Oscillatory Motion of a Camphor Grain in a One-Dimensional Finite Region

TL;DR

This study analyzes how a camphor grain moves in a finite water channel, showing it oscillates or stays still depending on channel length and resistance, with a mathematical model explaining the bifurcation behavior.

Contribution

The paper presents an analytical reduction of the boundary-influenced motion of a camphor grain to an ODE, revealing bifurcation phenomena in a confined self-propelled system.

Findings

Oscillatory motion occurs depending on channel length and resistance.

A Hopf bifurcation explains the transition between rest and oscillation.

The model captures boundary effects analytically.

Abstract

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.