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Daan’s two-process model is known to be one of the most powerful models, covering various situations from free-running to sleep deprivation. In this study, bifurcation properties of the model dynamics are clarified using a circle map. In addition, the two-process model is applied to simulate the two distinct patterns (the period-prolonged: I and the period-shortened: II) of sleep-wake cycles during internal desynchronizations. We propose the novel interpretation that types I and II could be regarded as mutual entrainments between the body temperature rhythm and the sleep-wake cycle with the period ratios 3:4 and 3:2, respectively. From the bifurcation properties of the two-process model obtained above, the model is shown to be able to generate the respective type of mutual entrainment with an appropriate range of a parameter value, i.e, a gap between thresholds. The variable behavior of human circadian rhythm is …