Entrainment in harmonically forced continuous and impulsive Goodwin's oscillators

Abstract

The Goodwin oscillator is a simple yet illustrative model of a biochemical system with a stable limit cycle. Considered as a prototypical biological oscillator, Goodwin's model is broadly used e.g. to describe circadian rhythms, hormonal cycles, self-oscillatory metabolic pathways.These periodic or non-periodic oscillations are self-sustained; at the same time, they are entrainable by external periodic signals, adjusting the characteristics of the autonomous oscillatory behavior. Mathematical analysis of entrainment phenomena, i.e. nonlinear phenomena imposed by periodic exogenous signals, remains an open problem. This paper presents a comparative analysis of forced dynamics arising in two versions of Goodwin's oscillator: the classical continuous oscillator and a more recent impulsive one, e.g. capturing pulsatile secretion of hormones. The main finding of this study is that while the continuous oscillator is always forced to a periodic solution by a sufficiently large exogenous signal amplitude, the impulsive one commonly exhibits a quasiperiodic or chaotic behavior thus highlighting the role of non-smooth dynamics in entrainment.