文档介绍：Ordinary differential equations
and
Dynamical Systems
Gerald Teschl
Gerald Teschl
Fakult¨atf¨urMathematik
Nordbergstraße 15
Universit¨atWien
1090 Wien, Austria
E-mail: Gerald.******@
URL: ./˜gerald/
2000 Mathematics subject classification. 34-01
Abstract. This book provides an introduction to ordinary differential equa-
tions and dynamical systems. We start with some simple examples of explic-
itly 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 investigate 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–Bendixsontheorem 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: March 25, 2009
Copyright
c 2000–2009 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 13
§. Qualitative analysis of first-order equati