Recurrence plots for characterizing random dynamical systems

The recurrence plot was originally proposed for visualizing time series data. As recurrence plots have mainly been used for analyzing time series generated from nonlinear deterministic systems, it is not well known whether they can be applied to gain insight into analyzing time series generated from a random dynamical system, in which stochastic components play a central role. In this study, we demonstrate that a recurrence plot can provide new viewpoints for the stochasticity in the underlying dynamics. In particular, we present three theorems: the first theorem demonstrates that a recurrence plot can eventually establish one-to-one correspondence with a joint set of initial conditions and a series of stochastic inputs if the underlying dynamics is expansive and topologically transitive; the second theorem distinguishes deterministic and stochastic systems; and the third theorem enables the second theorem to be used for a shorter time series. Moreover, we propose a stochasticity test based on a recurrence plot. The theorems and stochasticity test are verified by numerical examples as well as real datasets. (C) 2020 The Author(s). Published by Elsevier B.V.