  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.
