Ion motion in a periodic radio frequency (RF) quadrupole electric field is analysed theoretically by the matrix method and direct trajectory calculation. General properties of the ion motion: stability condition, oscillation spectrum and secular frequency are derived analytically from the elements of the transformation matrix. Stability diagrams for ion motion in the Paul ion trap are presented for rectangular waveforms with different duty cycles (duration of pulse over period). Calculation of the secular frequencies of the ion motion in the ion trap is performed for the first time. The relation of radial and axial secular frequencies along the RF scan line was found to be practically identical in both the square waveform and harmonic voltage cases. Pulse shape distortions, due to resistive-inductive-capacitive filtering in real devices, are considered. Stability diagrams of ion motion in the ion trap with distorted voltage waveforms are calculated. Distortion of the waveform is shown to introduce minor changes in the diagram shape with respect to the diagram for an ideal square wave. Within the first stable region, distortion of the waveform does not lead to any auxiliary parametric resonances of the ion motion. Ion trapping with a pure random pulsed voltage is investigated by means of direct trajectory simulations. It is shown that, in this case, the ion motion can be conditionally stable for a considerable length of time.