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.