Shear Diversity Prevents Collective Synchronization

Large ensembles of heterogeneous oscillators often exhibit collective synchronization as a result of mutual interactions. If the oscillators have distributed natural frequencies and common shear (or nonisochronicity), the transition from incoherence to collective synchronization is known to occur at large enough values of the coupling strength. However, here we demonstrate that shear diversity cannot be counterbalanced by diffusive coupling leading to synchronization. We present the first analytical results for the Kuramoto model with distributed shear and show that the onset of collective synchronization is impossible if the width of the shear distribution exceeds a precise threshold.

Figure 1

Phase diagram for (a) Lorentzian and (b) Gaussian PDFs, $g(\omega )$ and $h(q)$. (a) Solid line: synchronization critical coupling, Eq. (15). Inset: $R^2$ as a function of $K$; the solid line is $R^2=(K-K_c)/K$. (b) Solid line: critical coupling given by Eq. (17). In both (a) and (b) the symbols correspond to numerical results obtained using an ensemble of SL oscillators, Eq. (2), with (a) $N=40000$, $\delta =0.01,{\omega }_0=q_0=\frac{1}{2}$ and (b) $N=22500$, $\sigma =0.02$, ${\omega }_0=q_0=0$. Parameters ${\omega }_j$ and $q_j$ were deterministically selected to represent $p(\omega ,q)$, and averages were done over time.

Figure 2

(a),(b) Sketch of the possible bifurcation scenarios for large $K$ values, depending on the sign of $J$, Eq. (21). Solid (dashed) lines indicate stable (unstable) solutions. (c) Bifurcation diagram for Lorentzian PDFs at $K/\delta =40$, obtained from Eq. (14).