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 was composed of a cantilever beam, two piezo-electric layers at the root and two magnets at the end, and was subjected to harmonic excitations at different bias angles. The energy harvesting system exhibited intrawell, periodic, and chaotic snap-through vibrational patterns under different excitation frequencies at different bias angles explored. To better understand the multiple dynamic behaviors of the system corresponding to different power outputs, we identify the output voltage response by the methods of multiscale entropy (MSE) 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.