Clock networks containing the same central architectures may vary drastically in their potential to oscillate, raising the question of what controls robustness, one of the essential functions of an oscillator. We computationally generate an atlas of oscillators and found that, while core topologies are critical for oscillations, local structures substantially modulate the degree of robustness. Strikingly, two local structures, incoherent and coherent inputs, can modify a core topology to promote and attenuate its robustness, additively. The findings underscore the importance of local modifications to the performance of the whole network. It may explain why auxiliary structures not required for oscillations are evolutionary conserved. We also extend this computational framework to search hidden network motifs for other clock functions, such as tunability that relates to the capabilities of a clock to adjust timing to external cues. Experimentally, we developed an artificial cell system in water-in-oil microemulsions, within which we reconstitute mitotic cell cycles that can perform self-sustained oscillations for 30 to 40 cycles over multiple days. The oscillation profiles, such as period, amplitude, and shape, can be quantitatively varied with the concentrations of clock regulators, energy levels, droplet sizes, and circuit design. Such innate flexibility makes it crucial to studying clock functions of tunability and stochasticity at the single-cell level. Combined with a pressure-driven multi-channel tuning setup and long-term time-lapse fluorescence microscopy, this system enables a high-throughput exploration in multi-dimension continuous parameter space and single-cell analysis of the clock dynamics and functions. We integrate this experimental platform with mathematical modeling to elucidate the topology-function relation of biological clocks. With FRET and optogenetics, we also investigate spatiotemporal cell-cycle dynamics in both homogeneous and heterogeneous microenvironments by reconstructing subcellular compartments.