Statistical Complexity of Low- and High-Dimensional Systems

We suggest a new method for the analysis of experimental time series that can distinguish high-dimensional dynamics from stochastic motion. It is based on the idea of statistical complexity, that is, the Shannon entropy of the so-called ϵ-machine (a Markov-type model of the observed time series). This approach has been recently demonstrated to be efficient for making a distinction between a molecular trajectory in water and noise. In this paper, we analyse the difference between chaos and noise using the Chirikov-Taylor standard map as an example in order to elucidate the basic mechanism that makes the value of complexity in deterministic systems high. In particular, we show that the value of statistical complexity is high for the case of chaos and attains zero value for the case of stochastic noise. We further study the Markov property of the data generated by the standard map to clarify the role of long-time memory in differentiating the cases of deterministic systems and stochastic motion.