Transport phenomena n chaotic dynamical systems: classical ratchets and intermittent dynamics.

Dynamical systems exhibit an extremely rich variety of behaviors with regards to transport properties. A full understanding of how dynamics precisely determines the nature of transport is still not fully accomplished and there is a wide set of systems for which such properties are quite subtle. The subject of the first part of this Thesis is the ratchet effect, namely the generation of transport with a preferred direction in systems without a net driving force or even against a small applied bias. This property has recently attracted much attention but, while stochastic ratchets are rather well understood, purely deterministic ratchets are much more tricky. The system I have worked on is a family of low dimensional Hamiltonian maps, defined on a cylindrical phase space. The second part of the Thesis is about weakly chaotic systems and their dynamical properties. The phase space of typical area-preserving maps reveals the co-existence of chaotic trajectories and islands of regular motion. Even if the interest focuses on statistical properties of motion on the chaotic component, the influence of regular structures cannot be neglected because trajectories that approach them can be trapped for very long time, before being reinjected into the chaotic sea. The chaotic dynamics is thus interspersed by laminar segments where the system behaves as an integrable one. This behavior is called intermittency and it strongly influences the long-time properties of quantities like correlations decay or recurrence time statistics, which typically present a power-law tail, instead of the asymptotic exponential decay of fully hyperbolic systems. In unbounded systems intermittency affects also transport moments, generating anomalous diffusion processes, in contrast to the normal diffusion observed for fully hyperbolic systems. My work on intermittency is about a two-dimensional map whose transport moments display an anomalous behavior. The investigations of both parts of the Thesis is carried on through theoretical analysis and numerical simulations.