Rapid advances in single-cell RNA sequencing (scRNA-seq) technology have enabled the investigation of gene expression changes at the single-cell level, particularly for elucidating the heterogeneity among cells and complex biological processes. This technique reveals subtle molecular differences within individual cells, thereby offering a unique viewpoint for the investigation of cell cycle progression, cellular differentiation, and disease pathogenesis. However, accurately identifying and analyzing cell cycle dynamics in scRNA-seq data remains challenging due to the complexity of the data and the subtle differences between cell states. To address this challenge, we developed the integrated Sinusoidal and Piecewise AutoEncoder (SPAE), an autoencoder-based piecewise linear model, for characterizing the cell cycle dynamics and cell states in scRNA-seq data. Compared with existing methods, SPAE demonstrates substantially improved accuracy and robustness in cell cycle characterization. Additionally, SPAE can accurately predict cancer cell cycle transitions and effectively facilitate the removal of cell cycle effects from gene expression data. SPAE is available for non-commercial use at https://github.com/YaJahn/SPAE.