Finite-dimensional reducibility of time-delay systems under pulse-modulated feedback

The paper deals with further development and refinement of recent results on hybrid systems with impulsive feedback and a time delay in the continuous part. Such a closed-loop system can be considered as an impulsive and time-delayed version of the Goodwin oscillator, which is well known in mathematical biology. It also arises in mathematical modeling of non-basal regulation of endocrine systems as the pulsatile Smith model. The key property in the analysis of the system dynamics is finite-dimensional (FD) reducibility of the linear infinite-dimensional part that is studied in depth in the present paper. Under FD-reducibility of the linear continuous part, reductions of the hybrid dynamics of the considered system to a discrete and a delay-free formulation are suggested. Conditions for the existence and local stability of periodic solutions with m impulses in the least period (m-cycles) are derived.