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A self-propelled particle in a two-dimensional axisymmetric system, such as aparticle in a central force field or confined in a circular region, may showrotational or oscillatory motion. These motions do not require asymmetry of theparticle or the boundary, but arise through spontaneous symmetry breaking. Wepropose a generic model for a self-propelled particle in a two-dimensionalaxisymmetric system. A weakly nonlinear analysis establishes criteria fordetermining rotational or oscillatory motion.

General Criteria for Determining Rotation or Oscillation in a Two-dimensional Axisymmetric System