Traveling wave in a three-dimensional array of conformist and contrarian oscillators

We consider a system of conformist and contrarian oscillators coupled locally in a three-dimensional cubic lattice and explore collective behavior of the system. The conformist oscillators attractively interact with the neighbor oscillators and therefore tend to be aligned with the neighbors' phase. The contrarian oscillators interact repulsively with the neighbors and therefore tend to be out of phase with them. In this paper, we investigate whether many peculiar dynamics that have been observed in the mean-field system with global coupling can emerge even with local coupling. In particular, we pay attention to the possibility that a traveling wave may arise. We find that the traveling wave occurs due to coupling asymmetry and not by global coupling; this observation confirms that the global coupling is not essential to the occurrence of a traveling wave in the system. The traveling wave can be a mechanism for the coherent rhythm generation of the circadian clock or of hormone secretion in biological systems under local coupling.

Figure 1

Schematic plot of $N=8^3$ conformist (blue, dark) oscillators with attractive coupling and contrarian (red, light) oscillators with repulsive coupling. From left to right, conformist fractions are $p=0.2,0.5$, and $0.8$.

Figure 2

Snapshots of the oscillators' phase at a certain time $t$ for $Q=0.05$, where the blue (dark) points represent the conformist oscillators, and the red (light) points represent the contrarians for the system of $N={16}^3$ oscillators for (a) $p=0.5$, incoherent state, (b) $0.92$, traveling-wave state, and (c) $0.96,\pi$ state. The radial positions of the oscillators are randomly assigned within some range (green circles) to prevent overlaps between the oscillators having the same phase. The average phase difference between dense conformist (blue arrow) and sparse contrarian clusters (red arrow) are represented as $\delta$ : $\delta <\pi$ for the traveling-wave state, and $\delta =\pi$ for the $\pi$ state. The phase-distribution function $P({\varphi }_i,t)$ at different times $t$, where the same color as the above panel is used for the conformist and contrarian oscillators for the corresponding state: (d) $p=0.5$, incoherent state, (e) $0.92$, traveling-wave state, and (f) $0.96,\pi$ state. Note that left peaks represent conformist clusters, and right peaks represent contrarian clusters in panels (e) and (f). The data are also shown by a contour map, with heights indicated by the color bar.

Figure 3

Wave speed $\Omega$ is plotted as a function of the fraction $p$ for various values of $Q(\equiv |K^{-}|/K^{+})$. The data have been averaged over 20 ensembles with different initial conditions, and the error bars represent the statistical deviations. Lines are drawn to guide the eye.

Figure 4

Wave speed $\Omega$ plotted as a function of $p$ for $Q=0.05$, varying the system size $L$. The data for each $L$ have been averaged over 20 ensembles with different initial conditions, and the error bars represent the statistical deviations. Lines are drawn to guide the eye.

Figure 5

Order parameters of the entire system $R$ (black circles), the conformists $r_1$ (blue diamonds), and the contrarians $r_2$ (red triangles) are shown as a function of $p$ for $Q=0.05$, where the system size $N={16}^3$ is chosen. The data have been averaged over 20 different configurations, and the error bars represent the statistical deviations. Lines are drawn to guide the eye.

Figure 6

(a) Order parameter $R$ is plotted as a function of $p$ for various system size $L$ for $Q=0.05$. The data have been averaged over 20 configurations with different initial conditions, and the error bars represent the statistical deviations. Lines are drawn to guide the eye. Successive slope $\Delta R/\Delta p$ for the transition at $p\simeq 0.7$ is also shown as a function of $p$, varying the values of $L$ (see inset). (b) The transition points $p_c^{\left(1\right)}$ (lower) and $p_c^{\left(2\right)}$ (upper) for the two transitions as a function of $1/L$; linear fitting of the data suggests $p_c^{\left(1\right)}=0.772$ and $p_c^{\left(2\right)}=0.953$ at the thermodynamic limit $L\to \infty$.

Figure 7

Order parameter $R$ shown as a function of $p$ for various values of $Q$ from $0.01$ to $0.5$. System size $N={16}^3$ is chosen. The data have been averaged over 20 different configurations, and the error bars represent the statistical deviations. Lines are drawn to guide the eye.