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Nonlinear synchrotron oscillations are present in virtually all accelerators (for economic reasons). For RF parameters likely to be used in 600-1000 GeV accelerators, or in high frequency (high harmonic) RF systems at lower energies, these nonlinearities give rise to numerous resonances, including basic noncoupled resonances which the present study investigates both by analytical and computational methods. A relatively simple theory1 has been sufficient to explain qualitatively many effects, such as "propellers" and "beads" in phase space, for both third- and half-integral resonances. Some extensions of the work of reference 1 are reported. However, understanding some details of the Â¿ = Â¿ resonance was not possible without the use of Moser transformations.2 In particular, a null occurs in the resonance driving term as the synchronous particle phase Â¿s is varied, an effect predicted by the Moser theory. Our results indicate that suitably designed particle accelerators may operate in the vicinity of some phase oscillation resonances but that the resonances in question influence the desirable choice of Â¿s.