Cycles in Impulsive Goodwin's Oscillators of Arbitrary Order

TL;DR

This paper proves the existence and uniqueness of periodic solutions in impulsive Goodwin's oscillators of any order, extending previous models and providing conditions for their stability and behavior.

Contribution

It generalizes the analysis of Goodwin's oscillators to arbitrary order and establishes conditions for the existence and uniqueness of cycles, including for high-order systems.

Findings

Existence of positive periodic solutions for any order m >= 1.

Uniqueness of the 1-cycle for m <= 10.

Conditions for uniqueness in higher-order models.

Abstract

Existence of periodical solutions, i.e. cycles, in the Impulsive Goodwin's Oscillator (IGO) with the continuous part of an arbitrary order m is considered. The original IGO with a third-order continuous part is a hybrid model that portrays a chemical or biochemical system composed of three substances represented by their concentrations and arranged in a cascade. The first substance in the chain is introduced via an impulsive feedback where both the impulse frequency and weights are modulated by the measured output of the continuous part. It is shown that, under the standard assumptions on the IGO, a positive periodic solution with one firing of the pulse-modulated feedback in the least period also exists in models with any m >= 1. Furthermore, the uniqueness of this 1-cycle is proved for the IGO with m <= 10 whereas, for m > 10, the uniqueness can still be guaranteed under mild assumptions on the frequency modulation function.