The non-stationary characteristics of earthquake ground motion can significantly impact structural responses, and it is important to avoid the limitations of traditional response spectrum and power spectrum methods, which may fail to capture the temporal information of earthquake ground motion. This paper utilizes the fast wavelet transform method and local spectral theory to obtain earthquake wave samples and their local spectral characteristics at different scales. Dynamic time-history analysis is then employed to assess the contribution of earthquake wave samples corresponding to different wavelet components to the peak displacement response of the structure"s top node. The controlling components are identified, and the relationship between the energy values of the controlling wavelet components and the peak displacement response of the structure is analyzed. Additionally, the influence of local spectral characteristics of earthquake ground motion on the dynamic reliability of linear structures is examined. The results indicate that: The wavelet components controlling the linear structural responses are determined by the fundamental period of the structure, while the wavelet components controlling non-linear structural responses consist of low-frequency components in the neighboring periods of the fundamental period.The wavelet components with the highest energy values are not necessarily the controlling components, and their contribution to structural responses is significantly less than that of the controlling components. As the intensity of non-linearity increases, the low-frequency components in the neighboring periods of the fundamental period have a greater impact on structural responses, and this impact becomes more significant as the structural period increases. Removing a key wavelet component from the earthquake wave can lead to a significant reduction in failure probability, and the time at which the failure probability enters a stable segment is significantly advanced. The change is most pronounced after removing the controlling components.