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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