  We study the noisy Kuramoto model for two interacting communities of
oscillators, where we allow the interaction in and between communities to be
positive or negative (but not zero). We find that, in the thermodynamic limit
where the size of the two communities tends to infinity, this model exhibits
non-symmetric synchronized solutions that bifurcate from the symmetric
synchronized solution corresponding to the one-community noisy Kuramoto model,
even in the case where the phase difference between the communities is zero and
the interaction strengths are symmetric. The solutions are given by fixed
points of a dynamical system. We find a critical condition for existence of a
bifurcation line, as well as a pair of equations determining the bifurcation
line as a function of the interaction strengths. Using the latter we are able
to classify the types of solutions that are possible and thereby identify the
phase diagram of the system. We also analyze properties of the bifurcation line
in the phase diagram and its derivatives, calculate the asymptotics, and
analyze the synchronization level on the bifurcation line. Part of the proofs
are numerically assisted. Lastly, we present some simulations illustrating the
stability of the various solutions as well as the possible transitions between
these solutions.
