Many oscillatory systems of great interest exhibit pulsing behavior. The analysis of such oscillators has historically utilized a constant-phase model such as the Kuramoto equation to describe their dynamics. These models accurately describe the behavior of pulsing oscillators on larger timescales, but to not explicitly capture the pulsing nature of the system being analyzed. Indeed, the Kuramoto model and its derivatives abstract the pulsing dynamics and instead use a constantly advancing phase, thereby blurring the specific dynamics in order to fit to an analytically tractable framework. In this thesis, a novel modification is presented by introducing a phase-dependence to the frequency of such oscillators. Consequently, this modification induces clear pulsing behavior, and thus introduces new dynamics such as nonlinear phase progressions that more accurately reflect the nature of such systems. The analysis of this new system of equations is presented and the discovery of a heretofore unknown phenomenon termed periodic stability is described in which the phase-locked state of the system oscillates between stability and instability at a frequency determined by the mean phase. The implications of this periodic stability on the system such as oscillations in the coherence are discussed. The theoretical predictions made by this novel analysis are simulated numerically, and extended to real experimental systems such as electrical Wien-Bridge oscillators and neurons; systems previously described using the abstract Kuramoto model. Finally, novel dynamics of a population of these oscillators such as wave self-organization are presented. The results of this work thus have clear implications on all real systems described presently by the Kuramoto model.