Resumen de Numerical computation of invariant objects with wavelets
David Romero Sanchez
In certain classes of dynamical systems invariant sets with a strange geometry appear. For example, under certain conditions, the iteration of two-dimensional quasi-periodically forced skew product $(\theta_{n+1}, x_{n+1})=(\Rot{\omega}{}(\theta_n),F_{\sigma,\varepsilon}(\theta_n, x_n))$ where $\Rot{\omega}{}(\theta)=\theta+\omega$, $\omega\in\R\backslash\Q$ and $\sigma,\varepsilon\in\R$ gives us Strange Non-Chaotic Attractors, $\varphi$.
To obtain analytical approximation of these objects it seems more natural to use wavelets instead of the more usual Fourier approach due to its adaptability. The aim of this thesis is to describe an efficient algorithm for the semi-analitical computation of the invariant object, using both Daubechies and Haar wavelets, by means of the numerical computation of the wavelet coefficients.
The aim for this exercise is twofold. From one side to be able to study possible bifurcations or zoom in the "pinching zone" of the object. From the other side try to get estimates of the regularity of the object. The study of this regularity depending on parameters, for a certain models of skew products, may give another point of view to the fractalization routes described in the literature and that are currently under discussion.
To perform such exercise(s), firstly, we have translated the $\R$-Daubechies wavelets language to $\SI$. After that, we have carried out two different strategies to get the wavelet coefficients $\PER{\lD}$. The first one based on the Fast Wavelet Transform. The other, solve the Invariance Equation $ \varphi(\Rot{\omega}{}(\theta)) = F_{\sigma,\varepsilon}(\theta,\varphi(\theta)) $ using the Newton's method. From such coefficients $\PER{\lD}$ we get (numerical) estimations for the aforesaid proposed questions.
