Existence of hysteresis in the Kuramoto model with bimodal frequency distributions

We investigate the transition to synchronization in the Kuramoto model with bimodal distributions of the natural frequencies. Previous studies have concluded that the model exhibits a hysteretic phase transition if the bimodal distribution is close to a unimodal one due to the shallowness of the central dip. Here we show that proximity to the unimodal-bimodal border does not necessarily imply hysteresis when the width, but not the depth, of the central dip tends to zero. We draw this conclusion from a detailed study of the Kuramoto model with a suitable family of bimodal distributions.

Figure 1

(Color online) Examples of bimodal frequency distributions given by Eq. (5) with $\delta =1$. Left panel: $\xi =\gamma$ [what implies $g\left(0\right)=0$]. Note that as $\gamma$ decreases the maxima of the distribution become closer. For all these distributions (with $\xi =\gamma$) the route to synchronization as $K$ is increased from zero is $\text{I}\to \text{SW}\to \text{PS}$ (cf. Fig. 3). Right panel: two examples with $\xi <\gamma$. The distribution depicted with a continuous line has well separated peaks and shows a transition $\text{I}\to \text{SW}\to \text{PS}$, whereas the other distribution is closer to the unimodal limit [Eq. (7)] and presents hysteresis in the route to synchronization (cf. Fig. 4).

Figure 2

The parameter space of distribution [Eq. (5)] [not defined above the bisectrix $\xi =\gamma$ neither at point (1,1)]. Function [Eq. (5)] is unimodal below line B and bimodal above it (shaded regions). Three lines signal the loci of codimension-two bifurcations (A, B, and D) projected on the $\left(\gamma ,\xi \right)$ plane. Between lines D and B (dark gray region) the transition to synchronization involves hysteresis.

Figure 3

Phase diagram for $\xi =\gamma$. For this case the synchronization transition never involves hysteresis. The solid lines mark the saddle-node (SNIC) [from Eq. (26)] and the Hopf [Eq. (24)] bifurcations. Symbols correspond to the numerical estimation of the bifurcation lines via numerical integration of the original Eq. (1) with $N=2000$.

Figure 4

Phase diagram for $\xi =0.5$. Solid lines mark the bifurcations: saddle-node off the limit cycle (SN), SNIC, Hopf bifurcation [Eq. (24)], heteroclinic bifurcation (found numerically using the reduced equations), and pitchfork bifurcation [Eq. (30)]. Three big circles signal the codimension-two points: (a) Takens-Bogdanov, (b) degenerate pitchfork, and (d) saddle-node separatrix loop. The open symbols correspond to different bifurcations found by numerical integration of Eq. (1) with $N=2000$. Filled symbols inside each region indicate parameter values for the phase portraits in Fig. 5.

Figure 5

Phase portraits in (rotated) $x_1,x_2$ coordinates for qualitatively different cases. Each panel corresponds to a value of $\gamma$ and $K$ at the position of a filled symbol in Fig. 4. (a) and (b) Partial synchronization with $K=4$ and (a) $\gamma =0.6$ and (b) $\gamma =0.75$; (c) coexistence SW/PS—$\gamma =0.67$, $K=3.45$; (d) coexistence I/PS—$\gamma =0.7$, $K=3.3$; (e) SW—$\gamma =0.6$, $K=3.5$; and (f) I—$\gamma =0.6$, $K=2.5$.