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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?

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1 Answer 1

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If in $$ \dot \phi=-\Delta ω+C\sin\phi $$ $C$ is large enough so that $|Δω|<C$, 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.

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