Ordinary Differential Equations

by Hale, Jack K.

• ISBN13: 9780486472119
• ISBN10: 0486472116
• Format: Paperback
• Copyright: 2009-05-21
• Publisher: Dover Publications

Summary

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.

Author Biography

A longtime professor at Brown University, Jack K. Hale is a Regents Professor Emeritus at the Georgia Institute of Technology.

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
Hill’s 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
Duffing’s equation with small damping and small harmonic forcing
The subharmonic of order 3 for Duffing’s 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
Wazewski’s principle
Remarks and suggestions for further study
APPENDIX
ALMOST PERIODIC FUNCTIONS
REFERENCES
INDEX