This paper reports a numerical study of two-dimensional periodically perturbed flow past a cylinder. Both harmonic and non-harmonic perturbation waveforms of the inflow velocity are considered for a maximum instantaneous Reynolds number of 180. Phase portraits of the lift force are employed to identify the dynamical state of the cylinder wake and determine the range of kinematical parameters for which primary synchronization occurs, that is the regime where vortex formation is phase-locked to the subharmonic of the perturbation frequency. The effect of different perturbation waveforms on the synchronization range and on patterns of vortex formation is examined in detail over the normalized amplitude–frequency space. It is shown that systematic shifts of the synchronization range, towards either higher or lower frequencies, can be attained by imposing different perturbation waveforms. As a consequence, in certain regions of the parameter space it is possible to obtain multiple periodic and/or quasi-periodic wake states for waveforms of exactly the same amplitude and frequency. For the range of parameters where synchronization occurs, different vortex patterns are attained in the wake involving the shedding of solitary and binary vortices, or mixtures thereof, which can be controlled by the perturbation waveform. The phenomenology of these patterns, which result from modification of the basic Kármán mode in the unperturbed wake, is discussed and explained in terms of the generation of circulation that is induced by perturbations in the relative velocity. Vortex patterns from cylinders oscillating either in line with or transverse to a free stream are recast in the framework of the relative velocity.