Computer assisted proof of the existence of the Lorenz attractor in the Shimizu-Morioka system

Maciej J. Capiński, Dmitry Turaev, Piotr Zgliczyński

Research output: Contribution to journalArticlepeer-review

11 Scopus citations

Abstract

We prove that the Shimizu-Morioka system has a Lorenz attractor for an open set of parameter values. For the proof we employ a criterion proposed by Shilnikov, which allows to conclude the existence of the attractor by examination of the behaviour of only one orbit. The needed properties of the orbit are established by using computer assisted numerics. Our result is also applied to the study of local bifurcations of triply degenerate periodic points of three-dimensional maps. It provides a formal proof of the birth of discrete Lorenz attractors at various global bifurcations.

Original languageEnglish
Pages (from-to)5410-5440
Number of pages31
JournalNonlinearity
Volume31
Issue number12
DOIs
StatePublished - 30 Oct 2018
Externally publishedYes

Keywords

  • 37D05
  • 37D10
  • 37G35
  • Lorenz attractor
  • Shimizu-Morioka system
  • computer assisted proof Mathematics Subject Classification numbers: 34C23

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