1st Edition

Method of Averaging for Differential Equations on an Infinite Interval Theory and Applications

By Vladimir Burd Copyright 2007
    356 Pages 12 B/W Illustrations
    by Chapman & Hall

    356 Pages
    by Chapman & Hall

    In recent years, mathematicians have detailed simpler proofs of known theorems, have identified new applications of the method of averaging, and have obtained many new results of these applications. Encompassing these novel aspects, Method of Averaging of the Infinite Interval: Theory and Applications rigorously explains the modern theory of the method of averaging and provides a solid understanding of the results obtained when applying this theory.

    The book starts with the less complicated theory of averaging linear differential equations (LDEs), focusing on almost periodic functions. It describes stability theory and Shtokalo's method, and examines various applications, including parametric resonance and the construction of asymptotics. After establishing this foundation, the author goes on to explore nonlinear equations. He studies standard form systems in which the right-hand side of a system is proportional to a small parameter and proves theorems similar to Banfi's theorem. The final chapters are devoted to systems with a rapidly rotating phase.

    Covering an important asymptotic method of differential equations, this book provides a thorough understanding of the method of averaging theory and its resulting applications.

    PREFACE

    AVERAGING OF LINEAR DIFFERENTIAL EQUATIONS
    Periodic and Almost Periodic Functions. Brief Introduction
    Bounded Solutions
    Lemmas on Regularity and Stability
    Parametric Resonance in Linear Systems
    Higher Approximations. Shtokalo Method
    Linear Differential Equations with Fast and Slow Time
    Asymptotic Integration
    Singularly Perturbed Equations

    AVERAGING OF NONLINEAR SYSTEMS
    Systems in Standard Form. First Approximation
    Systems in Standard Form. First Examples
    Pendulum Systems with an Oscillating Pivot
    Higher Approximations of the Method of Averaging
    Averaging and Stability
    Systems with a Rapidly Rotating Phase
    Systems with a Fast Phase. Resonant Periodic Oscillations
    Systems with Slowly Varying Parameters

    APPENDICES
    Almost Periodic Functions
    Stability of the Solutions of Differential Equations
    Some Elementary Facts from the Functional Analysis

    REFERENCES
    INDEX

    Biography

    Vladimir Burd

    ". . . a very readable book . . . clearly written book can be highly recommended to students with interests in ordinary differential equations. Non-experts and researchers in natural sciences will also find interesting methods that are useful in applications."

    – In EMS Newsletter, March 2008

    "Covering an important asymptotic method of differential equations, this book provides a thorough understanding of the method of averaging theory and its resulting applications."

    – in L’Enseignment Math, 2007