  Synchronization in a frequency-weighted Kuramoto model with a uniform
frequency distribution is studied. We plot the bifurcation diagram and identify
the asymptotic coherent states. Numerical simulations show that the system
undergoes two first-order transitions in both the forward and backward
directions. Apart from the trivial phase-locked state, a novel nonstationary
coherent state, i.e., a nontrivial standing wave state is observed and
characterized. In this state, oscillators inside the coherent clusters are not
frequency-locked as they would be in the usual standing wave state. Instead,
their average frequencies are locked to a constant. The critical coupling
strength from the incoherent state to the nontrivial standing wave state can be
obtained by performing linear stability analysis. The theoretical results are
supported by the numerical simulations.
