Nonlinear Differential EquationsPDF电子书下载

CHAPTER 1 PRELIMINARY CONSIDERATIONS 1

1．Linear Second-order Equations 1

2．Some Nonlinear Second-order Equations 16

3．The Initial-value Problem 25

4．Higher-order Systems 28

5．The Taxicab Geometry 35

6．Continuous,Differentiable,Analytic,and Lipschitz Vector Functions 38

CHAPTER 2 THE EXISTENCE AND THE UNIQUENESS OF A SOLUTION OF THE INITIAL-VALUE PROBLEM 41

1．The Cauchy-Lipschitz Existence Theorem 41

2．The Uniqueness Theorem 46

3．Continuity with Respect to Initial Values 48

4．The Cauchy-Peano Existence Theorem 49

5．An Existence and Uniqueness Theorem for Nonautonomous Systems 56

CHAPTER 3 PROPERTIES OP SOLUTIONS 60

1．Extension of the Trajectories 60

2．Continuity Properties of Trajectories 62

3．The Poincaré Expansion Theorem 67

4．Differentiability of Solutions 72

CHAPTER 4 PROPERTIES OF LINEAR SYSTEMS 79

1．Bases and the Principal Matrix Solution 79

2．The Linear Inhomogeneous Equation 91

3．Linear Homogeneous Equations with Constant Coefficients 98

4．Linear Systems with Periodic Coefficients 106

5．Asymptotic Behavior of Solutions of Linear Systems 115

CHAPTER 5 STABILITY IN NONLINEAR SYSTEMS 129

1．The Concept of Stability 129

2．Stability of Singular Points of Autonomous Systems 131

3．Stability of Singular Points of Nonautonomous Systems 137

4．Stability of Singular Solutions of Implicit Equations 144

5．Stability of Nonsingular Solutions 149

6．Direct Stability Method 160

CHAPTER 6 TWO-DIMENSIONAL SYSTEMS 170

1．Critical Points of Autonomous Systems 170

2．Properties of Limit Cycles 179

CHAPTER 7 PERTURBATIONS OF PERIODIC SOLUTIONS 194

1．Perturbations of Periodic Solutions in Non-autonomous Systems 194

2．Periodic Solutions of Nonautonomous Quasi-harmonic Equations 203

3．Perturbations of Periodic Solutions in Autonomous Systems 209

4．Periodic Solutions of Autonomous Quasi-harmonic Equations 215

CHAPTER 8 A GENERAL ASYMPTOTIC METHOD 220

1．Introduction 220

2．The Mathieu Equation 221

3．The Free Oscillations of the van der Pol Equation 236

4．The Forced Oscillations of the van der Pol Equation 240

5．The Forced Oscillations of the Duffing Equation 251

GENERAL REFERENCES 262

Index 263