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1 Basic Concepts 1

1 Phase Spaces and Phase Flows 1

2 Vector Fields on the Line 11

3 Phase Flows on the Line 19

4 Vector Fields and Phase Flows in the Plane 24

5 Nonautonomous Equations 28

6 The Tangent Space 33

2 Basic Theorems 48

7 The Vector Field near a Nonsingular Point 48

8 Applications to the Nonautonomous Case 56

9 Applications to Equations of Higher Order 59

10 Phase Curves of Autonomous Systems 68

11 The Directional Derivative.First Integrals 72

12 Conservative Systems with One Degree of Freedom 79

3 Linear Systems 95

13 Linear Problems 95

14 The Exponential of an Operator 97

15 Properties of the Exponential 104

16 The Determinant of the Exponential 111

17 The Case of Distinct Real Eigenvalues 115

18 Complexification and Decomplexification 119

19 Linear Equations with a Complex Phase Space 124

20 Complexification of a Real Linear Equation 129

21 Classification of Singular Points of Linear Systems 139

22 Topological Classification of Singular Points 143

23 Stability of Equilibrium Positions 154

24 The Case of Purely Imaginary Eigenvalues 160

25 The Case of Multiple Eigenvalues 167

26 More on Quasi-Polynomials 176

27 Nonautonomous Linear Equations 188

28 Linear Equations with Periodic Coefficients 199

29 Variation of Constants 208

4 Proofs of the Basic Theorems 211

30 Contraction Mappings 211

31 The Existence, Uniqueness, and Continuity Theorems 213

32 The Differentiability Theorem 223

5 Differential Equationson Manifolds 233

33 Differentiable Manifolds 233

34 The Tangent Bundle.Vector Fields on a Manifold 243

35 The Phase Flow Determined by a Vector Field 250

36 The Index of a Singular Point of a Vector Field 254

Sample Examination Problems 269

Bibliography 273

Index 275