Measuring synchronization in coupled model systems: A comparison of different approaches.

Synchronization between coupled chaotic systems has been a topic of increasing interest since the early 1990s (for an overview cf. Ref. [1]). Since synchronization phenomena can manifest themselves in many different ways, various concepts for its description have been offered (e.g., identical, phase and generalized synchronization). Corresponding to and extending this variety of concepts, many different approaches have been proposed aiming at a quantification of the degree of synchronization between two systems. However, in the literature on the quantification of synchronization almost exclusively one single measure is applied either to model systems or to real data. Only rarely different measures are used to analyze the same system and thus a comprehensive comparison of all these different approaches is still missing. Furthermore, investigations on model systems are typically carried out by analyzing long and noise free time series (typically at least in the order of 10^5 data points). Therefore it is far more challenging to investigate the dependence between coupled model systems by applying bivariate measures to short artificial time series (in the order of 10^3 data points) with or without additive noise. The aim of this study [2] is to evaluate whether the analysis of model systems can contribute useful information in order to decide which measure of synchronization is most suitable for an application to field data. To this aim ten different measures of synchronization are employed to analyze short bivariate time series generated from three coupled model systems with different individual properties (e.g., power spectra, dimension). Model systems comprise coupled Hénon maps and coupled Rössler and Lorenz systems. Measures of synchronization include symmetric ones like linear cross correlation, mutual information [3] and different indices of phase synchronization (with the phases extracted either using Hilbert or wavelet transform) [4,5] as well as asymmetric ones like transfer entropy [6], two related state space approaches quantifying non-linear interdependencies [7] and event synchronization [8]. As a means of validating the dependence on the coupling strength the maximum Lyapunov exponent was employed. For each measure the synchronization between the first components of three unidirectionally coupled model systems was calculated for monotonously increasing coupling strengths. To compare the different measures in their capability to distinguish between different degrees of coupling for every model system a degree of ordering M is defined to yield its maximum value 1 for a measure for which higher values of coupling strength necessarily lead to higher values of synchronization. Furthermore, the robustness against noise was evaluated by means of the noise-to-signal ratio NSR ?noise /?signal. For each measure and every system the critical noise-to-signal ratio NSRC was defined as the noise-to-signal ratio for which the order M for the first time falls below times the value obtained for the case without noise. Finally, to investigate to which extent the different measures of synchronization carry independent and non-redundant information, both correlation and cluster analyses were performed. Based on the results from all three systems contaminated with white noise first the correlation coefficients were determined. Furthermore, we applied the recently introduced mutual information clustering (MIC) [9].