Transients and asymptotics in granular phase space

The dynamics of dissipative dynamical systems can be described by the sequential appearance of two different regimes. From a given initial condition, one first observes transient behavior characterized by a high degree of contraction of volumes in phase space. This is followed by an asymptotic regime with one or several attractors into which trajectories inject after long times. There is however, no sharp crossover between these two regimes and the identification of either one depends on the precision of measurement. In order to investigate these issues, we studied the dynamics of contracting integer maps. We found out that for the cases which in the continuum limit correspond to bifurcations, transients consists of two regimes sharply separated by a crossover point which displays universal scaling with the size of the set. Moreover, their average lengths display power law dependence on the accuracy of their measurement. This behavior persists away from bifurcation but with a different scaling law. In addition, we studied deterministic diffusion on finite sets and obtained analytic expressions for the mean square displacement in the long time limit.