Dynamics of the Tyson-Hong-Thron-Novak circadian oscillator model

We study the dynamics of a circadian oscillator model which was proposed by Tyson, Hong, Thron and Novak. This model indicates a molecular mechanism for the circadian rhythm in Drosophila. After giving a detailed study of the equilibria, we further investigate the effects of the rates of mRNA degradation and synthesis. When the rate of mRNA degradation is rather fast, we prove that there are no periodic orbits in this model. When the rate of mRNA degradation is slow enough, this model is transformed into a slow-fast system. Then based on the geometric singular perturbation theory, we prove the existence of canard explosion, relaxation oscillations, homoclinic/heteroclinic orbits and saddle-node bifurcations as the rates of mRNA degradation and synthesis change. Finally, we give the biological interpretation of the obtained results and point out that this model can be transformed into a Li\'enard-like equation, which could be helpful to investigate the dynamics of the general case.