Discontinuous phase transition in the Kuramoto model with asymmetric dynamic interaction

We investigate the critical behavior of the modified Kuramoto model with an asymmetric dynamic interaction which has been proposed to explain the difference between the synchronized frequency and the average intrinsic frequency. We find that the discontinuous phase transition arises when oscillators interact only with other oscillators whose phases are ahead. From the comparison with the conventional Kuramoto model in which the interaction possesses phase reflection symmetry, we conclude that the dynamical symmetry breaking and the dynamic change in interaction structure play important roles in changing the transition nature from continuous to discontinuous ones.

Figure 1

(a) Schematic diagram for the dynamic interaction. The black circle denotes the $i\mathrm{th}$ oscillator with phase ${\varphi }_i\left(t\right)$, and the red (blue) ones denote the oscillators with phases ahead of (behind) ${\varphi }_i\left(t\right)$ in the dynamic interaction set ${\mathrm{\Lambda }}_i\left(t\right)$ at time $t$. The $i\mathrm{th}$ oscillator interacts with $|{\mathrm{\Lambda }}_i\left(t\right)|=3$ oscillators denoted with red and blue circles. The gray circles denote the other oscillators which are not elements of ${\mathrm{\Lambda }}_i\left(t\right)$. (b) Directed temporal network representation of the dynamic interaction. Each node denotes the corresponding oscillator in (a), and the links denote the interaction between oscillators. The red and blue arrows denote the links directed away from the $i\mathrm{th}$ node, and the black arrows denote the links from other nodes. The direction of the link is defined as being from $i$ to $j$ at time $t$ if $j\in {\mathrm{\Lambda }}_i\left(t\right)$. The number of elements in the set ${\mathrm{\Lambda }}_i\left(t\right)$ is $|{\mathrm{\Lambda }}_i\left(t\right)|$, which is interpreted as node $i$'s outdegree at time $t$.

Figure 2

The order parameter $\langle \overline{R}\rangle$ as a function of the coupling strength $K$ for system sizes $N=25,50,100,200$, and 400 at (a) $({\lambda }_f,{\lambda }_b)=(1/2,1/5)$ and (b) $({\lambda }_f,{\lambda }_b)=(1/4,0)$. Note that the interaction is not symmetric (${\lambda }_f\ne {\lambda }_b$) for both (a) and (b), but the order parameter $\langle \overline{R}\rangle$ exhibits sharply distinct behaviors.

Figure 3

(a) and (b) The probability distribution $P\left(\right|\mathrm{\Lambda }|/N)$ of the scaled outdegree $\left|\mathrm{\Lambda }\right|/N$ for system size $N=400$ at (a) $({\lambda }_f,{\lambda }_b)=$ $(1/2,1/5)$ and (b) $(1/4,0)$. (c) and (d) The average scaled outdegree $\langle \overline{\left|\mathrm{\Lambda }\right|}\rangle /N$ at various systems sizes versus the coupling strength $K$ at (c) $({\lambda }_f,{\lambda }_b)=$ $(1/2,1/5)$ and (d) $(1/4,0)$. Depending on the value of ${\lambda }_b$, the probability distribution in (a) and (b) and the average scaled outdegree in (c) and (d) exhibit very different behaviors.

Figure 4

(a) The order parameter $\langle \overline{R}\rangle$ and (b) the normalized average outdegree $\langle \overline{\left|\mathrm{\Lambda }\right|}\rangle$ as functions of the scaled coupling strength $\tilde{K}$ for $({\lambda }_f,{\lambda }_b)=(1/4,0)$. [The same data as in Figs. 2 and 3 have been used.]

Figure 5

The probability distribution $P\left(R\right)$ of the order parameter $R$ for system size $N=400$ at (a) $({\lambda }_f,{\lambda }_b)=$ $(1/2,1/5)$ and (b) $(1/4,0)$. The positions of peaks $R_{\mathrm{peak}}$ in $P\left(R\right)$ are shown in (c) and (d), which correspond to (a) and (b), respectively. In (d), note that two peaks coexist in the shaded region around $\tilde{K}\approx 3.5$.

Figure 6

(a) Peak position in the $(R,\tilde{K})$ plane for different system sizes and (b) the probability density of the order parameter near $(R,\tilde{K})=(0.6,3.5)$.