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.
Table of Contents
AVERAGING OF LINEAR DIFFERENTIAL EQUATIONS
Periodic and Almost Periodic Functions. Brief Introduction
Lemmas on Regularity and Stability
Parametric Resonance in Linear Systems
Higher Approximations. Shtokalo Method
Linear Differential Equations with Fast and Slow Time
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
Almost Periodic Functions
Stability of the Solutions of Differential Equations
Some Elementary Facts from the Functional Analysis
". . . 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