Frequency precision of two-dimensional lattices of coupled oscillators with spiral patterns

Two-dimensional lattices of $N$ synchronized oscillators with reactive coupling are considered as high-precision frequency sources in the case where a spiral pattern is formed. The improvement of the frequency precision is shown to be independent of $N$ for large $N$, unlike the case of purely dissipative coupling where the improvement is proportional to $N$, but instead depends on just those oscillators in the core of the spiral that acts as the source region of the waves. Our conclusions are based on numerical simulations of up to $N=29929$ oscillators and analytic results for a continuum approximation to the lattice in an infinite system. We derive an expression for the dependence of the frequency precision on the reactive component of the coupling constant, depending on a single parameter given by fitting the frequency of the spiral waves to the numerical simulations.

Figure 1

Plot of $R/k^2$ derived from the analytic continuum approximation, with $R$ the relative frequency precision and $k=\sqrt{\Omega \gamma }$ with $\Omega$ the spiral frequency, as a function of the reactive coupling constant $\gamma$.

Figure 2

Spiral on lattice of $N=29929$ and $\gamma =0.45$ after evolution to convergence of $r<{10}^{-3}$. Colored squares are individual oscillators with phase given by the colorbar.

Figure 3

Relative frequency precision $R$ as function of the number of oscillators $N$ in a square lattice for a selection of values of the reactive coupling constant $\gamma$ evenly spaced between 0.29 and 0.43 ($\gamma$ increases up the diagram). For small $\gamma$ and $N$, the data are consistent with $R\propto N^{-1}$ (dashed line), as would be found for the model without reactive coupling. For large enough $N$, the data is consistent with a value of $R$ independent of system size.

Figure 4

Adjoint eigenvector on a lattice of $N=29929$ oscillators and $\gamma =0.4$: $2\pi \rho a\left(\rho \right)$ is plotted on a logarithmic scale as a function of the distance $\rho$ from the center of the spiral, which is chosen as the interpolated point with maximum $a$. Numerical data are shown in gray; black curve is Eq. (15) fitted for $k$ and a multiplicative constant. Fits performed against points $\rho <80$ to avoid the edge effects, which are apparent for $\rho \gtrsim 80$.

Figure 5

Spiral parameter $k$ as a function of reactive coupling constant $\gamma$ for system sizes $14884\le N\le 29929$: circles, large-$N$ numerical data for which edge effects are ignorable ($k\sqrt{N}>4$); pluses, remaining numerical data; solid line, Hagan relationship Eq. (21) fitted to circles giving $\alpha \approx 0.840$; blue line, empirical exponential fitted to all data points.

Figure 6

Relative frequency precision $R$ as a function of $\gamma$ for system sizes $14884\le N\le 29929$: circles, large-$N$ numerical data for which edge effects are ignorable ($k\sqrt{N}>4$); pulses, remaining numerical data; solid line, analytic prediction Eqs. (20) and (21) with $\alpha \approx 0.840$; dashed line, empirical fit for $R=a{k}^b$ with Eq. (21) for $k\left(\gamma \right)$.