Fig. 2: Validation of an empirical measure of turbulence in a ring of coupled oscillators.

The figure demonstrated the suitability of different measures for capturing turbulence by using the same four points, where the first (D = 0.0011) is deep in the oscillatory regime, the second is at the maximum of turbulence (D = 0.0524), the third is in the turbulent region but close to the edge of the noisy regime (D = 0.1671), while the fourth point is in the noisy regime at the edge of turbulence (D = 0.1805). a The mean value across time of the global Kuramoto order parameter is not sensitive as shown from the boxplots (across 100 trials). b Equally, global metastability is not able to capture turbulence. c In contrast, spatiotemporal metastability at the fine spatial scale detect turbulence perfectly. d However, using this measure at the coarse spatial scale fails to significantly detect turbulence. e Examples of the spacetime evolution in specific trials of the simulation of R, where the two middle panels clearly show turbulence, reflecting the vorticity of the local synchronisation. f Spacetime evolution snapshots of the phases. g The measure of edge metastability perfectly captures the turbulent regime using only coarse spatial observations, as shown in detail in the magnified inset showing a close up of the last three boxplots across the 100 trials. h The measure of Edge Spacetime Predictability (ESP) characterises the information transfer correlation across space and time and shows significant differences between the turbulent and noisy regimes. Furthermore, the inset shows the correlation in the turbulent regime between the fine spacetime metastability and the coarse ESP. i Examples of the evolution in time of the edge metastability for one trial at each of the four timepoints. The error bars depict the standard error.