Kuramoto Lab: Locating the Threshold

Module 2: Synchronization and Collective Behaviour

This page is the interpretation step. You compare the transition in your Bifurcation Diagram with the critical coupling $K_c = 2/(\pi g(0))$ from Critical Coupling.

In your animation script from Animation, set the coupling strength just below and just above the critical value to see the difference in the dynamics. Below $K_c$ the oscillators remain incoherent. Above $K_c$ they start to synchronize and form a cluster on the unit circle, as in Figure 1.

Figure 1: Comparison between the theoretical critical coupling $K_c$ and the empirical transition point observed in simulations. Individual oscillators are blue dots and the order parameter is the red vector.

Exercise

Find $K_c$ from $K_c = 2/(\pi g(0))$ for your normal frequencies with $\sigma = 1$. Estimate the empirical transition point from your points: the value of $K$ where $r$ starts to rise clearly above zero. Alternatively, fit the near-onset scaling $r \approx \sqrt{16/(\pi K_c^3)}\,\sqrt{\mu/(-g''(0))}$ of Bifurcation Diagram to the points just above the transition. Do the two estimates agree?

Solution

For a normal density $g(0) = 1/\sqrt{2\pi}$, so $K_c = 2\sqrt{2\pi}/\pi = \sqrt{8/\pi} \approx 1.60$. With $N = 1000$ and 20 values of $K$ between 0 and 5, $r_\infty$ stays below about $0.1$ up to $K \approx 1.3$ and jumps to about $0.6$ between $K \approx 1.6$ and $K \approx 1.8$. The grid spacing is about $0.26$, so the empirical threshold agrees with $K_c$ within one grid step. The exact $K$ of the jump moves with the seed, because each sample of frequencies differs from $g$ (Finite-Size Effects).

Exercise

The empirical points in the normal case lie close to zero for $K < K_c$ but not exactly at zero. Give two reasons.

Solution

First, a finite population has $r$ of order $N^{-1/2}$ even when incoherent (about $0.045$ for the $N = 500$ figures of Bifurcation Diagram, $0.03$ for $N = 1000$). Second, the time window may be too short for $r(t)$ to settle near $K_c$, where the approach to the steady state is slow. Finite-Size Effects studies the first effect.

--- title: "Kuramoto Lab: Locating the Threshold" subtitle: "Module 2: Synchronization and Collective Behaviour" format: html --- This page is the interpretation step. You compare the transition in your [Bifurcation Diagram](kuramoto-diagram.qmd) with the critical coupling $K_c = 2/(\pi g(0))$ from [Critical Coupling](critical-coupling.qmd). In your animation script from [Animation](kuramoto-animation.qmd), set the coupling strength just below and just above the critical value to see the difference in the dynamics. Below $K_c$ the oscillators remain incoherent. Above $K_c$ they start to synchronize and form a cluster on the unit circle, as in @fig-kuramoto-bifurcation-validate. ```{python} #| label: fig-kuramoto-bifurcation-validate #| fig-cap: 'Comparison between the theoretical critical coupling $K_c$ and the empirical transition point observed in simulations. Individual oscillators are blue dots and the order parameter is the red vector.' #| fig-width: 7 #| fig-height: 4 #| echo: false import numpy as np import matplotlib.pyplot as plt from scipy.integrate import solve_ivp from amlab.odes_coupled.kuramoto import ( initialize_oscillators, kuramoto_ode_meanfield, kuramoto_order_parameter, ) theta0, omega = initialize_oscillators( 100, distribution="normal", scale_omega=0.5, scale_phase=1.0, seed=1 ) fig, axs = plt.subplots(1, 2, figsize=(7, 4)) for ax, k, title in zip(axs, [0.1, 10.0], ["K < Kc", "K > Kc"]): sol = solve_ivp(kuramoto_ode_meanfield, (0, 100), theta0, args=(omega, k)) theta_end = sol.y[:, -1] _, _, rc, rs = kuramoto_order_parameter(theta_end) ax.add_artist(plt.Circle((0, 0), 1, color="lightgray", fill=False)) ax.scatter(np.cos(theta_end), np.sin(theta_end), s=50, color="blue", alpha=0.5) ax.plot([0, rc], [0, rs], color="red", linewidth=2) ax.plot([rc], [rs], "ro", markersize=6) ax.set_title(title) ax.set_xlabel("Cos(theta)") ax.set_xlim(-1.1, 1.1) ax.set_ylim(-1.1, 1.1) ax.set_aspect("equal") ax.grid(True) axs[0].set_ylabel("Sin(theta)") plt.tight_layout() plt.show() ``` ::: {.callout-note} ## Exercise Find $K_c$ from $K_c = 2/(\pi g(0))$ for your normal frequencies with $\sigma = 1$. Estimate the empirical transition point from your points: the value of $K$ where $r$ starts to rise clearly above zero. Alternatively, fit the near-onset scaling $r \approx \sqrt{16/(\pi K_c^3)}\,\sqrt{\mu/(-g''(0))}$ of [Bifurcation Diagram](kuramoto-diagram.qmd) to the points just above the transition. Do the two estimates agree? ::: ::: {.callout-tip collapse="true"} ## Solution For a normal density $g(0) = 1/\sqrt{2\pi}$, so $K_c = 2\sqrt{2\pi}/\pi = \sqrt{8/\pi} \approx 1.60$. With $N = 1000$ and 20 values of $K$ between 0 and 5, $r_\infty$ stays below about $0.1$ up to $K \approx 1.3$ and jumps to about $0.6$ between $K \approx 1.6$ and $K \approx 1.8$. The grid spacing is about $0.26$, so the empirical threshold agrees with $K_c$ within one grid step. The exact $K$ of the jump moves with the seed, because each sample of frequencies differs from $g$ ([Finite-Size Effects](kuramoto-finite-size.qmd)). ::: ::: {.callout-note} ## Exercise The empirical points in the normal case lie close to zero for $K < K_c$ but not exactly at zero. Give two reasons. ::: ::: {.callout-tip collapse="true"} ## Solution First, a finite population has $r$ of order $N^{-1/2}$ even when incoherent (about $0.045$ for the $N = 500$ figures of [Bifurcation Diagram](kuramoto-diagram.qmd), $0.03$ for $N = 1000$). Second, the time window may be too short for $r(t)$ to settle near $K_c$, where the approach to the steady state is slow. [Finite-Size Effects](kuramoto-finite-size.qmd) studies the first effect. :::