文档介绍：Ordinary differential equations
and
Dynamical Systems
Gerald Teschl
Gerald Teschl
Institut f¨urMathematik
Nordbergstraße 15
Universit¨atWien
1090 Wien, Austria
E-mail: Gerald.******@
URL: ./~gerald/
1991 Mathematics subject classification. 34-01
Abstract. This manuscript provides an introduction to ordinary differential
equations and dynamical systems. We start with some simple examples
of explicitly solvable equations. Then we prove the fundamental results
concerning the initial value problem: existence, uniqueness, extensibility,
dependence on initial conditions. Furthermore we consider linear equations,
the Floquet theorem, and the autonomous linear flow.
Then we establish the Frobenius method for linear equations in -
plex domain and investigates Sturm–Liouville type boundary value problems
including oscillation theory.
Next we introduce the concept of a dynamical system and discuss sta-
bility including the stable manifold and the Hartman–Grobman theorem for
both continuous and discrete systems.
We prove the Poincar´e–Bendixson theorem and investigate several ex-
amples of planar systems from classical mechanics, ecology, and electrical
engineering. Moreover, attractors, Hamiltonian systems, the KAM theorem,
and periodic solutions are discussed as well.
Finally, there is an introduction to chaos. Beginning with the basics for
iterated interval maps and ending with the Smale–Birkhoff theorem and the
Melnikov method for homoclinic orbits.
Keywords and phrases. Ordinary differential equations, dynamical systems,
Sturm-Liouville equations.
Typeset by AMS-LATEX and Makeindex.
Version: October 21, 2004
Copyright
c 2000-2004 by Gerald Teschl
Contents
Preface vii
Part 1. Classical theory
Chapter 1. Introduction 3
§. Newton’s equations 3
§. Classification of differential equations 6
§. First order autonomous equations 8
§. Finding explicit solutions 11
§. Qualitative analysis of first order