Entrainment of Goodwin´s oscillators by periodic exogenous signals

The circadian pacemakers, which have been discovered in most of living organisms, are known to be entrainable by the environmental exogenuous cues, or zeitgebers (“time givers”). If the influence of an exogenous periodic excitation is sufficiently long, the internal circadian “clock” adjusts the phase and period to the external signal. The pattern of zeitgeber´s oscillations is then reproduced for a long duration by the pacemaker, even when isolated from the environment. In recent years, the phenomenon of entrainment of biological clocks has been the subject of extensive experimental research, mainly focusing on revelation of dependencies between the “free-run” and disturbed circadian cycles. Meanwhile, the mathematical model of entrainment is still far from being well understood; to the best of the authors´ knowledge, neither conventional mathematical definitions of entrainment nor mathematical proofs of this property for realistic models of circadian oscillators have been elaborated. In this paper, we make a substantial effort towards filling this gap, by considering dynamics of Goodwin-type oscillators under periodic excitations. We show that any periodic external signal gives birth to a periodic solution of the same period, which is in general non-unique. Along with increasing signal amplitude, the forced solution becomes not only unique but also asymptotically stable. This sheds light on the mathematical meaning of the entrainment phenomena and confirms the entrainability of circadian rhythms by sufficiently strong periodic signals, reported in the biological literature.