Bifurcation phenomena of two self-propelled camphor disks on an annular field depending on system length

Mode selection and bifurcation of a synchronized motion involving two symmetric self-propelled objects in a periodic one-dimensional domain were investigated numerically and experimentally by using camphor disks placed on an annular water channel. Newton's equation of motion for each camphor disk, whose driving force was the difference in surface tension, and a reaction-diffusion equation for camphor molecules on water were used in the numerical calculations. Among various dynamical behaviors found numerically, four kinds of synchronized motions (reversal oscillation, stop-and-move rotation, equally spaced rotation, and clustered rotation) were also observed in experiments by changing the diameter of the water channel. The mode bifurcation of these motions, including their coexistence, were clarified numerically and analytically in terms of the number density of the disk. These results suggest that the present mathematical model and the analysis of the equations can be worthwhile in understanding the characteristic features of motion, e.g., synchronization, collective motion, and their mode bifurcation.

Figure 1

Phase diagram obtained numerically for Eqs.(4) with $N=2$ (a) and the magnified diagram of the shaded region (b). The system length $L$ and the viscosity constant $\mu$ were varied as bifurcation parameters. We neglected the range $0\le L<1$, which was physically meaningless since the disks overlapped each other. The other parameters were fixed as $r_0=0.5,{\gamma }_0=1.0,{\gamma }_1=0.0,a=0.05$. For the numerical simulation, a finite difference method was used with $\mathrm{\Delta }x={10}^{-3}$, and $\mathrm{\Delta }t={10}^{-3}$, under periodic boundary conditions $u(-L,t)=u(L,t),u_x(-L,t)=u_x(L,t)$.

Figure 2

Typical spatiotemporal plots for each behavior in the phase diagram (Fig. 1): (a) stationary state (STA), (b) rotation (ROT), (c) symmetric oscillation (sOSC), (d) cluster (CLUS), (e) symmetric oscillatory rotation (sOSCR), (f) asymmetric oscillation (aOSC), (g) asymmetric oscillatory rotation (aOSCR), (h) irregular motion (IRREG). The phases of the disks ${\theta }_i\left(t\right)=\pi x_i\left(t\right)/L$ ($i=1,2$) are shown, together with the phase difference $\mathrm{\Delta }\theta =|{\theta }_2-{\theta }_1|$. The values of $L$ and $\mu$ used for the above plots are $(L,\mu )=$ (a) $(4,0.100)$, (b) $(40,0.080)$, (c) $(32,0.030)$, (d) $(48,0.010)$, (e) $(8,0.006)$, (f) $(4,0.015)$, (g) $(4,0.010)$, and (h) $(8,0.015)$.

Figure 3

Bifurcation curves for the constant rotation solutions (STA, ROT, and CLUS) obtained analytically from Eqs. (6) and (7). (a) As $L$ is fixed and $\mu$ is varied, the stationary-state and rotation solutions typically undergo pitchfork, Hopf, and saddle-node bifurcations, which are represented by triangles, squares, and diamonds, respectively. The black, gray, and dotted curves denote the stationary-state, rotation, and cluster solutions, respectively. Stable (resp. unstable) solutions are depicted by solid (resp. broken) curves. (b) When $L$ is also varied, five bifurcation curves are plotted: the Hopf and pitchfork bifurcation points of the stationary-state (resp. rotation) solutions, which are shown by the filled (resp. open) squares and triangles in (a), form the black (resp. gray) solid and broken curves, respectively, and the saddle-node points of the rotation solutions, which are shown by the open diamonds in (a), form the dot-dash curve. Note that the four arrows (i), (ii), (iii), and (iv) correspond to those in (a).

Figure 4

Snapshots of experimental results for a typical synchronized motion of two camphor disks ($\alpha$ and $\beta$) on annular water channels with different inner diameters $[\mathit{ID}=$ (a) 30, (b) 20, (c) 60, and (d) 70 mm] (top view). White and gray arrows denote the directions of motion for two camphor disks, $\alpha$ and $\beta$, respectively. Note that, throughout the experiments, the diameters of the camphor disks and the channel widths were fixed as 10 and 20 mm, respectively.

Figure 5

Time evolution of (1) ($\mathrm{\Delta }\theta =|{\theta }_{\alpha }-{\theta }_{\beta }|$) and (2) ${\theta }_{\alpha }$ (black line) and ${\theta }_{\beta }$ (gray line) for different diameters of the water channels $[\mathit{ID}=$ (a) 30, (b) 20, (c) 60, (d) 70 mm]. The data for (a–d) correspond to those for (a–d) in Fig. 4, respectively. The angles ${\theta }_{\alpha }$ and ${\theta }_{\beta }$ are the description of the polar coordinate for the motion of the camphor disks, $\alpha$ and $\beta$, respectively.

Figure 6

Schematic figure for distinguishing between Motions I and II. In the experiments, the two motions were distinguished by the average value of the quantity $\mathrm{\Delta }{\theta }_{-}/\mathrm{\Delta }{\theta }_{+}$ where $\mathrm{\Delta }{\theta }_{+}$ and $\mathrm{\Delta }{\theta }_{-}$ were the angles of the counterclockwise and clockwise rotation made by a camphor disk, respectively. We assumed that $\mathrm{\Delta }{\theta }_{-}\le \mathrm{\Delta }{\theta }_{+}$. The motions were classified as Motion I when $0.3\le \mathrm{\Delta }{\theta }_{-}/\mathrm{\Delta }{\theta }_{+}\le 1$ and as Motion II when $\mathrm{\Delta }{\theta }_{-}/\mathrm{\Delta }{\theta }_{+}<0.3$.

Figure 7

Superposition of the phase diagrams (Fig. 1) and the bifurcation curves in Fig. 3. Note that the bifurcation curves are all represented by black solid curves, and the phase boundaries by black broken curves. We see that the Hopf (resp. pitchfork) bifurcation curve for the ROT solution, which are denoted by the gray solid (resp. gray broken) curve in Fig. 3, almost coincides with the ROT-sOSCR (resp. ROT-CLUS) boundary.

Figure 8

(a) Bifurcation curve of the ROT, sOSCR and CLUS solutions for $N=2, \mu =0.0002$, and $L=0$–60. The vertical axis denotes the maximal value of the velocity of the two disks. The black solid and broken curves, respectively, indicate stable and unstable ROT solutions, the gray solid curve denotes stable sOSCR solution, and the black dotted curve does stable CLUS solution. The curves for ROT and CLUS solutions were obtained analytically from Eqs. (6) and (7), while the one for sOSCR solution was obtained by numerical simulation. (b) Congestion motion obtained numerically for Eqs. (4) for $N=3$ (left) and $N=6$ (right). In both cases, $L$ is chosen from the range $\left[N{L}_1/2,N{L}_2/2\right]$. One representative trajectory is drawn by a bolder line. Congestion waves are observed that propagate in the opposite direction of the rotation. (c) Billiard motion obtained numerically for Eqs. (4) for $N=3$ (left) and $N=6$ (right). In both cases, $L$ is chosen from the range $\left[N{L}_2/2,N{L}_3/2\right]$. One representative trajectory is drawn by a bolder line. The disks rotate in the same direction, except for one which rotates in the opposite direction, almost collide with one of the other disks, and then change direction. In this manner, the collision wave propagates in the opposite direction of the rotation.