Permutation-information-theory approach to unveil delay dynamics from time-series analysis

In this paper an approach to identify delay phenomena from time series is developed. We show that it is possible to perform a reliable time delay identification by using quantifiers derived from information theory, more precisely, permutation entropy and permutation statistical complexity. These quantifiers show clear extrema when the embedding delay τ of the symbolic reconstruction matches the characteristic time delay τ(S) of the system. Numerical data originating from a time delay system based on the well-known Mackey-Glass equations operating in the chaotic regime were used as test beds. We show that our method is straightforward to apply and robust to additive observational and dynamical noise. Moreover, we find that the identification of the time delay is even more efficient in a noise environment. Our permutation approach is also able to recover the time delay in systems with low feedback rate or high nonlinearity.