Synchronization of coupled active rotators by common noise

We study the effect of common noise on coupled active rotators. While such a noise always facilitates synchrony, coupling may be attractive (synchronizing) or repulsive (desynchronizing). We develop an analytical approach based on a transformation to approximate angle-action variables and averaging over fast rotations. For identical rotators, we describe a transition from full to partial synchrony at a critical value of repulsive coupling. For nonidentical rotators, the most nontrivial effect occurs at moderate repulsive coupling, where a juxtaposition of phase locking with frequency repulsion (anti-entrainment) is observed. We show that the frequency repulsion obeys a nontrivial power law.

Figure 1

The average value $\langle cos\mathrm{\Phi }\rangle$ determining the Lyapunov exponent for the order parameter $I$ (or $J$) [see Eqs. (24) and (23)] is plotted as a function of the system parameters (filled surface). The asymptotic behavior $\langle cos\mathrm{\Phi }\rangle \approx 0.5B{\sigma }^2/({\mathrm{\Omega }}_0^2-B^2)$ corresponding to Eq. (19) (wire frame) is compared against the exact formula (24).

Figure 2

The time-average value of the order parameter $\langle I\rangle$ for an ensemble of nonidentical oscillators vs coupling strength is plotted for ${\mathrm{\Omega }}_0=10$, $J_0=2.5$, and values of the frequency band half width $\gamma /{\sigma }^2=0$ (circles), ${10}^{-6}$, ${10}^{-5}$, ${10}^{-4}$, ${10}^{-3}$, ${10}^{-2}$ (solid lines, from top to bottom). Solid lines represents the results of numerical simulation for Eqs. (6) and (7) with $I$ determined by Eq. (9). The analytical estimates (26) plotted with dashed lines appear to be in a good agreement with the results of numerical simulations.

Figure 3

(a) Frequencies in an ensemble of 41 active rotators with a Gaussian distribution of torque parameters $\mathrm{\Omega }$, in dependence on the coupling strength $\mu$. (b) The averaged order parameter $\langle R\rangle$. Parameters of the model: ${\mathrm{\Omega }}_0=10$, $B=2.5$, ${\sigma }^2=0.5$, the standard deviation of the distribution of torques $5\times {10}^{-4}$. One can see the repulsion of the frequencies in the whole range of negative values of $\mu$, with a maximal effect around $\mu =-0.022$.

Figure 4

The average frequency shift is plotted vs the natural frequency mismatch for $B=2.5$, ${\mathrm{\Omega }}_0=10$, ${\sigma }^2=0.5$, $\gamma =5\times {10}^{-4}$, $\mu =0.02,0.015,0.01,0.005,0,-0.005,-0.01,-0.015$ (from bottom to top on the right-hand side). The frequencies are obtained by virtue of direct numerical simulation of Eqs. (6, 7, 8).

Figure 5

The average frequency shift is plotted on a logarithmic scale vs the natural frequency mismatch for $B=2.5$, ${\mathrm{\Omega }}_0=10$, ${\sigma }^2=0.5$, $\mu =0.015$, 0.01, 0.005, 0, $-0.005$, $-0.01$, $-0.015$ (from bottom to top). Black solid lines: the results of direct numerical simulation for stochastic system (6, 7, 8) with $\gamma =5\times {10}^{-4}$; green circles: Eqs. (6, 7, 8) with $\gamma =0$; green dashed lines: analytical theory (29); blue crests: asymptotic law (30). The results are provided for both negative (a) and positive (b) values of $\nu$ as these cases are not identical. Deviations of the simulated frequencies from the asymptotic law are due to finiteness of the order parameter $I$ in simulations, while at the derivation of (29) and (30) we assumed $I\to \infty$. Red vertical dash-dotted lines: the natural torque shifts $\omega =\gamma$ and $\omega =3.08\gamma$, half of the rotators in the ensemble with the Lorentzian distribution of torques have $\left|\omega \right|>\gamma$, and 20%—$\left|\omega \right|>3.08\gamma$. With these lines one can see that the approximation $I\to \infty$, adopted at the derivation of the analytical expressions (29) and (30), which neglects the finiteness of $I$ and misses the transition to a linear law for $\omega \to 0$ (can be seen for the black solid lines), is relevant for a considerable fraction of the rotators in the ensemble.

Figure 6

The average frequency shift (black solid lines) and order parameter $\langle I\rangle$ (blue dashed line) are plotted vs common noise strength $\sigma$ for an ensemble with fixed inherent parameters $B=2.5$, ${\mathrm{\Omega }}_0=10$, $\mu =-0.02$, and $\gamma =5\times {10}^{-4}$: the results of numerical simulation of stochastic system (6, 7, 8) for $\omega /\gamma =0.25$, 0.5, 1, 2, 4.

Figure 7

The dynamics of the angle differences ${\phi }_N\left(t\right)-{\phi }_1\left(t\right)$ between the most fast and most slow rotators in a population illustrated in Fig. 3. Green: uncoupled rotators $\mu =0$; red: attractive coupling $\mu =0.005$; blue: weak repulsive coupling $\mu =-0.01$; magenta: strong repulsive coupling $\mu =-0.1$. Only in the latter case there are no phase slips, in all other cases one can clearly see long epochs where ${\phi }_N\left(t\right)-{\phi }_1\left(t\right)\approx 2\pi m$.

Figure 8

Statistics of the time intervals between the slips, for different values of coupling strength: $\mu =0.005$, $\mu =0$, and $\mu =-0.01$. Parameters of the population of rotators are the same as in Fig. 3.