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 › peer-review

84 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

ASJC Scopus subject areas

Mathematics(all)

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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",

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

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T1 - Periodic solutions of planar systems with two delays

AU - Ruan, Shigui

AU - Wei, Junjie

PY - 1999/1/1

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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.

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

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