Critical Coupling

Module 2: Synchronization and Collective Behaviour

This page summarizes Kuramoto’s self-consistency argument for the limit $N \to \infty$, following Strogatz (2000, sec. 4). Work in the rotating frame, so $g$ is symmetric about $0$.

Steady states with constant $r$

Look for steady solutions in which $r$ and $\psi$ are constant. In the rotating frame, with $g$ symmetric, you can choose the origin of phases so that $\psi = 0$ (Strogatz 2000). Then the mean-field equation reads

\[ \dot\theta_i = \omega_i - K r \sin\theta_i. \tag{1}\]

Each oscillator is a nonuniform oscillator (Flows on the Circle) with $a = Kr$. The population splits in two groups:

Locked oscillators, $|\omega_i| \le Kr$, sit at a stable fixed point with $\sin\theta_i = \omega_i/(Kr)$ and $|\theta_i| \le \pi/2$.
Drifting oscillators, $|\omega_i| > Kr$, circulate around the circle. Their stationary distribution is symmetric under $\theta \to \theta + \pi$ together with $\omega \to -\omega$, so they add nothing to $r$ (Strogatz 2000).

Self-consistency

Only the locked oscillators contribute to $r$. Using $\omega = Kr\sin\theta$ to change variables,

\[ r = K r \int_{-\pi/2}^{\pi/2} \cos^2\theta \; g(Kr\sin\theta)\, d\theta. \tag{2}\]

$r = 0$ always solves Equation 2: this is the incoherent state. A partially locked state with $r > 0$ must satisfy

\[ 1 = K \int_{-\pi/2}^{\pi/2} \cos^2\theta \; g(Kr\sin\theta)\, d\theta. \tag{3}\]

Letting $r \to 0^+$ and using $\int_{-\pi/2}^{\pi/2}\cos^2\theta\,d\theta = \pi/2$ gives the threshold where this branch starts:

\[ K_c = \frac{2}{\pi g(0)}. \tag{4}\]

This is Kuramoto’s formula for the critical coupling (Kuramoto 2003; Strogatz 2000).

Two cases you will use

Lorentzian frequencies. For $g(\omega) = \gamma / [\pi(\gamma^2 + \omega^2)]$, Equation 4 gives $K_c = 2\gamma$ and Equation 3 can be solved exactly:

\[ r = \sqrt{1 - \frac{K_c}{K}}, \qquad K \ge K_c. \tag{5}\]

Normal frequencies. For a normal density with standard deviation $\sigma$, $K_c = 2\sqrt{2\pi}\,\sigma/\pi = \sqrt{8/\pi}\,\sigma \approx 1.596\,\sigma$. There is no closed form for $r(K)$, so you solve Equation 3 numerically.

Square-root onset

For any smooth $g$ with $g''(0) < 0$, expanding Equation 3 for small $r$ gives

\[ r \approx \sqrt{\frac{16}{\pi K_c^3}}\sqrt{\frac{\mu}{-g''(0)}}, \qquad \mu = \frac{K - K_c}{K_c}, \tag{6}\]

so $r$ grows like $\sqrt{K - K_c}$ just above threshold (Strogatz 2000). The transition from incoherence to partial locking is continuous. The same square-root law appears at a supercritical pitchfork bifurcation.

Figure 1: Order parameter in the limit $N \to \infty$. Solid: exact Lorentzian result Equation 5 with $\gamma = 1$ ($K_c = 2$). Dashed: near-onset approximation Equation 6 for a normal density with $\sigma = 1$ ($K_c \approx 1.60$), valid only close to $K_c$.

What the argument does not show

Kuramoto’s argument finds the steady states. It does not prove which of them are stable, nor that the population approaches them. These questions were settled only by much later work, which Strogatz (2000) reviews. Finite $N$ adds fluctuations of order $N^{-1/2}$ on top of these curves (Finite-Size Effects).

Exercise

Check that Equation 6 agrees with the Lorentzian result Equation 5 near $K_c$.

Solution

For $\gamma = 1$, $g(0) = 1/\pi$, $K_c = 2$ and $g''(0) = -2/\pi$. Then Equation 6 gives $r \approx \sqrt{16/(8\pi)}\,\sqrt{\mu\pi/2} = \sqrt{\mu}$. The exact result is $r = \sqrt{(K - K_c)/K} = \sqrt{\mu K_c/K}$, which tends to $\sqrt{\mu}$ as $K \to K_c$.

Exercise

Your frequencies are normal with $\sigma = 0.5$. What is $K_c$? What happens to $K_c$ if you double the spread of frequencies?

Solution

$K_c = \sqrt{8/\pi}\times 0.5 \approx 0.80$. Since $g(0) \propto 1/\sigma$, $K_c$ is proportional to $\sigma$. Doubling the spread doubles $K_c$: a more diverse population needs stronger coupling to synchronize.

References

Kuramoto, Yoshiki. 2003. Chemical Oscillations, Waves, and Turbulence. Dover Publications.

Strogatz, Steven H. 2000. “From Kuramoto to Crawford: Exploring the Onset of Synchronization in Populations of Coupled Oscillators.” Physica D: Nonlinear Phenomena 143: 1–20.

--- title: "Critical Coupling" subtitle: "Module 2: Synchronization and Collective Behaviour" format: html --- This page summarizes Kuramoto's self-consistency argument for the limit $N \to \infty$, following @strogatz2000kuramoto [Section 4]. Work in the rotating frame, so $g$ is symmetric about $0$. ## Steady states with constant $r$ Look for steady solutions in which $r$ and $\psi$ are constant. In the rotating frame, with $g$ symmetric, you can choose the origin of phases so that $\psi = 0$ [@strogatz2000kuramoto]. Then the mean-field equation reads $$ \dot\theta_i = \omega_i - K r \sin\theta_i. $$ {#eq-mf-steady} Each oscillator is a nonuniform oscillator ([Flows on the Circle](flows-on-the-circle.qmd)) with $a = Kr$. The population splits in two groups: - **Locked** oscillators, $|\omega_i| \le Kr$, sit at a stable fixed point with $\sin\theta_i = \omega_i/(Kr)$ and $|\theta_i| \le \pi/2$. - **Drifting** oscillators, $|\omega_i| > Kr$, circulate around the circle. Their stationary distribution is symmetric under $\theta \to \theta + \pi$ together with $\omega \to -\omega$, so they add nothing to $r$ [@strogatz2000kuramoto]. ## Self-consistency Only the locked oscillators contribute to $r$. Using $\omega = Kr\sin\theta$ to change variables, $$ r = K r \int_{-\pi/2}^{\pi/2} \cos^2\theta \; g(Kr\sin\theta)\, d\theta. $$ {#eq-self-consistency} $r = 0$ always solves @eq-self-consistency: this is the **incoherent state**. A partially locked state with $r > 0$ must satisfy $$ 1 = K \int_{-\pi/2}^{\pi/2} \cos^2\theta \; g(Kr\sin\theta)\, d\theta. $$ {#eq-self-consistency-r} Letting $r \to 0^+$ and using $\int_{-\pi/2}^{\pi/2}\cos^2\theta\,d\theta = \pi/2$ gives the threshold where this branch starts: $$ K_c = \frac{2}{\pi g(0)}. $$ {#eq-kc} This is Kuramoto's formula for the critical coupling [@kuramoto2003chemical; @strogatz2000kuramoto]. ## Two cases you will use **Lorentzian frequencies.** For $g(\omega) = \gamma / [\pi(\gamma^2 + \omega^2)]$, @eq-kc gives $K_c = 2\gamma$ and @eq-self-consistency-r can be solved exactly: $$ r = \sqrt{1 - \frac{K_c}{K}}, \qquad K \ge K_c. $$ {#eq-r-lorentz} **Normal frequencies.** For a normal density with standard deviation $\sigma$, $K_c = 2\sqrt{2\pi}\,\sigma/\pi = \sqrt{8/\pi}\,\sigma \approx 1.596\,\sigma$. There is no closed form for $r(K)$, so you solve @eq-self-consistency-r numerically. ## Square-root onset For any smooth $g$ with $g''(0) < 0$, expanding @eq-self-consistency-r for small $r$ gives $$ r \approx \sqrt{\frac{16}{\pi K_c^3}}\sqrt{\frac{\mu}{-g''(0)}}, \qquad \mu = \frac{K - K_c}{K_c}, $$ {#eq-r-onset} so $r$ grows like $\sqrt{K - K_c}$ just above threshold [@strogatz2000kuramoto]. The transition from incoherence to partial locking is continuous. The same square-root law appears at a supercritical pitchfork bifurcation. ```{python} #| label: fig-critical-coupling #| fig-cap: "Order parameter in the limit $N \\to \\infty$. Solid: exact Lorentzian result @eq-r-lorentz with $\\gamma = 1$ ($K_c = 2$). Dashed: near-onset approximation @eq-r-onset for a normal density with $\\sigma = 1$ ($K_c \\approx 1.60$), valid only close to $K_c$." #| fig-width: 6 #| fig-height: 3.5 #| echo: false import numpy as np import matplotlib.pyplot as plt from amlab.synchronization.kuramoto_sweep import critical_coupling k = np.linspace(0, 6, 600) kc_lor = critical_coupling(1 / np.pi) r_lor = np.where(k > kc_lor, np.sqrt(np.clip(1 - kc_lor / np.maximum(k, 1e-12), 0, None)), 0.0) sigma = 1.0 g0 = 1 / (np.sqrt(2 * np.pi) * sigma) g2 = -g0 / sigma**2 kc_norm = critical_coupling(g0) mu = (k - kc_norm) / kc_norm r_norm = np.where(k > kc_norm, np.sqrt(16 / (np.pi * kc_norm**3) * np.clip(mu, 0, None) / (-g2)), 0.0) mask = (k < kc_norm) | (r_norm < 0.8) fig, ax = plt.subplots(figsize=(6, 3.5)) ax.plot(k, r_lor, label=r"Lorentzian, $\gamma = 1$") ax.plot(k[mask], r_norm[mask], "--", label=r"Normal, $\sigma = 1$ (near onset)") ax.axvline(kc_lor, color="C0", linewidth=0.8, linestyle=":") ax.axvline(kc_norm, color="C1", linewidth=0.8, linestyle=":") ax.set_xlabel("Coupling strength K") ax.set_ylabel(r"$r_\infty$") ax.set_ylim(0, 1) ax.legend() plt.tight_layout() plt.show() ``` ## What the argument does not show Kuramoto's argument finds the steady states. It does not prove which of them are stable, nor that the population approaches them. These questions were settled only by much later work, which @strogatz2000kuramoto reviews. Finite $N$ adds fluctuations of order $N^{-1/2}$ on top of these curves ([Finite-Size Effects](kuramoto-finite-size.qmd)). ::: {.callout-note} ## Exercise Check that @eq-r-onset agrees with the Lorentzian result @eq-r-lorentz near $K_c$. ::: ::: {.callout-tip collapse="true"} ## Solution For $\gamma = 1$, $g(0) = 1/\pi$, $K_c = 2$ and $g''(0) = -2/\pi$. Then @eq-r-onset gives $r \approx \sqrt{16/(8\pi)}\,\sqrt{\mu\pi/2} = \sqrt{\mu}$. The exact result is $r = \sqrt{(K - K_c)/K} = \sqrt{\mu K_c/K}$, which tends to $\sqrt{\mu}$ as $K \to K_c$. ::: ::: {.callout-note} ## Exercise Your frequencies are normal with $\sigma = 0.5$. What is $K_c$? What happens to $K_c$ if you double the spread of frequencies? ::: ::: {.callout-tip collapse="true"} ## Solution $K_c = \sqrt{8/\pi}\times 0.5 \approx 0.80$. Since $g(0) \propto 1/\sigma$, $K_c$ is proportional to $\sigma$. Doubling the spread doubles $K_c$: a more diverse population needs stronger coupling to synchronize. :::

Steady states with constant \(r\)

Self-consistency

Two cases you will use

Square-root onset

What the argument does not show

References