Nonlinear Differential Equations

PREFACE......Page 5 CONTENTS......Page 9 1. Linear Second-order Equations......Page 11 2. Some Nonlinear Second-order Equations......Page 26 3. The Initial-value Problem......Page 35 4. Higher-order Systems......Page 38 5. The Taxicab Geometry......Page 45 6. Continuous......Page 0 1. The Cauchy-Lipschitz Existence Theorem......Page 51 2. The Uniqueness Theorem......Page 56 3. Continuity with Respect to Initial Values......Page 58 4. The Cauchy-Peano Existence Theorem......Page 59 5. An Existence and Uniqueness Theorem for Nonautonomous Systems......Page 66 1. Extension of the Trajectories......Page 70 2. Continuity Properties of Trajectories......Page 72 3. The Poincare Expansion Theorem......Page 77 4. Differentiability of Solutions......Page 82 1. Bases and the Principal Matrix Solution......Page 89 2. The Linear Inhomogeneous Equation......Page 101 3. Linear Homogeneous Equations with Constant......Page 108 4. Linear Systems with Periodic Coefficients......Page 116 5. Asymptotic Behavior of Solutions of Linear Systems......Page 125 1. The Concept of Stability......Page 139 2. Stability of Singular Points of Autonomous Systems......Page 141 3. Stability of Singular Points of N onautonomous Systems......Page 147 4. Stability of Singular Solutions of Implicit Equations......Page 154 5. Stability of Nonsingular Solutions......Page 159 6. Direct Stability Method......Page 170 1. Critical Points of Autonomous Systems......Page 180 2. Properties of Limit Cycles......Page 189 1. Perturbations of Periodic Solutions in Nonautonomous Systems......Page 204 2. Periodic Solutions of Nonautonomous Quasiharmonic Equations......Page 213 3. Perturbations of Periodic Solutions in Autonomous Sytems......Page 219 4. Periodic Solutions of Autonomous Quasi-harmonic Equations......Page 225 I. Introduction......Page 230 2. The Mathieu Equation......Page 231 3. The Free Oscillations of the van der Pol Equation......Page 246 4. The Forced Oscillations of the van der Pol Equation......Page 250 5. The Forced Oscillations of the Duffing Equation......Page 261 GENERAL REFERENCES......Page 272 Index......Page 273