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In this paper, the dynamical analysis of a periodic potential subject to noise without and with the periodic signal is presented. The third-order differential equation of the stationary probability density (SPD) is derived for the periodic potential, which driven by multiplicative dichotomous and additive white noise. It is found that the SPD undergoes a transition from the unimodal to the bimodal structure with the increase of multiplicative noise intensity. The power spectra density (PSD) and the qualify factor show the peak structure when a suitable dose of noise intensity is added. That is, the coherence resonance (CR) appears. Meanwhile, the average input energy per period and the amplitude of average response are calculated to quantify stochastic resonance (SR). The curve of the average input energy per period shows multiple extrema as a function of multiplicative noise intensity at the small fixed additive noise intensity. This phenomenon indicates the stochastic multi-resonance happens for this case.