Recurrence Times in Quasi-Periodic Motion: Statistical Properties,Role of Cell Size,Parameter Dependence

We investigate the probabilistic properties of recurrence times for the simplest form of aperiodic deterministic dynamics, quasi-periodic motion. Previous results using number theory techniques predict two fundamental recurrence times for uniform quasi-periodic motion on a two-dimensional torus, while no analogous analytic result seems to exist for higher dimensional tori. The two-dimensional uniform case is reanalyzed from a more geometric point of view and new, workable expressions are derived that enable us fully to understand and predict the recurrence phenomenon and to analyze its parameter dependence. Emphasis is placed on the statistical properties and, in particular, on the variability of recurrence times around their mean, in relation to local Farey tree structure. Higher-dimensional tori are considered, and seen to also display a high variability in their finite-time recurrence behavior. The results are finally extended to the non-uniform quasi-periodic case.