Synopses & Reviews
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
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.
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.
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