Extracting order parameters from global measurements with application to coupled electrochemical oscillators

The calculation of order parameters to characterize the degree of synchronization usually requires the measurement of time series of observable quantities for all individual dynamical elements. In this article, a new method of extracting the Kuramoto order from global measurements for weakly coupled, nearly identical oscillators is established and successfully tested with electrochemical populations of smooth, relaxational and chaotic oscillators. An experimental study on a larger array of coupled smooth oscillators was performed and the results confirmed the theoretical predictions on size effects including enhanced fluctuations around the synchronization transition. The proposed method opens up the possibility for the study of synchronization in large populations of interacting oscillators in many fields including biology where measurements are limited to global signals and few local sites.