Rotational motion of a camphor disk in a circular region

TL;DR

This study investigates the rotational motion of a camphor disk confined in a circular region, revealing that stable rotation emerges through a double-Hopf bifurcation, supported by theoretical, numerical, and experimental evidence.

Contribution

The paper develops a reduced mathematical model and identifies the bifurcation mechanism responsible for stable rotational motion of a camphor disk.

Findings

Rotational motion is stabilized via a double-Hopf bifurcation.

Theoretical predictions align with numerical simulations.

Experimental results confirm the model's validity.

Abstract

In a two-dimensional axisymmetric system, a symmetric self-propelled particle exhibits rotational or oscillatory motion from the consideration of the system symmetry. 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 well corresponded to the experimental results.