In: Recurrence Plots and Their Quantifications: Expanding Horizons, Eds.: C. L. Webber, Jr. and C. Ioana and N. Marwan, Springer, Cham, 137–153p. (2016) DOI:10.1007/978-3-319-29922-8_7

Splayed Recurrence Analysis of Iterated Dynamical Systems

C. L. Webber, Jr.

Splayed Recurrence Analysis (SRA) is a new method for identifying and quantifying recurrent events in iterated systems. The technique is fully applicable to difference equations, Poincar´ sections of continuous time series, and independent random events. Inspiration for SRA comes from American roulette wheel gaming. It has been postulated that non-random wheel determinism is introduced by unbalanced wheels (mechanical) and non-random repeated motions of house spinners (human). Primary data were taken from actual roulette outcomes in which ball landing slots were reported sequentially according to spin orders. These data were stored in a matrix [slot #, spin #] and lines were passed through all possible pairs of points in the matrix and extrapolated to the border. Centers of points falling exactly on these extended lines, including the initial pair, were scored as recurrent points. Necessarily, there were gaps between points which led to point-to-point intervals being splayed-out. Six variables were extracted from the recurrent points comprising lines: (1) number of recurrent points per line; (2) intervals between recurrent points; (3) lengths of lines; (4) slopes of lines; (5) entropy of line lengths; (6) density of recurrent points. Besides the American roulette data, these SRA strategies were also applied to natural random numbers, chaotic models, and natural phenomenon. No differences could be detected for roulette data and naturally occurring random processes. But SRA was able to detect non-random structures in mathematically chaotic systems as well as in eruption times of the Old Faithful geyser. Because the methodology does not depend upon embeddings and delays as required for nonlinear analyses, SRA is classified as fully linear.

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