Adler phase equation ($\dot{\theta }=\omega -ϵ\sin \theta$)

The relative-phase drift $\dot{\theta }=\omega -ϵ\sin \theta$ describes competing frequency mismatch and phase coupling. For $0<\omega <ϵ$, the equilibrium points are $\arcsin (\omega /ϵ)$ and $\pi -\arcsin (\omega /ϵ)$, stable and unstable respectively. They merge in a saddle-node bifurcation at $\omega =ϵ$; greater mismatch gives running phase dynamics. Adding Gaussian white noise gives Overdamped Langevin dynamics in a tilted washboard potential.

For $0<\omega <ϵ$, set $\alpha =\arcsin (\omega /ϵ)$ and $\kappa =\sqrt{{ϵ}^2-{\omega }^2}$. The neighboring saddles of the tilted washboard potential have barriers $\Delta V_{+}=2\kappa -\omega (\pi -2\alpha )$ and $\Delta V_{-}=2\kappa +\omega (\pi +2\alpha )$ above the minimum at $\alpha$. Their difference is $2\pi \omega$; both have curvature $-\kappa$, so forward and backward Kramers escape rates have equal leading prefactors.

A phase that continues winding rather than tending to a locked value. In the noise-free Adler phase equation, $\omega >ϵ>0$ makes the drift positive at every phase, so there are no equilibrium points. The associated tilted washboard potential decreases monotonically; wells vanish at the locking threshold.