Resumen de Transport phenomena and anomalous diffusion in conservative systems of low dimension
Narcís Miguel Baños
The general framework of the dissertation is about the dynamics of chaotic orbits in 2D area-preserving maps (APM) and 3D volume-preserving maps (VPM). The focus are the statistical properties of chaotic orbits in the presence of regular motion (as opposed to chaotic).
The 2D study is centered in the Chirikov standard map for large values of the parameter ($k>2\pi$ in the usual representation). The main object of study are accelerator modes (AM). The existence of a (suitably scaled) limit behaviour when $k\to\infty$ is proved, and their effect in the diffusive properties in the momentum is studied by means of massive numerical simulations. We give explicit lower bounds for growth of the diffusion coefficients under the hypotheses suggested by the numerical results. Also the effect of a single Cantorus is studied, focusing mainly in the existence and position of stability islands and the applicability of the Greene-MacKay renormalisation theory. All numerical observations are related to the geometry and evolution of stability islands as the parameters of the systems are varied.
The first generalization to 3D VPM of the diffusive phenomena due to AM is also studied. A proper map of the 3-torus is constructed, in a way that AM appear. The Hopf-Saddle-Center bifurcation is proposed as mechanism for the creation of AM. This gives rise to a region of bounded motion analogous to stability islands for APM. A preliminary numerical study suggests strong analogies between the 2D and 3D cases.
