Figure 1

(Color online) The time evolution of ${\phi }_0$ (top panels) and $A$ (bottom panels), which give the mean and the inverse width of the distribution, is shown for simulations of globally coupled arrays of $N=4000$ sites and compared to the numerical solution of Eqs. (20, 21). The left column represents modest coupling, $K=10$, while the right column shows $K=80$; ${\sigma }^2=3.5$ for all plots. The initial values $-A\left(0\right)$ are chosen to be $\gtrsim K∕{\sigma }^2$; specifically, $A\left(0\right)=-9.7$ in the left column and $A\left(0\right)=-58$ in the right column. The dark (blue) lines show data from lattice simulations, while the light (red) lines represent theoretical predictions. The very light (brown) curves in the lower right panel and inset represent the uncoupled dynamics given by Eq. (23) and the Riccati equation (22). The lower left inset shows the simulation results and the prediction of the mean-field theory (solid black line) in the stationary state, which is exact when $N\to \infty$. The inset in the bottom right figure shows a close up of the late time evolution. Note the different horizontal and vertical scales in the various panels.Reuse & Permissions

Figure 2

(Color online) The time evolution of the mean (top panels) and inverse width (bottom panels) is shown for simulations of locally coupled two-dimensional arrays of $N=64\times 64$ sites and compared to the numerical solution of Eqs. (20, 21). The left column represents modest coupling, $K=10$, while the right column shows results for $K=80$; ${\sigma }^2=3.5$ for all plots. The dark (blue) lines show data from lattice simulations, while the light (red) lines represent theoretical predictions. The initial values are $A\left(0\right)=-9.2$ (left column) and $-42$ (right column). The inset in the bottom right panel shows a close up of the early-time evolution.Reuse & Permissions

Figure 3

(Color online) The time evolution of the mean field (top panels) and the inverse width of the distribution (bottom panels) is shown for both locally (dark or blue) and globally (black) coupled simulations and compared to the numerical solution (light or red) of the large-$K$ theory. Again, the left column is for $K=10$ and the right for $K=80$. Here we choose $\mid A\left(0\right)\mid <1$ [specifically, $A\left(0\right)\sim -0.1$], and the transient dynamics are not well described for early times. The theory accurately captures the steady-state behavior of the globally coupled simulations in the large-$K$ regime. $N=4000$ (globally coupled) and $64\times 64$ (locally coupled), and ${\sigma }^2=3.5$ for all panels.Reuse & Permissions

Figure 4

(Color online) Multistability is captured by the large-$K$ theory. The left panel shows the dynamics of the mean field and the inverse width of the distribution for initial conditions leading to a disordered state [${\phi }_0\left(0\right)\approx 0.5$, $A\left(0\right)=-29$]. The right panel shows the corresponding plots for initial conditions leading to an ordered phase [${\phi }_0\left(0\right)\approx 4$, $A\left(0\right)=-55$]. The inset in the bottom left shows a close up of the steady-state behavior. Dark (blue): globally coupled simulations. Light (brown): Gaussian ansatz theory. Black: mean-field theory. In all plots, $K=80$, ${\sigma }^2=3.0$.Reuse & Permissions

Figure 5

(Color online) The time evolution of the limit cycle radius $r$ (top panels) and of the Gaussian ansatz coefficients (three subsequent panels). The dark (blue) curves are appropriate moment results from simulations of globally coupled arrays ($N=4000$ sites), and the light (red) curves are obtained from the solution of Eqs. (25, 26). $K=10$ in the left column and $K=80$ in the right column. Initial values for left column, $(A,E,M)=(-9.6,4.7,-9.7)$, and right column, $(A,E,M)=(-56,7,-79)$. The noise intensity ${\sigma }^2=3.5$ in all panels. The dashed line in the top panels is the steady-state radius predicted by multiscale analysis.Reuse & Permissions

Figure 6

(Color online) Same as Fig. 5, but now the simulations are for locally coupled arrays. Initial values for left column, $(A,E,M)=(-9.4,4.4,-9.6)$, and right column, $(A,E,M)=(-55,5,-76)$.Reuse & Permissions

Figure 7

(Color online) Evolution of the theoretical probability distribution function as obtained by substituting the numerical solution of Eqs. (25, 26) into Eq. (17). The distribution becomes symmetrical (circular) in ${\phi }_0$ and $z_0$ prior to reaching its steady-state radius. The evolution is only shown for the short time leading up to relaxation of the distribution shape. Over longer times, the distribution will continue a circular trajectory whose radius eventually reaches its steady-state value (see Fig. 8). $K=10$ and ${\sigma }^2=3.5$.Reuse & Permissions

Figure 8

(Color online) The phase-space portraits of lattice simulations (dark or blue) and the large-$K$ equations (25, 26) (light or red) show good agreement in terms of the steady-state limit cycle radius, though the theory slights underestimates $r$. Inset: oscillation frequency. $K=80$, ${\sigma }^2=3.5$, and $\omega =1$.Reuse & Permissions

Figure 9

(Color online) Comparison of the phase-space portraits of lattice simulations (dark or blue) and the large-$K$ equations (light or red) shows that the latter correctly predict multistable behavior. The right inset shows a close up of the indicated portion of the phase portrait. The black curve is the unstable limit cycle obtained from the multiscale analysis. $K=80$, ${\sigma }^2=3.5$, and $\omega =1$.Reuse & Permissions

Figure 10

The multiscale analysis predicts the onset of bistability to occur at ${\sigma }^2\approx 2.4$. Curves are, from thickest to thinnest, ${\sigma }^2=3.5,3.0,2.4,2.0$.Reuse & Permissions