Kolmogorov–Sinai entropy from recurrence times

Observing how long a dynamical system takes to return to some state is one of the most simple ways to model and quantify its dynamics from data series. This work proposes two formulas to estimate the KS entropy and a lower bound of it, a sort of Shannon’s entropy per unit of time, from the recurrence times of chaotic systems. One formula provides the KS entropy and is more theoretically oriented since one has to measure also the low probable very long returns. The other provides a lower bound for the KS entropy and is more experimentally oriented since one has to measure only the high probable short returns. These formulas are a consequence of the fact that the series of returns do contain the same information of the trajectory that generated it. That suggests that recurrence times might be valuable when making models of complex systems. Recurrence times measure the time interval a system takes to return to a neighborhood of some state, being that it was previously in some other state. Among the many ways time recurrences can be defined, two approaches that have recently attracted much attention are the first Poincaré recurrence times (FPRs) [ 1] and the recurrence plots (RPs) [ 2]. While Poincaré recurrences refer to the sequence of time intervals between two successive visits of a trajectory (or a signal) to one particular interval (or a volume if the trajectory is high dimensional), a recurrence plot refers to a visualization of the values of a square array which indicates how much time it takes for two points in a trajectory with M points to become neighbors again. Both techniques provide similar results but are more appropriately applicable in different contexts. While the FPRs are more appropriated to obtain exact dynamical quantities (Lyapunov exponents, dimensions, and the correlation function) of dynamical systems [ 3], the RPs are more oriented to estimate relevant quantities and statistical characteristics of data coming from complex systems [ 4]. partially supported by the “Fundação para a Ciência e Tecnologia” (FCT). partially supported by SFB555. partially supported by the “Fundação para a Ciência e Tecnologia” (FCT). partially supported by SFB555.