  We analyze the synchronization dynamics of the thermodynamically large
systems of globally coupled phase oscillators under Cauchy noise forcings with
bimodal distribution of frequencies and asymmetry between two distribution
components. The systems with the Cauchy noise admit the application of the
Ott-Antonsen ansatz, which has allowed us to study analytically synchronization
transitions both in the symmetric and asymmetric cases. The dynamics and the
transitions between various synchronous and asynchronous regimes are shown to
be very sensitive to the asymmetry degree whereas the scenario of the symmetry
breaking is universal and does not depend on the particular way to introduce
asymmetry, be it the unequal populations of modes in bimodal distribution, the
phase delay of the Kuramoto-Sakaguchi model, the different values of the
coupling constants, or the unequal noise levels in two modes. In particular, we
found that even small asymmetry may stabilize the stationary partially
synchronized state, and this may happen even for arbitrarily large frequency
difference between two distribution modes (oscillator subgroups). This effect
also results in the new type of bistability between two stationary partially
synchronized states: one with large level of global synchronization and
synchronization parity between two subgroups and another with lower
synchronization where the one subgroup is dominant, having higher internal
(subgroup) synchronization level and enforcing its oscillation frequency on the
second subgroup. For the four asymmetry types, the critical values of asymmetry
parameters were found analytically above which the bistability between
incoherent and partially synchronized states is no longer possible.
