  We derive a concise and general expression of the energy dissipation rate for
coupled oscillators rotating on circular trajectories by unifying the
nonequilibrium aspects with the nonlinear dynamics via stochastic
thermodynamics. In the framework of phase oscillator models, it is known that
the even and odd parts of the coupling function express the effect on
collective and relative dynamics, respectively. We reveal that the odd part
always decreases the dissipation upon synchronization, while the even part
yields a characteristic square-root change of the dissipation near the
bifurcation point whose sign depends on the specific system parameters. We
apply our theory to hydrodynamically coupled Stokes spheres rotating on
circular trajectories that can be interpreted as a simple model of
synchronization of coupled oscillators in a biophysical system. We show that
the coupled Stokes spheres gain the ability to do more work on the surrounding
fluid as the degree of phase synchronization increases.
