Figure 1

Regions (tongues) of full synchrony for the harmonically forced ensemble with global nonlinear coupling [see Eqs. (5, 6)] with $\beta =\text{const}$ and $A=1-\epsilon r^2$. With a variation in $\beta$, the tip of the tongue moves along a circle with radius $\epsilon -1$. The tongues are plotted for $\beta =0$, $\beta =0.25\pi$, $\beta =0.4\pi$, and $\beta =0.5\pi$; the value of coupling is $\epsilon =1.4$.Reuse & Permissions

Figure 2

(Color online) Simulation of a periodically forced ensemble [model Eq. (5, 6)] with $\beta =\text{const}=\pi /4$ and $\epsilon A\left(r,\epsilon \right)=1-\epsilon r^2$, for $\epsilon =1.4$. For each point in the parameter plane $\nu ,\mu$ the simulation was performed twice, with nearly uniform and nearly identical initial conditions for oscillator phases. Black dots denote parameter values when both initial conditions lead to full synchrony; gray (yellow in color) dots denote the parameters when only nearly identical initial conditions lead to a fully synchronized state, while nearly uniform initial conditions lead to asynchrony. The border of the synchronization region nicely corresponds to the theoretical curve (black bold line), obtained according to Eq. (28).Reuse & Permissions

Figure 3

Regions (tongues) of full synchrony for the harmonically forced ensemble with global nonlinear coupling [Eqs. (5, 6)] with $\beta ={\beta }_0+{\epsilon }^2{r}^2$ and $A=1$. With variation in $\epsilon$ the tip of the tongue moves along a spiral. The tongues are plotted for ${\beta }_0=0.475\pi$ and $\epsilon =0.8$, $\epsilon =1.4$, $\epsilon =1.7$, and $\epsilon =\sqrt{\pi +{\epsilon }_q^2}\approx 1.79$.Reuse & Permissions

Figure 4

(Color online) Simulation of a periodically forced ensemble Eqs. (5, 6) with $A\left(r,\epsilon \right)=1$ and $\beta ={\beta }_0+{\epsilon }^2{r}^2$, for ${\beta }_0=0.475\pi$ and $\epsilon =0.4$. Black bold curve is determined by Eq. (29). Gray (yellow in color) area corresponds to multistability (cf. Fig. 2).Reuse & Permissions

Figure 5

(Color online) Illustration of the mean-field locking in a periodically driven oscillator ensemble with nonlinearity in the amplitude function (see text for details). Left column: subcritical driving $\mu =0.2<{\mu }_c$. Right column: supercritical driving $\mu =0.6>{\mu }_c$. In the lower plots red dashed and black solid lines show the frequencies of the individual oscillators and of the mean field, respectively. For $\mu =0.2$ the force is not strong enough to induce full synchronization; it entrains the mean field, but not individual oscillators, so that $r<1$. For $\mu =0.6$ there are both domains of full synchrony and of MFL. Nonlinearity in the amplitude function $\epsilon A\left(\epsilon ,r\right)=1-\epsilon r^2$ and $\beta =0$.Reuse & Permissions

Figure 6

(Color online) Bifurcation diagram for the forced ensemble with the amplitude nonlinearity (see text for details). Domains of FS, MFL, and asynchrony (AS) are seen. The black dotted curve is the border of the FS area, obtained via Eq. (28). Black solid and red dashed curves at the borders of the MFL domain correspond to the loss of synchrony via saddle-node and Hopf bifurcations, respectively. The black filled circles are codimension-2 Takens-Bogdanov points; the bifurcation structure in the vicinity of these points is shown in Fig. 11. Also, near Hopf bifurcation lines there exist narrow areas of MFL regimes with modulation in amplitude and phase (libration) which are not shown here (see text for details). The mean-field amplitude and frequencies of mean field and individual oscillators along the horizontal dashed lines are shown in Fig. 5. Nonlinearity in the amplitude function $\epsilon A\left(\epsilon ,r\right)=1-\epsilon r^2$ and $\beta =0$.Reuse & Permissions

Figure 7

Illustration of the desynchronization transition at large forcing (here $\mu =0.8$). MFL corresponds to the fixed point (shown for $\nu =-0.82$). With the decrease of forcing frequency $\nu$ the limit cycle appears; however, first it is small and the deviation of phase difference $\delta$ is bounded (libration). It means that on average the frequency of the mean field is still locked to the force. With further decrease in $\nu$, phase difference becomes unbounded (this takes place at $\nu \approx -0.841$) and the synchrony gets lost. Nonlinearity in the amplitude function $\epsilon A\left(\epsilon ,r\right)=1-\epsilon r^2$ and $\beta =0$.Reuse & Permissions

Figure 8

(Color online) The same as in Fig. 6, but for $\beta =0.25\pi$. In addition to domains of FS, mean-field locking [gray (yellow) area], and AS, there are two domains where bistability MFL/AS is observed. Solid black and dashed red curves are lines of saddle node and Hopf bifurcations, respectively. Black circle shows the Takens-Bogdanov point (the second Takens-Bogdanov point lays outside of the shown range of $\nu$ and $\mu$).Reuse & Permissions

Figure 9

(Color online) The same as in Figs. 6, 8 but for the model with phase nonlinearity: $A=1$, $\beta ={\beta }_0+{\epsilon }^2{r}^2$, and ${\beta }_0=0.475\pi$.Reuse & Permissions

Figure 10

(Color online) Simulation of the ensemble of $N=1000$ oscillators with nonlinearity in the amplitude function and with different distribution of constants of motion; the distribution is parameterized by $q$ (see text). Amplitude of the force is $\mu =0.4$. (a): $\nu =0.4$. Black circle, solid red, and orange dashed lines correspond to $q=1$, $q=0.9$, and $q=0.7$, respectively. (b): $\nu =0.6$. Solid black and orange dashed lines correspond to $q=1$ and $q=0.7$, respectively.Reuse & Permissions

Figure 11

Sketch of the bifurcation lines around the left TB point in Fig. 6. Here $s{n}_1$ and $s{n}_2$ are lines of the saddle-node bifurcation; $H$ and $h$ are lines of Hopf and homoclinic bifurcation, respectively; the line labeled by $S$ marks the transition from librations to rotations. Domains I, II, and III correspond to fixed point solutions, librations, and rotations, respectively. In a very small domain IV we observe coexistence of two fixed point solutions, corresponding to two MFL states with different amplitudes of the mean field. The line $s{n}_1$ (not shown in Fig. 6) goes from one TB point to the other and separates domains with one stable fixed point (beyond this line) and one stable and two unstable fixed points (below this line).Reuse & Permissions