We study the dynamics of a circadian oscillator model proposed by Tyson, Hong, Thron and Novak. This model describes a molecular mechanism for the circadian rhythm in Drosophila. After giving a detailed study of its equilibria, we investigate the dynamics in the cases that the rate of mRNA degradation is sufficiently high or low. When the rate is sufficiently high, we prove that there are no periodic orbits in the region with biological meaning. When the rate is sufficiently low, this model is transformed into a slow–fast system. Then based on the geometric singular perturbation theory, we prove the existence of relaxation oscillations, canard explosion, saddle–node bifurcations, and the coexistence of two limit cycles in this model. These results are helpful to understand the effects of biophysical parameters on circadian oscillations. Finally, we give the biological interpretation of the results and point out that this model can be transformed into a Liénard-like equation, which could be helpful to investigate the dynamics of the general case.

我们研究了由Tyson，Hong，Thron和Novak提出的昼夜节律模型的动力学。该模型描述了果蝇昼夜节律的分子机制。在详细研究了其平衡性之后，我们研究了mRNA降解率足够高或很低的情况下的动力学。当速率足够高时，我们证明该区域内没有具有生物学意义的周期性轨道。当速率足够低时，此模型将转换为慢速-快速系统。然后，基于几何奇异摄动理论，我们证明了该模型中存在弛豫振荡，卡纳德爆炸，鞍形节点分叉以及两个极限环并存。这些结果有助于理解生物物理参数对昼夜节律振荡的影响。最后，