Due to their high sensitivity to excitations with low intensity, bistable energy harvesting systems have received significant attention. In practical applications, it is difficult to achieve a bistable energy harvester (BEH) with a perfectly symmetric potential energy function. Moreover, gravity acts to exert a significant influence on the output response of a BEH oscillator when excited at different bias angles. Therefore, the experimental output voltage time-series of an asymmetric potential BEH are examined in this paper. The BEH studied here is composed of a cantilever beam, two piezoelectric layers at the root and two magnets at the end, and subjected to harmonic excitations at different bias angles. The energy harvesting system exhibited intra-well, periodic and chaotic snap-through vibrational patterns under different excitation frequencies at different bias angles. To better understand the multiple dynamic behaviors of the system corresponding to different power outputs, we identified the output voltage response by the methods of multiscale entropy and recurrence plots. Results indicate that periodic and chaotic vibrational patterns can be readily distinguished by the methods employed. Furthermore, it is demonstrated that the bias angle had a significant influence on the output power of the asymmetric potential BEH.