Phase locking

Phase locking is the maintenance of a fixed phase relation (possibly an integer combination of phases) between coupled oscillations, or between a pattern and imposed forcing. The following spatial-forcing example occurs in an amplitude equation. For a real-coefficient conjugately forced amplitude equation, $A=\rho e^{i\varphi }$ obeys ${\rho }_T=(\mu +\beta \cos 2\varphi -\lambda {\rho }^2)\rho$ and ${\varphi }_T=-\beta \sin 2\varphi$. Thus forcing selects stable phases $0,\pi$ for $\beta >0$ and $\pi /2,3\pi /2$ for $\beta <0$, when a nonzero amplitude exists. At $\beta =0$ this pinning disappears and the phase is neutral.