Local methods in nonlinear differential equations

✍ Scribed by Alexander D. Bruno ; translated from the Russian by William Hovingh and Courtney S. Coleman.

• Publisher: Springer-Verlag,
• Year: c1989.
• Tongue: English
• Leaves: 357
• Category: Library

✦ Synopsis

The method of normal forms is usually attributed to Poincare
although some of the basic ideas of the method can be found
in earlier works of Jacobi, Briot and Bouquet.
In this book, A. D. Bruno gives an account of the work of these
mathematicians and further developments as well as the
results of his own extensive investigation of the subject.
The book begins with a thorough presentation of the analytical
techniques necessary for the implementation of the theory as
well as an extensive description of the geometry of the
Newton polygon. It then proceeds to discuss the normal form
of systems of ordinary differential equations giving many
specific applications of the theory. An underlying theme of the
book is the unifying nature of the method of normal forms
in the study of the local properties of ordinary differential
equations.
The second part of the book shows how the method of normal
forms yields tools for studying bifurcations, in particular
classical results of Lyapunov concerning families of periodic
orbits in the neighborhood of equilibrium points of
Hamiltonian systems as well as the more modern results
concerning families of quasiperiodic orbits obtained by
Kolmogorov, Arnold and Moser.
The book is intended for mathematicians, theoretical
mechanicians, and physicists. It is suitable for advanced under­
graduate and graduate students.

✦ Table of Contents

Cover
Local Methods in Nonlinear Differential Equations
Introduction to the English Edition by Dr. Steven Wiggins
Table of Contents
Table of Contents
Basic Notation
Introduction
Chapter I Foundations of the Local Method
§1. Linear Inequalities in the Plane
§2. Zeros of an Analytic Function
§ 3. Level Curves of an Analytic Function
Chapter II A System of Two Differential Equations
§ 1. Simple Points and Elementary Singularities
§2. Generalized Normal Forms
§3. A Nonelementary Singular Point
§4. On Distinguishing Between a Center and a Focus
Chapter III The Normal Form of a System of n Differential Equations
§1 . The Normalizing Transformation
§2. The Integration and Classification of Normal Forms
§3. Analytic Integral Sets
§4. The Normal Form and Methods of Averaging
Chapter IV On the Newton Polyhedron
§ 1. A System of Differential Equations
§2. Other Problems Using the Newton Polyhedron
Chapter V Applications of the Normal Form in Mechanics
§ 1. On the Motion of a Gyroscope in a Cardan Suspension
§ 2. On the Oscillations of a Satellite in the Plane of an Elliptical Orbit
References
Part II The Sets of Analyticity of a Normalizing Transformation
Preface
Table of Contents
Basic Notation
Introduction
§ 1. The Seminormal Form
§ 2. Questions of Convergence
§ 3. A Hamiltonian System
§4. Families of Periodic Solutions
§ 5. Integral Manifolds with Small Divisors
Author’s Comments (1986)
References
Subject Index
Backside Blurb