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1 Beaverton Mathematics- Differential Equations

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Ordinary Differential Equations

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Synopses & Reviews

Publisher Comments:

Based on a Brown University course in applied mathematics, this rigorous and demanding treatment focuses on specific analytical methods. It emphasizes nonlinear problems, acquainting readers with problems and techniques in ordinary differential equations. The material is presented in a manner that prepares students for informed research of differential equations, teaching them how to be more effective in studies of the current literature. In addressing the applied side of the subject, the text devotes considerable attention to specific analytical methods common to applications. 

Introductory chapters offer necessary background material by reviewing basic facts of analysis and covering the general properties of differential equations. Topics include two-dimensional systems, linear systems and linearization, perturbations of noncritical linear systems, simple oscillatory phenomena and the method of averaging, and behavior near a periodic orbit. Additional subjects include integral manifolds of equations with a small parameter, periodic systems with a small parameter, alternative problems for the solution of functional equations, and the direct method of Liapunov. Exercises appear at the end of each chapter, and the appendix contains a convenient reference for almost periodic functions.

Synopsis:

This rigorous treatment prepares readers for the study of differential equations and shows them how to research current literature. It emphasizes nonlinear problems and specific analytical methods. 1969 edition.

Synopsis:

Based on a Brown University course in applied mathematics, this text is designed to prepare readers for the study of differential equations and to show them how to conduct effective literature searches. A rigorous and demanding treatment, it emphasizes nonlinear problems and focuses on specific analytical methods. 1969 edition.

Table of Contents

MATHEMATICAL PRELIMINARIES

Banach spaces and examples

Linear transformations

Fixed point theorems

GENERAL PROPERTIES OF DIFFERENTIAL EQUATIONS

Existence

Continuation of solutions

Uniqueness and continuity properties

Continuous dependence and stability

Extension of the concept of a differential equation

Differential inequalities

Autonomous systems-generalities

Autonomous systems-limit sets, invariant sets

Remarks and suggestions for further study

TWO DIMENSIONAL SYSTEMS

Planar two dimensional systems-the Poincaré-Bendixson theory

Differential systems on a torus

Remarks and suggestions for further study

LINEAR SYSTEMS AND LINEARIZATION

General linear systems

Stability of linear and perturbed linear systems

nth Order scalar equations

Linear systems with constant coefficients

Two dimensional linear autonomous systems

The saddle point property

Linear periodic systems

Hills equation

Reciprocal systems

Canonical systems

Remarks and suggestion for further study

PERTURBATION OF NONCRITICAL LINEAR SYSTEMS

Nonhomogeneous linear systems

Weakly nonlinear equations-noncritical case

The general saddle point property

More general systems

The Duffing equation with large damping and large forcing

Remarks and extensions

SIMPLE OSCILLATORY PHENOMENA AND THE METHOD OF AVERAGING

Conservative systems

Nonconservative second order equations-limit cycles

Averaging

The forced van der Pol equation

Duffings equation with small damping and small harmonic forcing

The subharmonic of order 3 for Duffings equation

Damped excited pendulum with oscillating support

Exercises

Remarks and suggestions for further study

BEHAVIOR NEAR A PERIODIC ORBIT

Stability of a periodic orbit

Sufficient conditions for orbital stability in two dimensions

Autonomous perturbations

Remarks and suggestions for further study

INTEGRAL MANIFOLDS OF EQUATIONS WITH A SMALL PARAMETER

Methods of determining integral manifolds

Statement of results

A “nonhomgeneous linear” system

The mapping principle

Proof of Theorem 2.1

Stability of the perturbed manifold

Applications

Exercises

Remarks and suggestions for further study

PERIODIC SYSTEMS WITH A SMALL PARAMETER

A special system of equations

Almost linear systems

Periodic solutions of perturbed autonomous equations

Remarks and suggestions for further study

ALTERNATIVE PROBLEMS FOR THE SOLUTION OF FUNCTIONAL EQUATIONS

Equivalent equations

A generalization

Alternative problems

Alternative problems for periodic solutions

The Perron-Lettenmeyer theorem

Remarks and suggestions for further study

THE DIRECT METHOD OF LIAPUNOV

Sufficient conditions for stability and instability in autonomous systems

Circuits containing Esaki diodes

Sufficient conditions for stability in nonautonomous systems

The converse theorems for asymptotic stability

Implications of asymptotic stability

Wazewskis principle

Remarks and suggestions for further study

APPENDIX

ALMOST PERIODIC FUNCTIONS

REFERENCES

INDEX

Product Details

ISBN:
9780486472119
Author:
Hale, Jack K
Publisher:
Dover Publications
Author:
Hale, Jack K.
Author:
Mathematics
Author:
Jack K. Hal
Author:
E
Subject:
Differential Equations
Subject:
Mathematics-Differential Equations
Copyright:
Edition Description:
Trade Paper
Series:
Dover Books on Mathematics
Publication Date:
20090531
Binding:
TRADE PAPER
Grade Level:
General/trade
Language:
English
Illustrations:
Y
Pages:
384
Dimensions:
8.5 x 5.38 in 1 lb

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Related Subjects

History and Social Science » World History » General
Science and Mathematics » Astronomy » General
Science and Mathematics » Mathematics » Differential Equations
Science and Mathematics » Mathematics » General
Science and Mathematics » Physics » General

Ordinary Differential Equations Used Trade Paper
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Product details 384 pages Dover Publications - English 9780486472119 Reviews:
"Synopsis" by ,
This rigorous treatment prepares readers for the study of differential equations and shows them how to research current literature. It emphasizes nonlinear problems and specific analytical methods. 1969 edition.
"Synopsis" by ,
Based on a Brown University course in applied mathematics, this text is designed to prepare readers for the study of differential equations and to show them how to conduct effective literature searches. A rigorous and demanding treatment, it emphasizes nonlinear problems and focuses on specific analytical methods. 1969 edition.
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