A cyclic system with delay and its characteristic equation

Elena Braverman; Karel Hasik; Anatoli F. Ivanov; Sergei I. Trofimchuk

doi:10.3934/dcdss.2020001

Article Contents

2020, Volume 13, Issue 1: 1-29. Doi: 10.3934/dcdss.2020001

This issue Previous Article Preface: Delay differential equations with state-dependent delays and their applications Next Article Long-time behavior of positive solutions of a differential equation with state-dependent delay

A cyclic system with delay and its characteristic equation

1.
Department of Mathematics and Statistics, University of Calgary, 2500 University Drive N.W., Calgary, AB, Canada T2N 1N4
2.
Mathematical Institute, Silesian University, 746 01 Opava, Czech Republic
3.
Department of Mathematics, Pennsylvania State University, P.O. Box PSU, Lehman, PA 18627, USA
4.
Instituto de Matematica y Fisica, Universidad de Talca, Casilla 747, Talca, Chile

S. I. Trofimchuk is the corresponding author, e-mail: trofimch@inst-mat.utalca.cl
S. I. Trofimchuk is the corresponding author, e-mail: trofimch@inst-mat.utalca.cl

Received: February 28, 2017

Revised: June 30, 2017

Published: January 2020

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Abstract

A nonlinear cyclic system with delay and the overall negative feedback is considered. The characteristic equation of the linearized system is studied in detail. Sufficient conditions for the oscillation of all solutions and for the existence of monotone solutions are derived in terms of roots of the characteristic equation.

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Mathematics Subject Classification: Primary: 34K06, 34K11, 34K13; Secondary: 34K20, 34K25.

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Figure 1. Graphs of $ \Theta_0(\omega) = \sum_{j = 1}^n\theta_j $ and $ y = -\omega\tau+\pi(2k-1) $, $ k\in\mathbb N $ for $ n = 6 $ (upper) and $ n = 10 $ (lower).

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