Impulsive Goodwin oscillator with large delay: Periodic oscillations, bistability, and attractors

The role of the time delay in the dynamics of a hybrid model of pulsatile feedback endocrine regulation is investigated. The model in hand can be seen as an impulsive and delayed version of the popular in computational biology Goodwin Oscillator, where the feedback is implemented by means of pulse modulation. The value of the time delay is related to the duration of the time interval between the firing times of the feedback impulses. Under the assumption of a cascade structure of the continuous part of the model, the hybrid dynamics of the closed-loop system are shown to be governed by a discrete mapping propagating through the firing times of the impulsive feedback. Conditions for existence and stability of periodic solutions of the model are obtained. Bifurcation analysis of the mapping reveals the phenomenon of bistability arising for larger time delay values but not observed for the smaller ones.