Finding recurrence in chaos: New quantifier helps predict repeating events

Anyone who has ever endured a stock market cycle, experienced déjà vu, or lived on a fault line has some understanding that recurrence lies at the heart of many chaotic processes. While it is often difficult to know precisely when and how something will recur, tools provide us a way to predict roughly when repeating events will happen. A paper on chaos theory and complex systems highlights a new theoretical approach to understanding — and potentially one day predicting — recurring events using complex data sets.

Drawing on concepts of information entropy, the authors present a new quantifier for better understanding recurrence in nonlinear time series. The approach uses a concept they have called microstates, which are used as small binary submatrices to determine the probabilities of events within a larger recurrence matrix.

Sergio Lopes, one of the paper’s authors, compares forming microstates from the specific interplay of complex data to forming words from alphabets. “If you limit the number of letters you use, or the combinations that are allowed, still many words can be created, but it loses a lot of complexity,” he said.

The method, which can be applied to discrete and continuous systems, uses small segments of data to yield results for short time series that are consistent with longer ones, a crucial feature for fields such as climatology and seismology, where data is often undersampled.

The group’s approach offers several advantages over traditional entropy quantifiers, including demanding smaller computational times to analyze recurrence matrices, good correlations with the maximum Lyapunov exponents, and a weak dependence on the vicinity threshold parameter.

Source: “Quantifying entropy using recurrence matrix microstates,” by Gilberto Corso, Thiago de Lima Prado, Gustavo Zampier dos Santos Lima, Juergen Kurths, and Sergio Roberto Lopes, Chaos (2018). The article can be accessed at https://doi.org/10.1063/1.5042026 .