General Criteria for Determining Rotation or Oscillation in a   Two-dimensional Axisymmetric System
  A self-propelled particle in a two-dimensional axisymmetric system, such as a particle in a central force field or confined in a circular region, may show rotational or oscillatory motion. These motions do not require asymmetry of the particle or the boundary, but arise through spontaneous symmetry breaking. We propose a generic model for a self-propelled particle in a two-dimensional axisymmetric system. A weakly nonlinear analysis establishes criteria for determining rotational or oscillatory motion. 
Yuki Koyano, Natsuhiko Yoshinaga, Hiroyuki Kitahata
nlin.PS