Recurrence plots bridge deterministic systems and stochastic systems topologically and measure-theoretically

Abstract

We connect a common conventional value to quantify a recurrence plot with its motifs, which have recently been termed “recurrence triangles.” The common practical value we focus on is DET, which is the ratio of the points forming diagonal line segments of length 2 or longer within a recurrence plot. As a topological value, we use different recurrence triangles defined previously. As a measure-theoretic value, we define the typical recurrence triangle frequency dimension, which generally fluctuates around 1 when the underlying dynamics are governed by deterministic chaos. By contrast, the dimension becomes higher than 1 for a purely stochastic system. Additionally, the typical recurrence triangle frequency dimension correlates most precisely with DET among the above quantities. Our results show that (i) the common practice of using DET could be partly theoretically supported using recurrence triangles, and (ii) the variety of recurrence triangles behaves more consistently for identifying the strength of stochasticity for the underlying dynamics. The results in this study should be useful in checking basic properties for modeling a given time series.