Download Introduction the theory of persistence of quasiperiodic solutions

Introduction the theory of persistence of quasi­periodic solutions
(KAM theory).
Quasiperiodic functions are, roughly, functions which can be expressed
with a finite number of frequencies. They appear naturally in nature
when there are several independent processes each of them with a
natural frequency. They were considered since antiquity as models of
the motion of the planets.
The question we want to address is: If we find a quasi­periodic
solution as a solution of a differential equation, will similar
equations have one? In the case of the planets, it is clear that the
equations treating them separately have quasi=periodic solutions. It
is not at all clear whether the Newton equations that include an small
coupling will still have them.
In spite of being a desire since ancient times, the theory of
perturbations of these solutions was only understood in the 1950's and
1960's by the work of A. N. Kolmogorov, V. I. Arnold and J. K. Moser
(hence the acronym KAM theory). The theory is a beautiful blend of
analysis, geometry and even number theory. In recent years, this
results have gone from being just a beautiful mathematical theory to
being a practical computational tool.
The goal of this lectures is to present the background of the theory
and a few numerical implementations.
A good set of notes for the course is
ftp;ftp.ma.utexas.edu/pub/papers/llave/tutorial.pdf where you can
also find references to the original literature.
Besides the theoretical lectures, we will present implementations in
OCTAVE (a very simple to use, extensively documented, public domain
package very similar to Matlab). See www.octave.org. It works better
in GNU/Linux (Ubuntu, Debian, Mint, etc.) but it also works in
Windows, Mac.
Lecture 1: Background
Quasi­periodic functions. Analytic functions on the torus: Fourier series, Cauchy bounds
Number theory: Diophantine vectors. Area preserving maps of the annulus Lecture 2: Perturbative expansions and small divisors
Lecture 3: Moser's translated curve theorem for analytic maps. Lecture 4: Numerical work
Rafael de la Llave