Bifurcations and Singularities for Coupled Oscillators with Inertia and Frustration.

We prove that any nonzero inertia, however small, is able to change the nature of the synchronization transition in Kuramoto-like models, either from continuous to discontinuous or from discontinuous to continuous. This result is obtained through an unstable manifold expansion in the spirit of Crawford, which features singularities in the vicinity of the bifurcation. Far from being unwanted artifacts, these singularities actually control the qualitative behavior of the system. Our numerical tests fully support this picture.