http://www.mat.univie.ac.at/~gerald/ftp/book-ode/
Lecture Notes Gerald Teschl
Faculty of Mathematics
University of Vienna
??Math. Dep. Uni. Vienna ESI
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Ordinary Differential Equations
and
Dynamical Systems
Gerald Teschl
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 the complex domain and investigate Sturm-Liouville type boundary value problems including oscillation theory.
Next we introduce the concept of a dynamical system and discuss stability including the stable manifold and the Hartman-Grobman theorem for both continuous and discrete systems.
We prove the Poincare-Bendixson theorem and investigate several examples 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.
MSC: 34-01
Keywords: Ordinary differential equations, Dynamical systems, Sturm-Liouville equations.
Download
The text is available as postscript (7.2M), or pdf (3.2M) version. In addition, there is a notebook (1M) with the Mathematica code from the text plus some further extensions. To solve complex equations via the power series method you also need my package DiscreteMath`DiffEqs`. Any comments and bug reports are welcome!
Table of contents
Part 1: Classical theory
Introduction
Newtons equations
Classification of differential equations
First order equations
Finding explicit solutions
Qualitative analysis of first order equations
Qualitative analysis of first order periodic equations
Initial value problems
Fixed point theorems
The basic existence and uniqueness result
Some extensions
Dependence on the initial condition
Extensibility of solutions
Euler's method and the Peano theorem
Linear equations
The matrix exponential
Linear autonomous first order systems
Linear autonomous equations of order n
General linear first order systems
Periodic linear systems
Appendix: Jordan canonical form
Differential equations in the complex domain
The basic existence and uniqueness result
The Frobenius method for second order equations
Linear systems with singularities
The Frobenius method
Boundary value problems
Introduction
Symmetric compact operators
Regular Sturm-Liouville problems
Oscillation theory
Periodic operators
Part 2: Dynamical systems
Dynamical systems
Dynamical systems
The flow of an autonomous equation
Orbits and invariant sets
The Poincaré map
Stability of fixed points
Stability via Liapunov's method
Newton's equation in one dimension
Local behavior near fixed points
Stability of linear systems
Stable and unstable manifolds
The Hartman-Grobman theorem
Appendix: Integral equations
Planar dynamical systems
The Poincaré-Bendixson theorem
Examples from ecology
Examples from electrical engineering
Higher dimensional dynamical systems
Attracting sets
The Lorenz equation
Hamiltonian mechanics
Completely integrable Hamiltonian systems
The Kepler problem
The KAM theorem
Part 3: Chaos
Discrete dynamical systems
The logistic equation
Fixed and periodic points
Linear difference equations
Local behavior near fixed points
Discrete dynamical systems in one dimension
Period doubling
Sarkovskii's theorem
On the definition of chaos
Cantor sets and the tent map
Symbolic dynamics
Strange attractors/repellors and fractal sets
Homoclinic orbits as source for chaos
Periodic solutions
Stability of periodic solutions
The Poincare map
Stable and unstable manifolds
Melnikov's method for autonomous perturbations
Melnikov's method for nonautonomous perturbations
Chaos in higher dimensional systems
The Smale horseshoe
The Smale-Birkhoff homoclinic theorem
Melnikov's method for homoclinic orbits
Bibliography
Glossary of notations
Index
Copyright ? 2001-2008 by Gerald Teschl. Last modified 09-06-2008. |