Noise-induced stabilization of collective dynamics

We illustrate a counterintuitive effect of an additive stochastic force, which acts independently on each element of an ensemble of globally coupled oscillators. We show that a very small white noise not only broadens the clusters, wherever they are induced by the deterministic forces, but can also stabilize a linearly unstable collective periodic regime: self-consistent partial synchrony. With the help of microscopic simulations we are able to identify two noise-induced bifurcations. A macroscopic analysis, based on a perturbative solution of the associated nonlinear Fokker-Planck equation, confirms the numerical studies and allows determining the eigenvalues of the stability problem. We finally argue about the generality of the phenomenon.

Figure 1

Different order parameters versus ${\gamma }_1$ obtained from direct simulations with no noise at all (red circles) and $D=1.2\times {10}^{-4}$ (black crosses). These and all other simulations in this paper refer to ${\gamma }_2=\pi$. (a) Time averaged Kuramoto order parameter, (b) difference between macroscopic and microscopic frequencies, and (c) microscopic frequencies of the system, obtained by averaging over the single frequencies of all the oscillators. Simulations correspond to the integration of Eq. (1) for $N=1000$ oscillators. Each order parameter has been computed for $1\times {10}^5$ time units after discarding a transient of $1\times {10}^4$ time units.

Figure 2

Phase diagram in the plane $({\gamma }_1,D)$. In region A only SCPS is stable; B indicates where only SD is stable, and C is a region of bistability. Continuous and dashed black lines separating regions A, B, and C correspond to super- and subcritical Hopf bifurcation, respectively, whereas the continuous red line separating regions B and C indicates a saddle node of periodic orbits bifurcation. These bifurcation lines have been obtained from direct simulations. Blue triangles are the results from the macroscopic description developed in Sec. 4. The two green circles indicate the points corresponding to simulations in Fig. 3 and the three purple squares correspond to the simulations of Fig. 4. Vertical dashed black lines indicate the range corresponding to Figs. 5 (${\gamma }_1=1.34$) and 6 (${\gamma }_1=1.42$). The four red diamonds show the parameter values of the potentials in Fig. 7.

Figure 3

Histograms for the density of oscillators at different times. System parameters are (a) ${\gamma }_1=1.25$ and $D=2\times {10}^{-5}$ and (b) ${\gamma }_1=1.35$ and $D=5\times {10}^{-5}$. The results have been obtained from direct simulations using $1\times {10}^4$ oscillators after discarding a transient of $1\times {10}^5$ time units.

Figure 4

Time series of the evolution of the Kuramoto order parameter for different values of noise and fixed ${\gamma }_1=1.34$ and ${\gamma }_2=\pi$. The dotted black line is the evolution of the deterministic system, the dashed red line corresponds to $D=6\times {10}^{-5}$, and the continuous blue line corresponds to $D=1.2\times {10}^{-4}$. Results correspond to simulations with $N=1\times {10}^4$ oscillators after discarding a transient of length of $9\times {10}^4$.

Figure 5

Different order parameters extracted from direct simulations of the biharmonic model versus the level of noise for ${\gamma }_1=1.34$. (a) The time-averaged Kuramoto order parameter $\langle {\tilde{P}}_D\left(1\right)\rangle$, (b) its standard deviation $\sigma$, and (c) the difference between the macroscopic frequency ${\mathrm{\Omega }}_D$ and the averaged microscopic frequencies ${\nu }_D$. Red circles, blue pluses, and black crosses correspond to $N=1000$, 4000, and 16 000 oscillators, respectively.

Figure 6

Different order parameters extracted from direct simulations of the biharmonic model versus level of noise for ${\gamma }_1=1.42$. (a) The time-averaged Kuramoto order parameter $\langle \left|{\tilde{P}}_D\left(1\right)\right|\rangle$, (b) its standard deviation $\sigma$, and (c) the difference between the macroscopic frequency ${\mathrm{\Omega }}_D$ and the averaged microscopic frequencies ${\nu }_D$. Simulations correspond to $N=1\times {10}^5$ oscillators.

Figure 7

Washboard potentials corresponding to Eq. (8) computed for a noise level of $D=6\times {10}^{-5}$. From top left to bottom left purple, green, blue, and amber correspond to ${\gamma }_1=1.15$, 1.16, 1.171, and 1.18, respectively. The inset shows the same results in an enlarged parameter range. The parameters used in each case have been computed numerically from direct simulations with $N=1\times {10}^4$ oscillators.

Figure 8

Shape of $P_0\left(\theta \right)$ (red line) and $p\left(\theta \right)$ (dashed blue line) for ${\gamma }_1=1.33$. The noise-induced perturbation $p$ has been arbitrarily rescaled. These functions correspond to the solutions of Eqs. (9) and (10), respectively.

Figure 9

Red circles show real and imaginary parts of the eigenvalues controlling stability of $\tilde{P}$ for ${\gamma }_1=1.33$ and $D=0$. Blue arrows show the directions where the eigenvalues are pushed when $D=1.7\times {10}^{-5}$. The inset is a zoom for $Re\left({\mathrm{\Lambda }}_D\right)>0$.

Figure 10

Eigenfunctions associated to the eigenvalues controlling the stability of SCPS for $\gamma =1.33$ and $D=0$. (a) Real and (b) imaginary parts of the eigenfunction associated to ${\mathrm{\Lambda }}_0^{\left(1\right)}$ (red lines), and first and second derivatives of $P_0$ suitably rescaled (dashed black lines), respectively. (c) Real and (d) imaginary parts of the eigenfunction associated to ${\mathrm{\Lambda }}_0^{\left(2\right)}$ (red lines), and third and fourth derivatives of $P_0$ suitably rescaled (dashed black lines), respectively.

Figure 11

Order parameter $\rho$ for different values of ${\gamma }_1$ for no noise (red circles) and $D=1\times {10}^{-6}$ (black crosses). The results have been obtained by direct simulations of the Rayleigh oscillators with 1000 units.