Comparing the locking threshold for rings and chains of oscillators

We present a case study of how topology can affect synchronization. Specifically, we consider arrays of phase oscillators coupled in a ring or a chain topology. Each ring is perfectly matched to a chain with the same initial conditions and the same random natural frequencies. The only difference is their boundary conditions: periodic for a ring and open for a chain. For both topologies, stable phase-locked states exist if and only if the spread or “width” of the natural frequencies is smaller than a critical value called the locking threshold (which depends on the boundary conditions and the particular realization of the frequencies). The central question is whether a ring synchronizes more readily than a chain. We show that it usually does, but not always. Rigorous bounds are derived for the ratio between the locking thresholds of a ring and its matched chain, for a variant of the Kuramoto model that also includes a wider family of models.

Figure 1

Schematic illustration of the relationship between standard coupling, telescopic coupling, and their agreement for odd coupling functions $f$.

Figure 2

Example showing a choice of $\mathrm{\Lambda }$ for a specific coupling function $f\left(x\right)=sin\left(x\right)+cos\left(3x\right)$, as indicated by the shaded region on the $x$ axis. The dashed lines illustrate how $\text{image}\left(f{|}_{\mathrm{\Lambda }}\right)=\text{image}\left(f\right)$, up to global extrema, while having $f^{\prime }{|}_{\mathrm{\Lambda }}\left(x\right)>0$.

Figure 3

Scatterplot comparing the critical width $\mathrm{\Gamma }$ for a chain and a ring of $N=25$ oscillators for a variety of coupling schemes and random realizations of the natural frequencies. For each data point, the corresponding ring and chain were matched, meaning that both were subject to the same initial conditions and natural frequencies. Initial phases ${\theta }_k\left(0\right)$ were chosen to be identically zero, and natural frequencies ${\eta }_k$ were drawn at random from a uniform distribution on $[-1,+1]$. Lines here represent a 1:1 ratio (dashed green line) and our theoretically predicted upper bound [solid red line defined by Eq. (13)]. Panel (a) has $f\left(x\right)=sin\left(x\right)$ and panel (b) has $f\left(x\right)=sin\left(x\right)+cos\left(3x\right)$, both under the telescopic coupling scheme of Eq. (5). Panels (c) and (d) on the right have $f\left(x\right)=sin(x+0.6)-sin\left(0.6\right)$. However, panel (c) uses telescopic coupling, whereas panel (d) follows the standard coupling Eqs. (4). Notice that the upper bound is always obeyed, but some data points lie below the lower dashed line, showing that it is not a strict bound. Values of $\mathrm{\Gamma }$ were estimated via a bisection technique combined with numerical integration, using a fourth-order Runge-Kutta method with a time step of 0.125, a transient time between $5\times {10}^2$ and $2\times {10}^3$ time units, and observation times of $5\times {10}^2$.

Figure 4

Same parameters, conditions, and coupling schemes as described in the caption of Fig. 3, except the natural frequencies ${\eta }_k$ were drawn from a standard Cauchy distribution instead of a uniform distribution. Although the data arrange themselves differently because of the altered choice of natural frequencies, they still obey the upper bound while only occasionally disobeying the lower bound.

Figure 5

A plot showing the log of the separation (as measured by the infinity norm) between the vector of ${\varphi }_k$'s for a chain and the same vector for a ring, given that they are subject to the same natural frequencies ${\eta }_k$, which were randomly drawn from a uniform distribution on $[-1,+1]$. We first calculate the locked solution for a chain, using an initial condition of all zeros. Then we use the final result of that calculation as the initial condition of the ring to allow for direct comparison. A coupling function $f\left(x\right)=-sin\left(x\right)$ was used. The straight line shows the best linear fit to the log-log plot, indicating that we are seeing a decay comparable to $O\left(N^{-1}\right)$. Values of the phases were computed by numerical integration, using a fourth-order Runge-Kutta method with a time step of 0.125, a transient time between $5\times {10}^2$ and ${10}^6$ time units, and observation times of ${10}^3$.