  We propose a thermodynamically consistent minimal model to study
synchronization which is made of driven and interacting three-state units. This
system exhibits at the mean-field level two bifurcations separating three
dynamical phases: a single stable fixed point, a stable limit cycle indicative
of synchronization, and multiple stable fixed points. These complex emergent
dynamical behaviors are understood at the level of the underlying linear
Markovian dynamics in terms of metastability, i.e. the appearance of gaps in
the upper real part of the spectrum of the Markov generator. Stochastic
thermodynamics is used to study the dissipated work across dynamical phases as
well as across scales. This dissipated work is found to be reduced by the
attractive interactions between the units and to nontrivially depend on the
system size. When operating as a work-to-work converter, we find that the
maximum power output is achieved far-from-equilibrium in the synchronization
regime and that the efficiency at maximum power is surprisingly close to the
linear regime prediction. Our work shows the way towards building a
thermodynamics of nonequilibrium phase transitions in conjunction to
bifurcation theory.
