A proof of Jones' conjecture

Jonathan Jaquette

doi:10.1016/j.jde.2018.09.017

A proof of Jones' conjecture

Jonathan Jaquette

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

12 Scopus citations

Abstract

In this paper, we prove that Wright's equation y^′(t)=−αy(t−1){1+y(t)} has a unique slowly oscillating periodic solution for parameter values α∈([Formula presented],1.9], up to time translation. This result proves Jones' Conjecture formulated in 1962, that there is a unique slowly oscillating periodic orbit for all α>[Formula presented]. Furthermore, there are no isolas of periodic solutions to Wright's equation; all periodic orbits arise from Hopf bifurcations.

Original language	English (US)
Pages (from-to)	3818-3859
Number of pages	42
Journal	Journal of Differential Equations
Volume	266
Issue number	6
DOIs	https://doi.org/10.1016/j.jde.2018.09.017
State	Published - Mar 5 2019
Externally published	Yes

All Science Journal Classification (ASJC) codes

Analysis
Applied Mathematics

Keywords

Branch and bound
Computer-assisted proofs
Delay differential equations
Jones' conjecture
Krawczyk method
Wright's equation

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10.1016/j.jde.2018.09.017

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title = "A proof of Jones' conjecture",

abstract = "In this paper, we prove that Wright's equation y′(t)=−αy(t−1)\{1+y(t)\} has a unique slowly oscillating periodic solution for parameter values α∈([Formula presented],1.9], up to time translation. This result proves Jones' Conjecture formulated in 1962, that there is a unique slowly oscillating periodic orbit for all α>[Formula presented]. Furthermore, there are no isolas of periodic solutions to Wright's equation; all periodic orbits arise from Hopf bifurcations.",

keywords = "Branch and bound, Computer-assisted proofs, Delay differential equations, Jones' conjecture, Krawczyk method, Wright's equation",

author = "Jonathan Jaquette",

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PY - 2019/3/5

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N2 - In this paper, we prove that Wright's equation y′(t)=−αy(t−1){1+y(t)} has a unique slowly oscillating periodic solution for parameter values α∈([Formula presented],1.9], up to time translation. This result proves Jones' Conjecture formulated in 1962, that there is a unique slowly oscillating periodic orbit for all α>[Formula presented]. Furthermore, there are no isolas of periodic solutions to Wright's equation; all periodic orbits arise from Hopf bifurcations.

AB - In this paper, we prove that Wright's equation y′(t)=−αy(t−1){1+y(t)} has a unique slowly oscillating periodic solution for parameter values α∈([Formula presented],1.9], up to time translation. This result proves Jones' Conjecture formulated in 1962, that there is a unique slowly oscillating periodic orbit for all α>[Formula presented]. Furthermore, there are no isolas of periodic solutions to Wright's equation; all periodic orbits arise from Hopf bifurcations.

KW - Branch and bound

KW - Computer-assisted proofs

KW - Delay differential equations

KW - Jones' conjecture

KW - Krawczyk method

KW - Wright's equation

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UR - https://www.scopus.com/pages/publications/85054525114#tab=citedBy

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DO - 10.1016/j.jde.2018.09.017

M3 - Article

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VL - 266

SP - 3818

EP - 3859

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 6

ER -

A proof of Jones' conjecture

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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