Bachelor's degree final project  v1.0
Faculty of Mathematics, University of Barcelona
Approximations of invariant curves of diffeomorphisms
Author:
Joan Gimeno
Date:
Spring 2014

What is the utility of the current library ?

Assumes that we have a discret system like that:

\[ \begin{cases} \bar x = f(x, \theta) + \varepsilon g(x,\theta)\\ \bar \theta = \theta + \omega, \end{cases} \]

where $ f, g $ are differential maps with the second variable on the torus.

A continuous map $ x\colon \mathbb{T} \rightarrow \mathbb{R} $ is an invariant curve of the discret system if, and only if,

\[ x(\theta + \omega) = f(x(\theta), \theta) + \varepsilon g(x(\theta),\theta) \quad \text{ for all } \theta. \]

The value $ \omega $ is known as the rotation number of the curve and it is common to suppose that is a irrational real number.

The library compute an approximation of some invariant curve of the discret system.