Fig. 1. Illustration of the emergent PDE approach for coupled oscillator systems.

a Initial condition of the Stuart-Landau ensemble, Eq. (1), colored with ascending imaginary part of W_k. b Trajectories obtained from integrating the initial conditions of (a) with the same color coding as in (a). The last snapshot is marked by black dots. c Zoom in to the upper part of (b), with the last snapshot marked by black dots. Above it, the last snapshot is color coded based on the ordering of the oscillators along the curve at that moment. d Zoom in on the top part of (b), but now with the new color coding. e Trajectories of the real part of the W_k, arranged by their initial values $\mathrm{Im}W$. f Trajectories of the real part of the W_k, arranged by the new color coding ϕ_i as in (d). (Finding ϕ_i is discussed in the text).