This thesis presents the results of studies on the problem of subharmonic oscillations in nonlinear Paul trap mass spectrometry. The objective of this thesis is to determine whether the subharmonic oscillations of ions in the trap could in any way affect the quality of mass spectrum in resonance ejection experiments. This is accomplished by studying the existence and stability criteria of these oscillations. This study is done for two casesone in which the auxiliary excitation frequency is close to thrice the ion axial secular frequency and other in which it is close to twice the ion axial secular frequency. Initially, the equations describing the ion motion in the presence of auxiliary excitation are derived. The equations describing the ion motion are then brought into a form easily amenable to analysis by techniques of perturbation theory. The necessary background and definitions to understand the basis for the thesis, along with a survey of results obtained in relevant areas in mass spectrometry and nonlinear dynamics is then developed. The first problem is the study of subharmonic oscillations when the auxiliary excitation frequency is in the vicinity of thrice the ion axial secular frequencies taken up. The application of the multiple time scales technique to the equations describing ion motion gives the slow flow equations, that describe the evolution of amplitudes of axial and radial oscillations. The expression for the steady state amplitudes of these oscillations are then derived. From these expressions the conditions for the existence of the oscillations are obtained in terms of the auxiliary amplitude and the frequency detuning. This is then followed by a detailed stability analysis for the subharmonic oscillation with a given amplitude and phase. The study ends with the discussion of the results obtained, the pertinent numerical studies and the relevance of this study to mass spectrometry. The second study is regarding the problem of subharmonic oscillations when the auxiliary excitation frequency is close to twice the ion axial secular frequency is analyzed. When PoincareLindstedt’s method is applied to the ion motion equations, the amplitude frequency relationship that describes the relation between the steady state subharmonic oscillation amplitude and the frequency detuning is obtained. The variation of the oscillation amplitude with the frequency detuning is then studied. Then follows the analysis of stability. The stability of subharmonic oscillation is analyzed using the results from the standard analysis of Hill’s equation of fourth order. This study ends with the discussion of the results obtained in the context of mass spectrometry. Finally, a summary of the results obtained is discussed.