Transition from quasiperiodicity to chaos for three coaxial vortex rings

D. Blackmore; O. Knio

doi:10.1002/zamm.20000801344

Transition from quasiperiodicity to chaos for three coaxial vortex rings

D. Blackmore^*, O. Knio

^*Corresponding author for this work

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

10 Scopus citations

Abstract

The dynamics of three coaxial vortex rings of strengths Γ₁, Γ₂ and Γ₃ in an ideal fluid is investigated. It is proved that if Γ_j, Γ_j + Γ_k and Γ₁ + Γ₂ + Γ₃ are not zero for all j, k = 1, 2, 3, then KAM and Poincaré-Birkhoff theory can be used to prove that if the distances among the rings are sufficiently small compared to the mean radius of the rings, there are many initial configurations of the rings that produce guasiperiodic or periodic motions. Moreover, it is shown that the motion become chaotic as the inter-ring distances are increased relative to the mean radius.

Original language	English (US)
Pages (from-to)	S173-S176
Journal	ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik
Volume	80
Issue number	4 SUPPL. 1
DOIs	https://doi.org/10.1002/zamm.20000801344
State	Published - 2000
Externally published	Yes

ASJC Scopus subject areas

Computational Mechanics
Applied Mathematics

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title = "Transition from quasiperiodicity to chaos for three coaxial vortex rings",

abstract = "The dynamics of three coaxial vortex rings of strengths Γ1, Γ2 and Γ3 in an ideal fluid is investigated. It is proved that if Γj, Γj + Γk and Γ1 + Γ2 + Γ3 are not zero for all j, k = 1, 2, 3, then KAM and Poincar{\'e}-Birkhoff theory can be used to prove that if the distances among the rings are sufficiently small compared to the mean radius of the rings, there are many initial configurations of the rings that produce guasiperiodic or periodic motions. Moreover, it is shown that the motion become chaotic as the inter-ring distances are increased relative to the mean radius.",

author = "D. Blackmore and O. Knio",

year = "2000",

doi = "10.1002/zamm.20000801344",

language = "English (US)",

volume = "80",

pages = "S173--S176",

journal = "ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik",

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TY - JOUR

T1 - Transition from quasiperiodicity to chaos for three coaxial vortex rings

AU - Blackmore, D.

AU - Knio, O.

PY - 2000

Y1 - 2000

N2 - The dynamics of three coaxial vortex rings of strengths Γ1, Γ2 and Γ3 in an ideal fluid is investigated. It is proved that if Γj, Γj + Γk and Γ1 + Γ2 + Γ3 are not zero for all j, k = 1, 2, 3, then KAM and Poincaré-Birkhoff theory can be used to prove that if the distances among the rings are sufficiently small compared to the mean radius of the rings, there are many initial configurations of the rings that produce guasiperiodic or periodic motions. Moreover, it is shown that the motion become chaotic as the inter-ring distances are increased relative to the mean radius.

AB - The dynamics of three coaxial vortex rings of strengths Γ1, Γ2 and Γ3 in an ideal fluid is investigated. It is proved that if Γj, Γj + Γk and Γ1 + Γ2 + Γ3 are not zero for all j, k = 1, 2, 3, then KAM and Poincaré-Birkhoff theory can be used to prove that if the distances among the rings are sufficiently small compared to the mean radius of the rings, there are many initial configurations of the rings that produce guasiperiodic or periodic motions. Moreover, it is shown that the motion become chaotic as the inter-ring distances are increased relative to the mean radius.

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U2 - 10.1002/zamm.20000801344

DO - 10.1002/zamm.20000801344

M3 - Article

AN - SCOPUS:23044521286

SN - 0044-2267

VL - 80

SP - S173-S176

JO - ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik

JF - ZAMM Zeitschrift fur Angewandte Mathematik und Mechanik

IS - 4 SUPPL. 1

ER -

Transition from quasiperiodicity to chaos for three coaxial vortex rings

Abstract

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