Running phase dynamics

Running phase dynamics means that an unwrapped phase keeps winding rather than approaching a locked value. For the deterministic Adler phase equation, $\omega >ϵ>0$ makes $\dot{\theta }=\omega -ϵ\sin \theta$ positive everywhere. There are no equilibrium points, and one full winding takesFor the integral, shift the angle so that the denominator is $\omega -ϵ\cos \varphi$ and use $u=\tan (\varphi /2)$. Splitting at the tangent's pole combines the two pieces intowhich has the displayed value. Thus the average winding speed is $2\pi /T=\sqrt{{\omega }^2-{ϵ}^2}$.

If $\omega =ϵ+\delta$ with small positive $\delta$, the average speed is asymptotic to $\sqrt{2ϵ\delta }$. Motion spends a long time near the vanished locking equilibria before completing each winding. The tilted washboard potential is monotone in this regime; noise-driven barrier hopping and deterministic running describe different mechanisms.