Singular unlocking transition in the Winfree model of coupled oscillators

The Winfree model consists of a population of globally coupled phase oscillators with randomly distributed natural frequencies. As the coupling strength and the spread of natural frequencies are varied, the various stable states of the model can undergo bifurcations, nearly all of which have been characterized previously. The one exception is the unlocking transition, in which the frequency-locked state disappears abruptly as the spread of natural frequencies exceeds a critical width. Viewed as a function of the coupling strength, this critical width defines a bifurcation curve in parameter space. For the special case where the frequency distribution is uniform, earlier work had uncovered a puzzling singularity in this bifurcation curve. Here we seek to understand what causes the singularity. Using the Poincaré-Lindstedt method of perturbation theory, we analyze the locked state and its associated unlocking transition, first for an arbitrary distribution of natural frequencies, and then for discrete systems of $N$ oscillators. We confirm that the bifurcation curve becomes singular for a continuum uniform distribution, yet find that it remains well behaved for any finite $N$, suggesting that the continuum limit is responsible for the singularity.

Figure 1

Stability diagram for the Winfree model (1), with coupling strength $\kappa$ and influence and sensitivity functions given by Eq. (3). The plot shows the regions of parameter space corresponding to different stable states of the system, for the case of $N=800$ oscillators with natural frequencies ${\omega }_i$ that are evenly spaced on the interval $[1-\Gamma ,1+\Gamma ]$. This choice of frequencies approximates a uniform distribution as $N\to \infty$. The goal of this paper is to understand the perturbative properties of the curve that bounds the “locking” region. (Diagram adapted from Ref. 41.)

Figure 2

(Color online) Graphical solution to Eq. (22), the self-consistency condition, for discrete populations. The curves show graphs of the right-hand side of Eq. (22) for four different population sizes $N$. In the example shown, the detunings were chosen according to a discrete uniform distribution given by Eqs. (6, 38). The horizontal dashed line represents the left-hand side of Eq. (22), a constant function of height $2{\gamma }_1$. Intersections between the dashed line and the curve correspond to locked states (solutions of the self-consistency equation), and determine the value of the maximum phase offset ${\varphi }_0\left(1\right)$ for such a state. Notice that for typical values of ${\gamma }_1$ there are two intersections for each $N$, indicating that there are two locked states (presumably one stable, and the other unstable). As ${\gamma }_1$ increases, these states approach each other and collide in a saddle-node bifurcation at the local maximum of the curve. Above this point, no locked solutions exist. The point marked by a small open circle corresponds to a saddle-node bifurcation for $N=4$, at $[{\gamma }_1^{\star },{\varphi }_0^{\star }\left(1\right)]=(0.884,0.516)$. As $N$ increases, the saddle-node value of ${\varphi }_0^{\star }\left(1\right)$ moves to the right and ultimately tends to $\pi ∕2$ as $N\to \infty$.

Figure 3

(Color online) Scaling laws for the discrete uniform distribution. (a) The maximum phase offset satisfies $1-\sin \left[{\varphi }_0^{\star }\left(1\right)\right]\sim c{N}^{-1}$, with $c$ given by Eq. (41). (b) The linear coefficient ${\gamma }_1^{\star }$ converges to $\pi ∕8$ for large $N$, as expected from the continuum limit (35). (c) The cubic coefficient ${\gamma }_3^{\star }$ is small and slowly becomes more negative for small values of $N$. (d) As $N$ increases, ${\gamma }_3^{\star }$ remains moderately small and negative, but eventually diverges according to the square-root scaling law (57).

Figure 4

(Color online) Scaling laws for discrete tentlike distributions. The curves show the $N$ dependence of (a) ${\varphi }_0^{\star }\left(1\right)$, (b) ${\gamma }_1^{\star }$, and (c) ${\gamma }_3^{\star }$. The curves in panel (a) and (b) are shown for $h_0=1∕2$ (dashed-dotted), $h_0=3∕4$ (dashed), and $h_0=1$ (solid). In (c), ${\gamma }_3^{\star }$ is shown for values of $h_0=1∕2$ (dashed-dotted, the limiting case of a discrete uniform distribution, which shows much weaker divergence than all the other curves), $h_0=3∕4$ (dashed, a typical piecewise-linear distribution, which shows $N^2$ divergence), and $h_0=0.95$ (solid).