Dynamical estimates of chaotic systems from Poincaré recurrences

We show a function that fits well the probability density of return times between two consecutive visits of a chaotic trajectory to finite size regions in phase space. It deviates from the exponential statistics by a small power-law term, a term that represents the deterministic manifestation of the dynamics. We also show how one can quickly and easily estimate the Kolmogorov–Sinai entropy and the short-term correlation function by realizing observations of high probable returns. Our analyses are performed numerically in the Hénon map and experimentally in a Chua’s circuit. Finally, we discuss how our approach can be used to treat the data coming from experimental complex systems and for technological applications.