1st Edition

# Nonlinear Ordinary Differential Equations

336 Pages
by CRC Press

336 Pages
by Routledge

Also available as eBook on:

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 Ordinary Differential Equations takes the reader from basic elementary notions to the point where the exciting and fascinating developments in the theory of nonlinear differential equations can be understood and appreciated.
Each chapter is self-contained, and includes a selection of problems together with some detailed workings within the main text. Nonlinear Ordinary Differential Equations helps develop an understanding of the subtle and sometimes unexpected properties of nonlinear systems and simultaneously introduces practical analytical techniques to analyze nonlinear phenomena. This excellent book gives a structured, systematic, and rigorous development of the basic theory from elementary concepts to a point where readers can utilize ideas in nonlinear 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
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
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