The cell division cycle of T. gondii tachyzoites shows remarkable differences compared to higher eukaryotes. Most notably, the G2 phase is absent, and the S-phase and mitosis occur concurrently with the assembly of the new daughters. Moreover, gene expression in T. gondii tachyzoites is periodic and follows a "just-in-time" principle, such that the timing of mRNA expression is a proxy for the cell biological events - a typical feature in Apicomplexa. This periodicity imposes topological constraints on the gene expression data manifold, which is observable in low-dimensional PCA/UMAP projections as a twisted torus. To study cell cycle progression, trajectory analysis is typically performed on the data manifold. However, available tools for trajectory analysis do not account for periodicity, resulting in incorrect inference of trajectory flows. We have developed a Fourier Transform based approach for analyzing periodic trajectories, allowing us to effectively capture the flow of replicating cells. Moreover, we present a stochastic traffic model to study the cell cycle progression flow that enables the identification of phase transition points and cell cycle checkpoints. Together, these tools enable the reconstruction of time-series of pseudo-synchronized parasites in a pseudo-time scale, which can be transformed into a real-time scale. Our algorithms are packaged into a toolbox which is available for download as a Python library from GitHub, Pypi, and Bioconda. The library features interactive functionality for seamless processing and analysis. We demonstrate the utility of our tool by studying the cell asexual replication cycle of T. gondii using single-cell multi-omics data. The analysis identified previously known and potentially new transition points and revealed that chromatin accessibility and transcriptional activity display variation in peak timing, hinting at gene classes that are activated/repressed by potentially similar mechanisms.