A proof of Wright's conjecture

Jan Bouwe van den Berg, Jonathan Jaquette*

*Corresponding author for this work

Research output: Contribution to JournalArticle

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 languageEnglish
Pages (from-to)7412-7462
Number of pages51
JournalJournal of Differential Equations
Volume264
Issue number12
Early online date26 Feb 2018
DOIs
Publication statusPublished - 15 Jun 2018

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Keywords

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

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