  We consider a variant of the Kuramoto model, in which all the oscillators are
now assumed to have the same natural frequency, but some of them are negatively
coupled to the mean field. These "contrarian" oscillators tend to align in
antiphase with the mean field, whereas the positively coupled "conformist"
oscillators favor an in-phase relationship. The interplay between these effects
can lead to rich dynamics. In addition to a splitting of the population into
two diametrically opposed factions, the system can also display traveling
waves, complete incoherence, and a blurred version of the two-faction state.
Exact solutions for these states and their bifurcations are obtained by means
of the Watanabe-Strogatz transformation and the Ott-Antonsen ansatz. Curiously,
this system of oscillators with identical frequencies turns out to exhibit more
complicated dynamics than its counterpart with heterogeneous natural
frequencies.
