Vibrational energy harvesting is a subject that has received much attention as a possible replacement for remote battery-operated sensor networks. We describe a vibrational energy harvester with an asymmetric buckled beam which is constructed out of commercially available components, and has demonstrated a significantly increased bandwidth compared to a device exhibiting linear resonance. This particular beam could not be mathematically modeled by existing analytic techniques, so a method is developed to produce a reduced order analytic model based on a finite element representation of the system. Moreover, we present an argument for why common single-mode Galerkin projection models are incapable of accurately reproducing snap-through behavior at higher than infinitesimal buckling levels. The model developed here demonstrates good agreement with the behavior exhibited by the experimental system around the parameter region of high power output, as evidenced by similar phase portraits and frequency response plots. Further, an argument is presented for why current trends towards testing non-linear systems with constant acceleration frequency sweeps are misleading, and an alternative comparison platform is suggested. The model is analyzed from a dynamical systems perspective, and it is shown that the transitions between high and low power output can be associated with a period doubling cascade or a boundary crisis where a chaotic attractor stabilizes through an intermittency transition. Chaotic behavior is observed to be closely related to the high power output region, but it is possible to have appreciable power output with a periodic response. Potential future work involves analyzing alternate beam configurations in search of an optimal solution to the high power output bandwidth problem.