The Kuramoto model has a second-order phase transition

up:: Kuramoto Model

(Strogatz)

The Kuramoto model has a Second-Order Phase Transition with respect to its coupling parameter $K$.

This means that, below some critical coupling threshold $K_c$, there can be no synchronization ($R=0$), no matter how long one waits or how many oscillators there are.

Only past this threshold can synchronization begin to happen (asymptotically in $t$ and $N$).

Derivation of threshold

One can find this critical threshold solving a self-consistent equation for the Kuramoto Model Order Parameter $R$, of the form

$$R=KR{\int }_{-\frac{\pi }{2}}^{\pi /2}{\cos }^2\theta g(\lambda R\sin \theta )d\theta$$

By making $R\to 0^{+}$¹, one finds the critical coupling

$$R_c=\frac{2}{\pi g(0)}$$

Derivation of second-order transition

Expanding $g(KR\sin \theta )$ around $0$ up to second order², and since $g^{\prime \prime }(0)=-\frac{2}{\pi }$, we have

$$R\sim \sqrt{\frac{8(\lambda -{\lambda }_c)}{{\lambda }_c^3}}\sim (\lambda -{\lambda }_c{)}^{1/2}$$

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References

• KURAMOTO, Y. Self-entrainment of a population of coupled non-linear oscillators. (H. Araki, Ed.), 1975.
• STROGATZ, S. H. Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering. 3. ed. CRC Press, 2019.
• 2011 Simons Lectures - Steven Strogatz, Coupled Oscillators That Synchronize Themselves - YouTube

Footnotes

1. And implicitly assuming continuity of $g$. ↩

2. Assuming $g(\omega )$ follows a Lorentzian distribution $g(\omega )=\frac{1}{\pi (1+{\omega }^2)}$ ($\gamma =1$), as (Kuramoto, 1975). ↩