The interaction between charged particles and circularly polarized electromagnetic waves propagating parallel to a static magnetic field is an important problem in classical plasma physics owing to possible energy exchange at the cyclotron resonance. The nonlinear motion of ions in such fields is known to be described by elliptic functions, but the physics is obscured by the complicated expressions. The properties of the particle trajectories are much more transparent in the perturbation–expansion solution derived here. For small wave amplitudes the solutions are valid for an extended time and the resonance effects are clearly evident. The initial phase between the particles and wave fields emerges as an important condition on the motion. Potential applications of the results in magnetospheric physics and plasma fusion devices are briefly discussed.