# Power series expansion of the order parameter in the Kuramoto model

In this review of the Kuramoto model, Eq. 14 is obtained by expanding the following integral in powers of $$K r$$,

$$r = K r \int_{-\pi/2}^{\pi/2}\cos^2(\theta) g(K r \sin{\theta}) \mathrm{d}\theta$$ where $$g(\omega)$$ is some unknown function (though is later taken to be the p.d.f. of the Lorentz distribution $$g(\omega) = (\gamma/\pi)/(\gamma^2 + \omega^2)$$, for some parameter $$\gamma$$). The asymptotic expression they obtain via the expansion is $$r \sim \sqrt{\frac{8(K-K_{c})}{-K_{c}^{3}g''(0)}}$$ where $$K_c = 2/(\pi g(0))$$. I don't see how you can expand this integral in powers of $$Kr$$ (I think this means a Taylor expansion in a new variable $$z=Kr$$ at $$z=0$$) without knowledge of $$g$$. Not much information is available, lots of other reviews appear to also omit this step. Is it a straightforward expansion? How is Eq. 14 obtained?

Presumably (because this is physics) $$g$$ is analytic in a neighborhood of the origin, so we can Taylor expand.

I think the really important part in deriving this equation for $$r$$ is the fact that the contribution from the first-order derivative term in $$g$$ is zero. Just underneath equation (10), they assume that the frequency distribution is an even function, i.e. $$g(\omega) = g(-\omega)$$. This implies that $$g'(0) = 0$$ and for that matter all the odd-order derivatives of $$g$$ are zero as well. You could also observe that $$\cos^2\theta\cdot\sin\theta$$ is an odd function, so even if $$g'(0)$$ were non-zero, that term would integrate to zero anyway.

If we Taylor expand to second order in $$Kr$$ we get:

\begin{align} 1 & = K\int_{-\pi/2}^{+\pi/2}\cos^2\theta\cdot g(Kr\sin\theta)d\theta \\ & = K\int_{-\pi/2}^{+\pi/2}\cos^2\theta\left(g(0) + \frac{g''(0)}{2}K^2r^2\sin^2\theta + \mathscr{O}(K^4r^4)\right)d\theta \end{align}

Do you see where to go from here?

• Yes that's great, thanks. Maybe they should say find a power series for $g$ in $K r \sin(\theta)$. Then you can write $K_c = 2/(\pi g(0))$ and rearrange, having integrated term by term.
– apg
Oct 21, 2022 at 17:00