Uncovering interaction of coupled oscillators from data

We develop a technique for reconstructing the phase dynamics equations for weakly coupled oscillators from data. We show how, starting from general scalar observables, one can first reconstruct the dynamics in terms of the corresponding protophases, and then, performing a transformation to the genuine, observable-independent phases, obtain an invariant description of the phase dynamics. We demonstrate that natural frequencies of oscillators can be recovered if several observations of coupled systems at different coupling strengths are available. We apply our theory to numerical examples and to a physical experiment with coupled metronomes.