimension 293

6.Differentiable Mappings 294

7.Remark 296

8.Submanifolds 296

9.An Example 297

34.The Tangent Bundle.Vector Fields on a Manifold 298

1.The Tangent Space 298

2.The Tangent Bundle 299

3.A Remark on Parallelizability 301

4.The Tangent Mapping 302

5.Vector Fields 303

35.The Phase Flow Defined by a Vector Field 304

1.Theorem 304

2.Construction of the Diffeomorphisms gt for Small t 305

3.The Construction of gt for any t 306

4.A Remark 307

36.The Indices of the Singular Points of a Vector Field 309

1.The Index of a Curve 309

2.Properties of the Index 310

3.Examples 310

4.The Index of a Singular Point of a Vector Field 312

5.The Theorem on the Sum of the Indices 313

6.The Sum of the Indices of the Singular Points on a Sphere 315

7.Justification 317

8.The Multidimensional Case 318

Examination Topics 323

Sample Examination Problems 324

Supplementary Problems 326

Subject Index 331