Explosive synchronization with asymmetric frequency distribution

In this work, we study the synchronization in a generalized Kuramoto model with frequency-weighted coupling. In particular, we focus on the situations in which the frequency distributions of oscillators are asymmetric. For typical unimodal frequency distributions, such as Lorentzian, Gaussian, triangle, and even special Rayleigh, we find that the synchronization transition in the model generally converts from the first order to the second order as the central frequency shifts toward positive direction. We characterize two interesting coherent states in the system: In the former, two phase-locking clusters are formed, rotating with the same frequency. They correspond to those oscillators with relatively high frequencies while the oscillators with relatively small frequencies are not entrained. In the latter, two phase-locking clusters rotate with different frequencies, leading to the oscillation of the order parameter. We further conduct theoretical analysis to reveal the relation between the asymmetric frequency distribution and the conversion of synchronization type, and justify the coherent states observed in the system.

Figure 1

The order parameter $r$ vs the coupling strength $\kappa$. It is shown that with the increase of ${\omega }_0$, the synchronization transition converts from the first-order type into the second-order one. From (a) to (d), ${\omega }_0=0.3,0.5,0.8,1.2$, respectively. The dashed lines correspond to $\kappa r=1$. Lorentzian FD is used for Figs. 1 to 6. $\mathrm{\Delta }=1$ and $N=10000$ are the same throughout this paper.

Figure 2

Characterization of the steady states in the forward continuity of the first-order transition. ${\omega }_0=0.5$. Rows (a) and (b) correspond to $\kappa =1.5$ and $3.5$, respectively. Columns 1–3 correspond to the order parameter, the distributions of the instantaneous phases and frequencies, respectively. Column 4 is the enlargement of column 3. Throughout this paper, the purple dashed line, such as that in (b2), denotes the boundary of synchronized clusters predicted by Eq. (9), and the purple straight lines, such as those in (a4) and (b4), denote the fluctuation range of instantaneous frequencies of oscillators predicted by Eq. (11).

Figure 3

Characterization of the steady states in the backward continuity of the first-order transition. ${\omega }_0=0.5$. Rows (a)–(c) correspond to $\kappa =3.5,2.08$, and $1.5$, respectively. The parameters and the arrangement of panels are the same as in Fig. 2.

Figure 4

Characterization of the steady states in the forward continuity of the second-order transition. ${\omega }_0=1.2$. Rows (a)–(d) correspond to $\kappa =1.5,1.6,1.9$, and $2.5$, respectively. The parameters and the arrangement of panels are the same as in Fig. 2.

Figure 5

Characterization of the steady states in the backward continuity of the second-order transition. ${\omega }_0=1.2$. Rows (a)–(d) correspond to $\kappa =2.5,1.9,1.6$, and $1.5$, respectively. The parameters and the arrangement of panels are the same as in Fig. 2.

Figure 6

Characterization of synchronization and the steady states in model (2) with other typical FDs. Rows (a)–(c) correspond to Gaussian, triangle, and Rayleigh FD, respectively. It is shown that the system generally converts from first-order transition to second-order one as the FD shifts towards positive direction. Columns 3 and 4 characterize the type-I and type-II coherent states corresponding to column 2, respectively. The inset plots the Rayleigh FD that is inherently asymmetric.

Figure 7

Illustration of the phase-locking condition for Eq. (2). The blue curve is $f\left({\omega }^{\prime }\right)$ that is defined in Eq. (9). The red dashed line is $\kappa r$, which intersects with $f\left({\omega }^{\prime }\right)$ and determines the phase-locking region for frequency ${\omega }^{\prime }$, as shown by the purple rectangle. (a) Corresponds to the situation of type-I coherent states. (b) Corresponds to the situation of type-II and oscillating coherent states. In Figs. 1 to 6, we have marked the theoretical synchronization areas by purple dashed lines. The theoretical results are well consistent with the numerical ones.

Figure 8

Characterization of the quasiperiodical property for the time series $x\left(t\right)=\text{Re}\left[r\right(t\left)\right]$ in oscillating coherent states. (a), (b) Correspond to the situations in Figs. 3(b1) and 4(c1), respectively. The first two “spectrum lines” correspond to the dominant frequencies of the time series.

Figure 9

(a) A schematic illustration for the phase-locking condition between two oscillators, i.e., Eqs. (12) and (13). (b) Characterization the conversion from the first-order transition to the second-order one by $\rho$ vs ${\omega }_0$ (the circles) and $d/d_{\max }$ vs ${\omega }_0$ (the squares). $d$ denotes the width of hysteresis loop, as shown in Fig. 1. $d_{\max }=2$ is the largest width of hysteresis loop when ${\omega }_0=0$. Note that different scalings are used for $\rho$ (the left axis) and $d/d_{\max }$ (the right axis) for better visualization. Other parameters are the same as in Fig. 1.