  The Kuramoto model exhibits different types of synchronization transitions
depending on the type of natural frequency distribution. To obtain these
results, the Kuramoto self-consistency equation (SCE) approach has been used
successfully. However, this approach affords only limited understanding of more
detailed properties such as the stability and finite size effect. Here, we
extend the SCE approach by introducing an effective potential, that is, an
integral version of the SCE. We examine the landscape of this effective
potential for second-order, first-order, and hybrid synchronization transitions
in the thermodynamic limit. In particular, for the hybrid transition, we find
that the minimum of effective potential displays a plateau across the region in
which the order parameter jumps. This result suggests that the effective free
energy can be used to determine a type of synchronization transition. For
finite systems, the effective potential contains local minima at which the
system can be trapped. Using numerical simulations, we determine the stability
of the system as a function of system size and simulation time.
