The motion of a self-propelled particle is affected by its surroundings, such as boundaries or external fields. In this paper, we investigated the bifurcation of the motion of a camphor grain, as a simple actual self-propelled system, confined in a one-dimensional finite region. A camphor grain exhibits oscillatory motion or remains at rest around the center position in a one-dimensional finite water channel, depending on the length of the water channel and the resistance coefficient. A mathematical model including the boundary effect is analytically reduced to an ordinary differential equation. Linear stability analysis reveals that the Hopf bifurcation occurs, reflecting the symmetry of the system.

We consider a rotor made of two camphor disks glued below the ends of a plastic stripe. The disks are floating on a water surface and the plastic stripe does not touch the surface. The system can rotate around a vertical axis located at the center of the stripe. The disks dissipate camphor molecules. The driving momentum comes from the nonuniformity of surface tension resulting from inhomogeneous surface concentration of camphor molecules around the disks. We investigate the stationary angular velocity as a function of rotor radius $\ell$. For large $\ell$ the angular velocity decreases for increasing $\ell$. At a specific value of $\ell$ the angular velocity reaches its maximum and, for short $\ell$ it rapidly decreases. Such behavior is confirmed by a simple numerical model. The model also predicts that there is a critical rotor size below which it does not rotate. Within the introduced model we analyze the type of this bifurcation.

The self-propelled motion with deformation of micrometer-sized soft matter in water has potential application not only for underwater carriers or probes in very narrow spaces but also for understanding cell locomotion in terms of non-equilibrium physics. As far as we know, there have been no reports about micrometer-sized self-propelled soft matter mimicking amoeboid motion underwater. Here, we report an artificial molecular system of underwater oil droplets exhibiting self-propelled motion with deformation as an initial experimental model. We describe the heterogeneity in a deformable self-propelled oil droplet system in aqueous and oil phases and at their interface based on the behavior and interaction of surfactant and oil molecules. The current results have great importance for scientific frontiers such as developing deformable micro-swimmers and exploring the emergence of self-locomotion of oil droplet-type protocells.

In a two-dimensional axisymmetric system, the system symmetry allows rotational or oscillatory motion as stable stationary motion for a symmetric self-propelled particle. In the present paper, we studied the motion of a camphor disk confined in a two-dimensional circular region. By reducing the mathematical model describing the dynamics of the motion of a camphor disk and the concentration field of camphor molecules on a water surface, we analyzed the reduced equations around a bifurcation point where the rest state at the center of the system becomes unstable. As a result, we found that rotational motion is stably realized through the double-Hopf bifurcation from the rest state. The theoretical results were confirmed by numerical calculation and corresponded well to the experimental results.

A simple model of an active colloid consisting of dumbbell-shaped particles that cyclically change their length without propelling themselves is proposed and analyzed. At nanoscales, it represents an idealization for bacterial cytoplasm or for a biomembrane with active protein inclusions. Our numerical simulations demonstrate that non-equilibrium conformational activity of particles can strongly affect diffusion and structural relaxation: while a passive colloid behaves as a glass, it gets progressively fluidized when the activity is turned on. Qualitatively, this agrees with experimental results on optical tracking of probe particles in bacterial and yeast cells where metabolism-induced fluidization of cytoplasm was observed.