1st Edition

Nonlinear Ordinary Differential Equations

By R. Grimshaw Copyright 1993

    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
    Poincaré-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.

    Biography

    Grimshaw, R.