Abstract
Synchronization is a widespread phenomenon that occurs among interacting oscillatory systems. It facilitates their temporal coordination and can lead to the emergence of spontaneous order. The detection of synchronization from the time series of such systems is of great importance for the understanding and prediction of their dynamics, and several methods for doing so have been introduced. However, the common case where the interacting systems have time-variable characteristic frequencies and coupling parameters, and may also be subject to continuous external perturbation and noise, still presents a major challenge. Here we apply recent developments in dynamical Bayesian inference to tackle these problems. In particular, we discuss how to detect phase slips and the existence of deterministic coupling from measured data, and we unify the concepts of phase synchronization and general synchronization. Starting from phase or state observables, we present methods for the detection of both phase and generalized synchronization. The consistency and equivalence of phase and generalized synchronization are further demonstrated, by the analysis of time series from analog electronic simulations of coupled nonautonomous van der Pol oscillators. We demonstrate that the detection methods work equally well on numerically simulated chaotic systems. In all the cases considered, we show that dynamical Bayesian inference can clearly identify noise-induced phase slips and distinguish coherence from intrinsic coupling-induced synchronization.
Highlights
Synchronization emerges from the interactions between oscillatory systems
We have demonstrated how phase synchronization (PS) and generalized synchronization (GS) can be detected by the use of Bayesian dynamical inference
The evaluation was based on the core definitions of PS and GS, and it led to inference of parameter values in the models on which the procedure was being demonstrated
Summary
Synchronization emerges from the interactions between oscillatory systems. It is a ubiquitous phenomenon that can involve either two or many systems, and it leads to the onset of coordinated dynamics and spontaneous order [1,2], with examples ranging from the synchronization of fireflies [3], through the cardiorespiratory system [4], to electrochemical oscillators [5]. GS is assessed through use of the state signals of the interacting systems: different methods have been developed based on nearest-neighbor mapping [7], mutual information [14], generalized angle [15], and statistical modeling [16]. The recently developed method for studying interoscillator interactions based on Bayesian dynamical inference [18,19] is especially well suited to the treatment of systems with time-varying parameters, including, in particular, those that are chronotaxic. Bayesian-based methods are readily applicable to the case of chaotic dynamical systems We illustrate the latter case by demonstrating satisfactory detection of PS and GS between coupled Rössler and Lorenz chaotic systems, with time-varying parameters and subject to noise. The Appendix describes succinctly how the measurement noise can be treated within the same inferential framework
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