Recurrence time statistics in deterministic and stochastic dynamical systems in continuous time: A comparison

Abstract

The dynamics of transitions between the cells of a finite phase-space partition is analyzed for deterministic and stochastic dynamical systems in continuous time. Special emphasis is placed on the dependence of mean recurrence time on the resolution \ensuremath{\tau} between successive observations, in the limit $\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\tau}}0.$ In deterministic systems the limit is found to exist, and to depend on only the intrinsic parameters of the underlying dynamics. In stochastic systems two different cases are identified, leading to a \ensuremath{\tau}-independent behavior and a ${\ensuremath{\tau}}^{1/2}$ behavior, depending on whether a finite speed of propagation of the signals exists or not. An extension of the results to the second moment of the recurrence time is outlined.