# phase locking of coupled oscillators

Consider two harmonic oscillators coupled like this: $$\frac{d\theta_1}{dt} = \omega_1+C_{12}\sin(\theta_2-\theta_1), \frac{d\theta_2}{dt} = \omega_2+C_{21}\sin(\theta_1-\theta_2).$$ Show that large $$C_{12} \text{ and } C_{21}$$ help achieving phase-locking.

My attempt: Consider $$\phi = \theta_1-\theta_2$$, then $$\frac{d\phi}{dt}=\frac{d\theta_1}{dt}-\frac{d\theta_2}{dt}=\omega_1+C_{12}\sin(\theta_2-\theta_1)-\omega_2-C_{21}\sin(\theta_1-\theta_2)=\omega_2-\omega_1+(C_{12}+C_{21})\sin(\theta_2-\theta_1).$$ Then the time when two oscillators phase lock is $$\frac{2\pi}{\omega_2-\omega_1+(C_{12}+C_{21})\sin(\theta_2-\theta_1)}$$

How do I show that large $$C_{12} \text{ and } C_{21}$$ help achieving phase-locking?

If in $$\dot \phi=-\Delta ω+C\sin\phi$$ $$C$$ is large enough so that $$|Δω|, then there are infinitely many periodically spaced stationary points at $$\sin ϕ_*=\frac{Δω}C$$. The solution for $$ϕ$$ is then bound between two of them, monotonically moving toward one of the boundaries. So with time $$ϕ=θ_1−θ_2$$ becomes almost constant.