Basin sizes depend on stable eigenvalues in the Kuramoto model

We show that for the Kuramoto model (with identical phase oscillators equally coupled), its global statistics and size of the basins of attraction can be estimated through the eigenvalues of all stable (frequency) synchronized states. This result is somehow unexpected since, by doing that, one could just use a local analysis to obtain the global dynamic properties. But recent works based on Koopman and Perron-Frobenius operators demonstrate that the global features of a nonlinear dynamical system, with some specific conditions, are somehow encoded in the local eigenvalues of its equilibrium states. Recognized numerical simulations in the literature reinforce our analytical results.

Figure 1

(a) Dependence of ${\mathrm{\Gamma }}_q$ with the winding number $q$, for different network sizes. By comparing this plot with that in Fig. 3 (inset) of Ref. [13], one can notice that ${\mathrm{\Gamma }}_q$ behaves as the linear size of the basin of attraction of the $q$ states. (b) Plot of ${\mathrm{\Gamma }}_q^N$, for $N=60$ and $R=2$ (black circles), and the fitted Gaussian curve (red line).

Figure 2

Standard deviation of the distribution of volumes of basins of attraction, divided by $\sqrt{N}$. The blue, red, green, magenta, and black marks refer, respectively, to numerical experiments performed in networks with $N=20$, 40, 60, 80, and 100 with $R/N\lesssim 0.1$. The solid black line is the theoretical result given by Eq. (13). Gray dashed line: A good agreement between our results and experimental data could be obtained by adding a constant $\varepsilon \approx 0.028$ to $\mathcal{F}\left(R\right)$.