Periodical Solutions in a Pulse-Modulated Model of Endocrine Regulation With Time-Delay

Discrete-time map for an impulsive Goodwin oscillator with a distributed delay

“…Hence, at most two firings of the pulse-modulated feedback can occur within the memory of the hereditary block. According to biological data, for the endocrine regulation of testosterone in the healthy reproductive human male, the inequalities in (20) appear to hold [4], which fact gives further credibility to the impulsive Goodwin oscillator. There are also medical observations indicating that larger time-delay values resulting, for instance, from blood transfusion or dialysis lead to significant dysfunctions of the endocrine system.…”

“…Consider an extension of the impulsive Goodwin oscillator treated in [4,8] to a class of hereditary systems…”

“…As already mentioned, the existence of a stable equilibrium is one of the main obstacles preventing a wider use of the mathematical model that is supposed to describe sustained oscillations. Following [4,7], let Φ 1 > max{τ, h − τ }, which inequality implies that…”

“…(2), the continuous part of the impulsive Goodwin oscillator was augmented with a pointwise time delay [3][4][5]7], i.e. a delayed version of system (3),…”

“…The pulsemodulated feedback exhorts only instantaneous action on the continuous part of the system and leaves it autonomous in the time intervals between the firing instants. The cascade structure of the infinite-dimensional continuous part enables the use of finite-dimensional (FD) reducibility, a notion introduced in [4]. It is instrumental in the analysis of time-delay inflicted phenomena by a reduction of the impulsive infinite-dimensional system in question to a discrete and finite-dimensional one.…”