An order parameter for synchronisation of angular velocities.

Order parameters have proved a vital tool for simplifying and understanding complex dynamics in a range of physical systems. However, applying these approaches to the high-dimensional time-dependent variability inherent to thermodynamically open systems-such as those found in neural networks and climate dynamics-remains a challenge. We introduce a novel order parameter based on alignment of component frequencies, in contrast to the widely used Kuramoto order parameter's alignment of process phases. We present numerical simulations comparing this new parameter to the Kuramoto order parameter in a range of models, including a prototypical phase-oscillator model relevant to many open systems, such as neuronal dynamics. These results show that the new parameter more accurately identifies synchronisation in conditions characteristic of open systems, revealing dynamics entirely missed by established methods. Supplementary Information: The online version contains supplementary material available at https://doi.org/10.1140/epjs/s11734-025-01984-3.