Periodic solutions of planar systems with two delays

Shigui Ruan; Junjie Wei

doi:10.1017/S0308210500031061

Periodic solutions of planar systems with two delays

Shigui Ruan, Junjie Wei

Research output: Contribution to journal › Article

83 Scopus citations

Abstract

In this paper, we consider a planar system with two delays: ẋ₁(t) = -a₀x₁(t) + a₁F₁(x₁(t - τ₁), x₂(t - τ₂)), ẋ₂(t) = -b₀x₂(t) + b₁F₂(x₁(t - τ₁), x₂(t - τ₂)). Firstly, linearized stability and local Hopf bifurcations are studied. Then, existence conditions for non-constant periodic solutions are derived using degree theory methods. Finally, a simple neural network model with two delays is analysed as an example.

Original language	English (US)
Pages (from-to)	1017-1032
Number of pages	16
Journal	Royal Society of Edinburgh - Proceedings A
Volume	129
Issue number	5
DOIs	https://doi.org/10.1017/S0308210500031061
State	Published - Jan 1 1999
Externally published	Yes

Fingerprint

ASJC Scopus subject areas

Mathematics(all)

Cite this

Ruan, S., & Wei, J. (1999). Periodic solutions of planar systems with two delays. Royal Society of Edinburgh - Proceedings A, 129(5), 1017-1032. https://doi.org/10.1017/S0308210500031061

@article{2781beddeb4841d5b63d7bcefe0147d5,

title = "Periodic solutions of planar systems with two delays",

abstract = "In this paper, we consider a planar system with two delays: ẋ1(t) = -a0x1(t) + a1F1(x1(t - τ1), x2(t - τ2)), ẋ2(t) = -b0x2(t) + b1F2(x1(t - τ1), x2(t - τ2)). Firstly, linearized stability and local Hopf bifurcations are studied. Then, existence conditions for non-constant periodic solutions are derived using degree theory methods. Finally, a simple neural network model with two delays is analysed as an example.",

author = "Shigui Ruan and Junjie Wei",

year = "1999",

month = jan,

day = "1",

doi = "10.1017/S0308210500031061",

language = "English (US)",

volume = "129",

pages = "1017--1032",

journal = "Proceedings of the Royal Society of Edinburgh Section A: Mathematics",

issn = "0308-2105",

publisher = "Cambridge University Press",

number = "5",

}

TY - JOUR

T1 - Periodic solutions of planar systems with two delays

AU - Ruan, Shigui

AU - Wei, Junjie

PY - 1999/1/1

Y1 - 1999/1/1

N2 - In this paper, we consider a planar system with two delays: ẋ1(t) = -a0x1(t) + a1F1(x1(t - τ1), x2(t - τ2)), ẋ2(t) = -b0x2(t) + b1F2(x1(t - τ1), x2(t - τ2)). Firstly, linearized stability and local Hopf bifurcations are studied. Then, existence conditions for non-constant periodic solutions are derived using degree theory methods. Finally, a simple neural network model with two delays is analysed as an example.

AB - In this paper, we consider a planar system with two delays: ẋ1(t) = -a0x1(t) + a1F1(x1(t - τ1), x2(t - τ2)), ẋ2(t) = -b0x2(t) + b1F2(x1(t - τ1), x2(t - τ2)). Firstly, linearized stability and local Hopf bifurcations are studied. Then, existence conditions for non-constant periodic solutions are derived using degree theory methods. Finally, a simple neural network model with two delays is analysed as an example.

UR - http://www.scopus.com/inward/record.url?scp=33746961598&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33746961598&partnerID=8YFLogxK

U2 - 10.1017/S0308210500031061

DO - 10.1017/S0308210500031061

M3 - Article

AN - SCOPUS:33746961598

VL - 129

SP - 1017

EP - 1032

JO - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

JF - Proceedings of the Royal Society of Edinburgh Section A: Mathematics

SN - 0308-2105

IS - 5

ER -

Access to Document

10.1017/S0308210500031061