Equivalence of coupled networks and networks with multimodal frequency distributions: Conditions for the bimodal and trimodal case

Populations of oscillators can display a variety of synchronization patterns depending on the oscillators' intrinsic coupling and the coupling between them. We consider two coupled symmetric (sub)populations with unimodal frequency distributions. If internal and external coupling strengths are identical, a change of variables transforms the system into a single population of oscillators whose natural frequencies are bimodally distributed. Otherwise an additional bifurcation parameter κ enters the dynamics. By using the Ott-Antonsen ansatz, we rigorously prove that κ does not lead to new bifurcations, but that a symmetric two-coupled-population network and a network with a symmetric bimodal frequency distribution are topologically equivalent. Seeking for generalizations, we further analyze a symmetric trimodal network vis-à-vis three coupled symmetric unimodal populations. Here, however, the equivalence with respect to stability, dynamics, and bifurcations of the two systems no longer holds.