In the first part, the equations of motion in a weakly corrugated, periodic magnetic field are linearized and solved by using paraxial approximation, to describe the model and the associated resonance condition. In the second part, the nonlinear evolution of the magnetic moment of resonant particles, in connection with their axial displacement is investigated analytically by using the multiple scale method. It is seen that the linear evolution is converted into a slow and periodic oscillation around the unperturbed value, with a considerable amplitude. The analytic expressions for the period and amplitude of the oscillations are derived and compared with the numerical simulations, which are also presented. Finally, the limitations of the paraxial approximation are concluded by investigating the numerical simulations, with actual field expressions. (C) 1997 American Institute of Physics.