Phase oscillator

A phase oscillator retains an oscillation's position within its cycle while suppressing its amplitude dynamics. Its simplest equation is $\dot{\theta }=\omega$, with phase understood modulo $2\pi$. This is useful when amplitude disturbances relax rapidly compared with the weak interactions being studied, leaving slow changes of relative phase as the important motion.

For two coupled oscillators, a reduced phase-difference equation can have a fixed point corresponding to phase locking. The Adler phase equation supplies a basic example. A locked phase difference permits the pair to share a mean frequency even when their uncoupled frequencies differ. If forcing is strong enough to change the oscillation's shape or amplitude substantially, the reduction needs further justification. Noise-driven phase slips likewise distinguish average synchronization from a perfectly fixed relative phase at all times.

A phase slip changes the continuously tracked relative phase by one cycle, usually $\pm 2\pi$, compared with a locked state. Tracking the angle only modulo $2\pi$ hides this accumulated winding: the wrapped phase can return to the same value after a full circuit. A jump caused solely by changing the plotting branch cut is not evidence of a dynamical phase slip.

In a noisy locked oscillator, fluctuations usually remain close to a stable phase, but a thermally activated phase slip crosses a barrier of the tilted washboard potential and settles in a neighboring well on the real-valued phase lift. The corresponding wrapped state can look unchanged even though the relative cycle count has increased or decreased by one.

If forward and backward slips have rare-event rates $r_{+}$ and $r_{-}$, their mean contribution to unwrapped phase drift is $2\pi (r_{+}-r_{-})$. This follows by multiplying the signed change per event by the expected event count per unit time. Equal rates can therefore give zero mean drift while individual slips still occur. The forward-backward bias of phase slips determines when a frequency mismatch makes one direction more likely.

A thermally activated phase slip is a noise-driven crossing between adjacent phase-locked wells. In the mobility-one Langevin equation$T_{\mathrm{eff}}$ has the units of the phase potential $V$. A barrier of height $\Delta V$ is rarely crossed when $\Delta V\gg T_{\mathrm{eff}}$.

For a smooth nondegenerate minimum $a$ and neighboring saddle $b$, the overdamped Kramers escape rate is asymptoticallyThe mean waiting time is approximately $1/r$, whereas local relaxation within the well occurs on the scale $1/V^{\prime \prime }(a)$. An intrawell phase autocorrelation approximation is useful only on timescales where escape remains rare; the barrier height alone does not specify this hierarchy.

For the Adler phase equation, the activation barriers of the Adler phase equation differ by $2\pi \omega$ in the two winding directions, while the leading curvature prefactors agree. Their rate ratio is therefore $r_{+}/r_{-}=e^{2\pi \omega /T_{\mathrm{eff}}}$ in the low-noise approximation. The barrier-crossing description fails at the deterministic locking threshold, where the minimum and saddle merge and the assumed nonzero curvatures vanish.

In a mobility-one noisy Adler phase equation, matching Kramers escape rate prefactors and a barrier difference $2\pi \omega$ give $p_{+}/p_{-}=e^{2\pi \omega /T_{\mathrm{eff}}}$. Positive mismatch favors forward winding. The ratio is a low-noise barrier-crossing result, not a claim that the particle remains in one well at arbitrarily long times.

The Adler phase equationdescribes a relative phase pulled by a coupling of strength $ϵ$ while a frequency mismatch $\omega$ drives it around the circle. For $|\omega |<ϵ$, an equilibrium satisfies $\sin \theta =\omega /ϵ$. Its linear growth rate is $-ϵ\cos \theta$, so the branch with positive cosine is stable and the opposite branch is unstable.

For $0<\omega <ϵ$, the stable and unstable values are $\arcsin (\omega /ϵ)$ and $\pi -\arcsin (\omega /ϵ)$ modulo $2\pi$. They merge at $\omega =ϵ$, a saddle-node bifurcation on the phase circle. The stable branch's relaxation rate is $\sqrt{{ϵ}^2-{\omega }^2}$, which tends to zero at the threshold. Greater positive mismatch gives running phase dynamics.

The drift is minus the derivative of the tilted washboard potential $V(\theta )=-\omega \theta -ϵ\cos \theta$. Although $V$ is not periodic, its derivative is; each increase of $2\pi$ lowers the potential by $2\pi \omega$. Adding Gaussian white noise produces overdamped Langevin dynamics. The activation barriers of the Adler phase equation then control rare thermally activated phase slips.

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.

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.