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 ？-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. 1. Introduction Statistical complexity is a measure that had been introduced by Crutchfield and Young in 1989 [1]. It has been proven useful for describing various complex systems, including those with hundreds of degrees of freedom [2]. According to our earlier paper [3], the statistical complexity of high-dimensional trajectories generated by the dynamics of an ensemble of water molecules grows up to the time scale of 1 microsecond, that is, an extremely long-time interval for a typical molecular dynamics simulation. Moreover, this property is much less pronounced for so-called surrogate time series that have exactly the same power spectrum and, hence, autocorrelation function as the original time series. For example, in Figure 1 we plot the dependence of statistical complexity on the length of the time series for the symbolic data obtained from a Poincaré the section of 3D velocities describing the motion of a hydrogen atom in an ensemble of 392 water molecules [3]. The details of computing the atomic trajectories as well as the method used for partitioning the phase space and obtaining a symbolic string from the initially floating point data can be found in [4]. In the same figure, we draw the curves calculated for so-called phase-shuffled surrogate time series [5], the data having identical autocorrelation function, and hence power spectrum as the original velocity trajectories. One can notice significant differences between the statistical complexity of the physical and the artificially generated data. Figure 1: Statistical complexity versus the (log of)