A note on stability of Mackey–Glass equations with two delays

Leonid Berezansky; Elena Braverman

doi:10.1016/j.jmaa.2017.01.050

A note on stability of Mackey–Glass equations with two delays

Leonid Berezansky, Elena Braverman

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

47 Scopus citations

Abstract

If the Mackey–Glass equation x˙(t)=r(t)[ax(h(t))1+xν(g(t))−x(t)] with a>1 and ν>0 incorporates not one but two variable delays, some new phenomena arise: there may exist non-oscillatory about the positive equilibrium unstable solutions, the effect of possible absolute stability for certain a and ν disappears. We obtain sufficient conditions for local and global stability of the positive equilibrium and illustrate the stability tests, as well as new effects of two different delays, with examples.

Original language	English
Pages (from-to)	1208-1228
Number of pages	21
Journal	Journal of Mathematical Analysis and Applications
Volume	450
Issue number	2
DOIs	https://doi.org/10.1016/j.jmaa.2017.01.050
State	Published - 15 Jun 2017

Keywords

Global stability
Instability
Local stability
Mackey–Glass equation
Non-oscillatory unstable solutions
Two delays

ASJC Scopus subject areas

Analysis
Applied Mathematics

Access to Document

10.1016/j.jmaa.2017.01.050

Cite this

@article{e56010b7d54d4bf4bf981f304ca6c641,

title = "A note on stability of Mackey–Glass equations with two delays",

abstract = "If the Mackey–Glass equation x˙(t)=r(t)[ax(h(t))1+xν(g(t))−x(t)] with a>1 and ν>0 incorporates not one but two variable delays, some new phenomena arise: there may exist non-oscillatory about the positive equilibrium unstable solutions, the effect of possible absolute stability for certain a and ν disappears. We obtain sufficient conditions for local and global stability of the positive equilibrium and illustrate the stability tests, as well as new effects of two different delays, with examples.",

keywords = "Global stability, Instability, Local stability, Mackey–Glass equation, Non-oscillatory unstable solutions, Two delays",

author = "Leonid Berezansky and Elena Braverman",

note = "Publisher Copyright: {\textcopyright} 2017 Elsevier Inc.",

year = "2017",

month = jun,

day = "15",

doi = "10.1016/j.jmaa.2017.01.050",

language = "English",

volume = "450",

pages = "1208--1228",

journal = "Journal of Mathematical Analysis and Applications",

issn = "0022-247X",

publisher = "Academic Press Inc.",

number = "2",

}

TY - JOUR

T1 - A note on stability of Mackey–Glass equations with two delays

AU - Berezansky, Leonid

AU - Braverman, Elena

PY - 2017/6/15

Y1 - 2017/6/15

N2 - If the Mackey–Glass equation x˙(t)=r(t)[ax(h(t))1+xν(g(t))−x(t)] with a>1 and ν>0 incorporates not one but two variable delays, some new phenomena arise: there may exist non-oscillatory about the positive equilibrium unstable solutions, the effect of possible absolute stability for certain a and ν disappears. We obtain sufficient conditions for local and global stability of the positive equilibrium and illustrate the stability tests, as well as new effects of two different delays, with examples.

AB - If the Mackey–Glass equation x˙(t)=r(t)[ax(h(t))1+xν(g(t))−x(t)] with a>1 and ν>0 incorporates not one but two variable delays, some new phenomena arise: there may exist non-oscillatory about the positive equilibrium unstable solutions, the effect of possible absolute stability for certain a and ν disappears. We obtain sufficient conditions for local and global stability of the positive equilibrium and illustrate the stability tests, as well as new effects of two different delays, with examples.

KW - Global stability

KW - Instability

KW - Local stability

KW - Mackey–Glass equation

KW - Non-oscillatory unstable solutions

KW - Two delays

UR - http://www.scopus.com/inward/record.url?scp=85011277672&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2017.01.050

DO - 10.1016/j.jmaa.2017.01.050

M3 - Article

AN - SCOPUS:85011277672

SN - 0022-247X

VL - 450

SP - 1208

EP - 1228

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 2

ER -

A note on stability of Mackey–Glass equations with two delays

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this