### Abstract

Wright's conjecture states that the origin is the global attractor for the delay differential equation y^{′}(t)=−αy(t−1)[1+y(t)] for all α∈(0,[Formula presented]] when y(t)>−1. This has been proven to be true for a subset of parameter values α. We extend the result to the full parameter range α∈(0,[Formula presented]], and thus prove Wright's conjecture to be true. Our approach relies on a careful investigation of the neighborhood of the Hopf bifurcation occurring at α=[Formula presented]. This analysis fills the gap left by complementary work on Wright's conjecture, which covers parameter values further away from the bifurcation point. Furthermore, we show that the branch of (slowly oscillating) periodic orbits originating from this Hopf bifurcation does not have any subsequent bifurcations (and in particular no folds) for α∈([Formula presented],[Formula presented]+6.830×10^{−3}]. When combined with other results, this proves that the branch of slowly oscillating solutions that originates from the Hopf bifurcation at α=[Formula presented] is globally parametrized by α>[Formula presented].

Original language | English |
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Pages (from-to) | 7412-7462 |

Number of pages | 51 |

Journal | Journal of Differential Equations |

Volume | 264 |

Issue number | 12 |

Early online date | 26 Feb 2018 |

DOIs | |

Publication status | Published - 15 Jun 2018 |

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### Keywords

- Delay differential equation
- Hopf bifurcation
- Newton–Kantorovich theorem
- Supercritical bifurcation branch
- Wright's conjecture

### Cite this

*Journal of Differential Equations*,

*264*(12), 7412-7462. https://doi.org/10.1016/j.jde.2018.02.018