Nonlinear Ordinary Differential Equations

INTRODUCTION Preliminary Notions First-Order Systems Uniqueness and Existence Theorems Dependence on Parameters, and Continuation LINEAR EQUATIONS Uniqueness and Existence Theorem for a Linear System Homogeneous Linear Systems Inhomogeneous Linear Systems Second-Order Linear Equations Linear Equations with Constant Coefficients LINEAR EQUATIONS WITH PERIODIC COEFFICIENTS Floquet Theory Parametric Resonance Perturbation Methods for the Mathieu Equation The Mathieu Equation with Damping STABILITY Preliminary Definitions Stability for Linear Systems Principle of Linearized Stability Stability for Autonomous Systems Liapunov Functions PLANE AUTONOMOUS SYSTEMS Critical Points Linear Plane, Autonomous Systems Nonlinear Perturbations of Plane, Autonomous Systems PERIODIC SOLUTIONS OF PLANE AUTONOMOUS SYSTEMS Preliminary Results The Index of a Critical Point Van der Pol Equation Conservative Systems PERTURBATION METHODS FOR PERIODIC SOLUTIONS Poincare-Lindstedt Method Stability PERTURBATION METHODS FOR FORCED OSCILLATIONS Non-Resonant Case Resonant Case Resonant Oscillations for Duffing's Equation Resonant Oscillations for Van der Pol's Equation AVERAGING METHODS Averaging Methods for Autonomous Equations Averaging Methods for Forced Oscillations Adiabetic Invariance Multi-Scale Methods ELEMENTARY BIFURCATION THEORY Preliminary Notions One-Dimensional Bifurcations Hopf Bifurcation HAMILTONIAN SYSTEMS Hamiltonian and Lagrangian Dynamics Liouville's Theorem Integral Invariants and Canonical Transformations Action-Angle Variables Action-Angle Variables: Perturbation Theory ANSWERS TO SELECTED PROBLEMS REFERENCES INDEX.