Entrainment of a Van der Pol-Type Circadian Pacemaker to Daylight Cycle

A van der Pol self-sustained oscillator with higher-order nonlinearity exhibits a rich dynamics, with multiple periodic attractors, and still the model allows analytical approximations. Some of these properties can be conveniently exploited in the framework of circadian oscillations. When interpreted as a biological oscillator that determines the alternation sleep/awake, the dynamic variable exhibits some interesting features that can be related to biological behavior. We analyze in the paper the phenomenon of entrainment of van der Pol-type circadian pacemaker to daylight cycle. We determine the amplitude and frequency of the circadian model without natural forcing light, and find that the agreement between analytical and numerical results hardly depends on the stiffness coefficient, μ of circadian oscillations. It is shown that a practical and precise drive which imitates the effects of the conditions of natural light can be introduced in the system and analytically treated. Considering the effects of forcing light on the sleep-aware cycle model, we find that a strong drive destroys the circadian oscillations and thus we confirm the importance of darkness for regular circadian oscillations. Moreover, we observe the reverse situation when we take into account the phase in the forcing light. For instance, for ϕ= π/ 2 and the duration of daylight DL= 12 h, the phenomenon of quenching of circadian oscillations disappears. The comparison between analytical treatment and numerical simulations of effects of light is discussed.