Forbidden and missing ordinal patterns in noisy chaotic time series

DARÍO G. PÉREZ; FELIPE OLIVARES; MIGUEL C. SORIANO; LUCIANO ZUNINO

Madrid

Conferencia; The 15th Experimental Chaos and Complexity Conference; 2018

Universidad Rey Juan Carlos

For deterministic one-dimensional maps, Amigó et al. Have conclusively demonstrated that not all possible ordinal patterns can effectively materialize into orbits, which in a sense makes these patterns forbidden. On the other hand, for stochastic processes the probability of observing individual patterns depends not only on the time series length but also on the correlations? structure. Then, the existence of a non-observed pattern does not qualify it as forbidden but only as missing. A similar observation also holds for the case of real data that always possess a stochastic component due to the omnipresence of observational noise. Thus, the existence of missing patterns could be either related to stochastic processes (correlated or uncorrelated) or to a deterministic nature. In this context, some open questions hold, how do forbidden/missing patterns behave in continuous systems, where the sampling time plays a crucial role? Which is the optimal sampling time to not over or under estimate the number of forbidden patterns? In order to answer these questions, we have initially analyzed the existence of forbidden/missing patterns from the well-known Mackey-Glass equation, operating in a high-dimensional chaotic regime contaminated with different amounts of observational noise. We found that in a noisy environment the number of forbidden patterns related to the ideal free noise chaotic dynamics goes to zero. We showed the decay of missing patterns as a function of the time series length for different samplings, i.e. Missing Patterns Spectrum, permits us to distinguish at what time scales the deterministic and stochastic features dominate the dynamics. On the other hand, we showed that the number of missing patterns as a function of the sampling time for a fixed length, reaches a maximum at a given time scale, which is a necessary condition for the presence of an intrinsic chaotic nature. Based on experiments of a paradigmatic opto-electronic oscillator, we demonstrated our previous hypotheses. In addition, numerical simulations of our experiment support our results.