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

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 orbi...

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Bibliographic Details
Published inNonlinearity Vol. 31; no. 12; pp. 5410 - 5440
Main Authors Capi ski, Maciej J, Turaev, Dmitry, Zgliczy ski, Piotr
Format Journal Article
LanguageEnglish
Published IOP Publishing 01.12.2018
Subjects
computer assisted proof
Lorenz attractor
Shimizu-Morioka system
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Summary: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.
Bibliography:London Mathematical Society
NON-102578.R2
ISSN:0951-7715
1361-6544
DOI:10.1088/1361-6544/aae032