Truncated series solution of the equation of motion of ions trapped in a radiofrequency quadrupole trap with superimposed octopolar potential

The power series method is used to solve numerically the equation of motion of ions in a radiofrequency quadrupole ion trap when a small octopolar potential is present. Every time step, the coordinates of the ion are represented by a truncated series of degree 15. The stability diagram of the trap is obtained by using the method to simulate the behavior of hundreds of ions up to a relatively large time. The nonlinear resonance lines are observed. The ion's oscillation frequencies are then studied. Their theoretical expression is derived and compared to the results of the truncated series solution.