Delay-induced dynamical phenomena in impulsive Goodwin´s oscillator: What we know so far

Impulsive Goodwin´s oscillator model is introduced to capture the dynamics of sustained periodic processes in endocrine systems controlled by episodic pulses of hormones. The model is hybrid and comprises a continuous subsystem describing the hormone concentrations operating under a discrete pulse-modulated feedback implemented by firing neurons. Time delays appear in mathematical models of endocrine systems due to the significant transport phenomena but also because of the time necessary to produce releasable hormone quantities. From a biological point of view, the neural control should be robust against the time delay to ensure the loop functionality over a wide range of inter-individual variability. The paper provides an overview of the currently available results and contributes a generalization of a Poincaré mapping approach to study complex dynamics of impulsive Goodwin oscillator. Both pointwise and distributed time delays are considered in a general framework based on the Poincaré mapping. Bifurcation analysis is utilized to illustrate the analytical results.