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Abstract

Identifying complex periodic windows surrounded by chaos in the two or higher dimensional parameter space of certain dynamical systems is
a challenging task for time series analysis based on complex network approaches. This holds particularly true for the case of shrimp structures,
where different bifurcations occur when crossing different domain boundaries. The corresponding dynamics often exhibit either period-
doubling when crossing the inner boundaries or, respectively, intermittency for outer boundaries. Numerically characterizing especially the
period-doubling route to chaos is difficult for most existing complex network based time series analysis approaches. Here, we propose to use
ordinal pattern transition networks (OPTNs) to characterize shrimp structures, making use of the fact that the transition behavior between
ordinal patterns encodes additional dynamical information that is not captured by traditional ordinal measures such as permutation entropy.
In particular, we compare three measures based on ordinal patterns: traditional permutation entropy εO, average amplitude fluctuations of
ordinal patterns 〈σ 〉, and OPTN out-link transition entropy εE. Our results demonstrate that among those three measures, εE performs best in
distinguishing chaotic from periodic time series in terms of classification accuracy. Therefore, we conclude that transition frequencies between
ordinal patterns encoded in the OPTN link weights provide complementary perspectives going beyond traditional methods of ordinal time
series analysis that are solely based on pattern occurrence frequencies.