Detecting regular dynamics from time series using permutations slopes

• Demonstration of the usefulness of the largest slopes for chaos detection.
• Detection of regular dynamics with zero entropy.
• Clear discernment between chaotic and regular dynamics.
• Confirmation of numerical estimation of the period of periodic dynamics.
• High robustness of the method against noise.

In this paper we present the entropy related to the largest slope of the permutation as an efficient approach for distinguishing between regular and non-regular dynamics, as well as the similarities between this method and the three-state test (3ST) algorithm. We theoretically establish that for suitably chosen delay times, permutations generated in the case of regular dynamics present the same largest slope if their order is greater than the period of the underlying orbit. This investigation helps making a clear decision (even in a noisy environment) in the detection of regular dynamics with large periods for which PE gives an arbitrary nonzero complexity measure.