One of the essential characteristics of an authentic circadian clock is that the free-running period sustains an approximately 24-hour cycle. When organisms are exposed to an external stimulus, the endogenous oscillators synchronize to the cycling environment signal in a process known as entrainment. These environmental cues perform an important role in resetting the phase and period of the circadian clock. A "generalized assumption" states that when an organism has a short period, it will experience a phase advance, while an organism with a long period experiences a phase delay. Despite widespread use, this positive relationship relating period to the phase of entrainment does not describe all known experimental data. We developed a two-step entrainment model to explain a broader range of results as well as provide more quantitative analysis. We prove existence and stability of periodic orbits and given analytical solutions of the range of entrainment, fit the phase trajectory over the entire entrainment process to data from a published study for 12 subjects in extended day cycles, i.e., longer than 24 h. Our simulations closely replicated the phase data and predicted correctly the phase of entrainment. We investigate the factors related to the rate of entrainment (ROE) and present the three-dimensional parameter spaces, illustrating the various behaviors of the phase of entrainment and ROE. Our findings can be applied to diagnostics and treatments for patients with sleep disorders caused by shift work or jet lag.
Keywords: Free-running period; Human circadian rhythm; Phase of entrainment; Range of entrainment; Rate of entrainment; Trajectory fitting.