 Phase locking

Phase locking maintains a fixed relation between the phases of oscillations, or between an oscillation and periodic forcing. For a one-to-one locked pair, ${\theta }_2-{\theta }_1$ is constant; more general locking can fix an integer combination $m{\theta }_1-n{\theta }_2$. Locking concerns the relative phases, so the individual oscillations can continue indefinitely.

For two equally coupled phase oscillators in the Kuramoto model, the difference satisfies $\dot{\delta }=\Delta \omega -K\sin \delta$. A stable locked value has $\sin \delta =\Delta \omega /K$ and $\cos \delta >0$ when $|\Delta \omega |<K$. The uncoupled frequencies need not coincide: coupling shifts both observed frequencies to the same value. Outside the locking range the difference keeps winding.

Spatial forcing can likewise pin the phase of a pattern. For the real-coefficient conjugately forced amplitude equationputting $A=\rho e^{i\varphi }$ with $\rho >0$ gives $\dot{\rho }=(\mu +\beta \cos 2\varphi -\lambda {\rho }^2)\rho$ and $\dot{\varphi }=-\beta \sin 2\varphi$. The phase-stable values are $0,\pi$ for $\beta >0$ and $\pi /2,3\pi /2$ for $\beta <0$. A physical nonzero branch also needs a positive squared amplitude $(\mu +\beta \cos 2\varphi )/\lambda$ when $\lambda \ne 0$; radial stability is a separate check. At $\beta =0$ the phase is neutral and this pinning disappears.