### Abstract

In this paper, we consider a planar system with two delays: ẋ_{1}(t) = -a_{0}x_{1}(t) + a_{1}F_{1}(x_{1}(t - τ_{1}), x_{2}(t - τ_{2})), ẋ_{2}(t) = -b_{0}x_{2}(t) + b_{1}F_{2}(x_{1}(t - τ_{1}), x_{2}(t - τ_{2})). 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 |

State | Published - 1999 |

Externally published | Yes |

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### ASJC Scopus subject areas

- Mathematics(all)
- Applied Mathematics

### Cite this

*Royal Society of Edinburgh - Proceedings A*,

*129*(5), 1017-1032.

**Periodic solutions of planar systems with two delays.** / Ruan, Shigui; Wei, Junjie.

Research output: Contribution to journal › Article

*Royal Society of Edinburgh - Proceedings A*, vol. 129, no. 5, pp. 1017-1032.

}

TY - JOUR

T1 - Periodic solutions of planar systems with two delays

AU - Ruan, Shigui

AU - Wei, Junjie

PY - 1999

Y1 - 1999

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

M3 - Article

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 -