Computer assisted proof of the existence of the Lorenz attractor in the Shimizu–Morioka system. (30th October 2018)

• Record Type: Journal Article
• Authors: Capiński, Maciej J; Turaev, Dmitry; Zgliczyński, Piotr
• Is Part Of: Nonlinearity. Volume 31:Number 12(2018:Dec.)
• Page Start: 5410
• Page End: 5440
• Publication Date: 2018-10-30
• Subjects: Lorenz attractor -- Shimizu–Morioka system -- computer assisted proof; 34C23 -- 37D05 -- 37D10 -- 37G35; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/aae032
• Languages: English
• ISSNs: 0951-7715

Abstract

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.