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Cycles in Impulsive Goodwin's Oscillators of Arbitrary Order

Anton V. Proskurnikov, Håkan Runvik, Alexander Medvedev

TL;DR

The paper investigates the existence and uniqueness of a 1-cycle periodic solution in the impulsive Goodwin's oscillator with a continuous part of order $m$, extending beyond prior results limited to low orders. It proves that a positive 1-cycle always exists for arbitrary $m$, and that the cycle is unique for $m\le 10$ under standard monotone modulation assumptions; for $m\ge 11$, multiple 1-cycles can occur, though uniqueness can be recovered by bounding $\\Phi'$ or by ensuring sufficiently sparse impulses. The analysis employs advanced techniques — notably divided differences and the Opitz formula — to express the key return-map derivatives and establish monotonicity properties when possible. These results illuminate the role of system order in sustaining pulsatile regulatory dynamics and provide tools applicable to a broader class of impulsive positive systems. The findings have implications for modeling pulsatile biological control and hybrid oscillators where the continuous dynamics can be of high order.

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.

Cycles in Impulsive Goodwin's Oscillators of Arbitrary Order

TL;DR

The paper investigates the existence and uniqueness of a 1-cycle periodic solution in the impulsive Goodwin's oscillator with a continuous part of order , extending beyond prior results limited to low orders. It proves that a positive 1-cycle always exists for arbitrary , and that the cycle is unique for under standard monotone modulation assumptions; for , multiple 1-cycles can occur, though uniqueness can be recovered by bounding or by ensuring sufficiently sparse impulses. The analysis employs advanced techniques — notably divided differences and the Opitz formula — to express the key return-map derivatives and establish monotonicity properties when possible. These results illuminate the role of system order in sustaining pulsatile regulatory dynamics and provide tools applicable to a broader class of impulsive positive systems. The findings have implications for modeling pulsatile biological control and hybrid oscillators where the continuous dynamics can be of high order.

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.
Paper Structure (12 sections, 66 equations, 1 figure, 1 table)

This paper contains 12 sections, 66 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. The impulsive Goodwin's oscillator
  3. Problem formulation: 1-cycle
  4. Main Results
  5. Proofs and Auxiliary Results
  6. Lemmas and proof of Theorem \ref{['thm.exist']}
  7. Lemmas and proofs of Theorem \ref{['thm.m9']}--Theorem \ref{['thm.m10']}
  8. An example of the IGO with multiple 1-cycles.
  9. Numerical example
  10. Conclusions
  11. Divided differences and Opitz formula
  12. Proof of Lemma \ref{['lem.Psi-props']}

Figures (1)

  • Figure 1: Graphs of $\Psi_k(x)$ for $k=9,10,14$.