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Nonlinear Ordinary Differential Equations.

"Ordinary differential equations have long been an important area of study because of their wide application in physics, engineering, biology, chemistry, ecology, and economics. Based on a series of lectures given at the Universities of Melbourne and New South Wales in Australia, Nonlinear Ordi...

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Bibliographic Details
Online Access: Full Text (via Taylor & Francis)
Main Author: Grimshaw, R.
Format: eBook
Language:English
Published: CRC Press, 2017.
Subjects:
Differential equations, Nonlinear.
Table of Contents:
  • 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
  • Poincar�-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.

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