<p>This paper presents a theoretical model of an asymmetric bistable shell, exhibiting the harmonic responses and snap-through transition phenomena systematically. The frequency band containing the first four natural frequencies acts as the critical domain for the snap-through motions. These motions exhibit diverse temporal features, including limit cycle oscillations, two-period/three-period/five-period/seven-period snap-through sequences, quasi-periodic transitions, and chaotic snap-through behaviors, all of which are proven to be loading-condition dependent. The diversity of multi-periodic snap-through trajectories and morphological characteristics—even at identical cycle numbers—emanates from the intricate synergy of nonlinear dynamic interactions and excitation parameter sensitivity. The harmonic response characteristics in dynamic snap-through motions act as nonlinear fingerprints, intricately encoding the system's evolutionary trajectory from ordered periodicity to chaotic complexity. Governed by resonance conditions, amplitude-dependent bifurcation sequences, and geometric nonlinearities, these characteristics offer profound mechanistic insights for the snap-through actuation and energy harvesting concerning the modes of snap-through. Excitation amplitude and frequency—vibration modes—harmonic responses—snap-through transitions form a complete closed loop, whereby the excitation (amplitude and frequency) and vibration modes jointly determine the type of harmonic responses, which serve as the prerequisite for snap-through transitions. Compared with prior work, the key novelty of this study resides in this closed loop. The theoretical model's accuracy can be validated by simulation and experimental results.</p>