This paper studies mechanisms of synchronisation and loss of synchrony among the three key oscillatory processes controlling sleep–wake cycles in the human brain: the 24-h circadian oscillator, the homeostatic sleep drive, and the environmental light–dark cycle. Synchronisation of these three rhythms promotes sleep and brain clearance and is critical for human health. Their desynchrony, on the other hand, is associated with impaired performance and disease development, including cancer, cardiovascular disease, and mental disorders. A biophysical model of arousal dynamics simulating sleep–wake cycles and circadian rhythms is used as the study system. It is based on established neurobiological mechanisms controlling sleep–wake transitions and incorporates the three oscillatory processes. Nonlinear dynamics methods and synchronisation theory are used to numerically investigate model dynamics under conditions that are not easily achievable in experiments. The role of homeostatic brain clearance rate in synchronisation is investigated, and selective turning on and off of coupling strengths between the oscillators allows us to determine their role in oscillators’ dynamics. We find that, in the absence of coupling between the circadian and homeostatic oscillators, the default state of the model corresponds to the endogenous homeostatic period that is far from $$\sim $$ 24-h rhythm of the circadian and light–dark cycles. Combined action of light and circadian oscillator on the homeostatic rhythm is required to achieve the typical sleep–wake pattern that is observed in young healthy people. Under 12-/12-h light–dark conditions with light at 80 lux, change of homeostatic clearance rate is found to induce two types of desynchronisation: (i) fast clearance rates $$\tau _H<58.1$$ h desynchronise the homeostatic oscillator from the circadian, while the circadian rhythm remains entrained to the light–dark cycle, and (ii) slow clearance rates $$\tau _H>69$$ h maintain synchronisation between the homeostatic and circadian oscillators, but the period of both is different from that of the light–dark cycle. Between these regimes, all three rhythms are synchronised under the studied conditions. The model predicts that the system is highly sensitive to external inputs to the neuronal populations of the sleep–wake switch, which affect the endogenous period of the homeostatic oscillator and can lead to complete loss of sleep. Model dynamics show that loss of synchronisation, which is traditionally ascribed solely to impairment of the circadian oscillator, can be caused by changes in the homeostatic clearance rate of the brain or external input to the neuronal populations of the sleep–wake switch. Thus, changes in circadian, homeostatic, and external factors (combined or specific) may be responsible for conditions of desynchronisation. This has significant implications for understanding individual variability in sleep–wake patterns and in mechanisms of sleep and circadian disorders, indicating that both the homeostatic and circadian mechanisms can be responsible for the same clinical or behavioural presentation of a disease.