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Transition from Quasiperiodicity to Chaos for Three Coaxial Vortex Rings
Author(s) -
Blackmore D.,
Knio O.
Publication year - 2000
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.20000801344
Subject(s) - quasiperiodicity , quasiperiodic function , coaxial , radius , vortex ring , chaotic , ring (chemistry) , physics , classical mechanics , ideal (ethics) , motion (physics) , vortex , mathematics , mechanics , condensed matter physics , chemistry , computer security , engineering , organic chemistry , artificial intelligence , computer science , electrical engineering , philosophy , epistemology
The dynamics of three coaxial vortex rings of strengths Γ 1 , Γ 2 and Γ 3 in an ideal fluid is investigated. It is proved that if Γ j , Γ j + Γ k and Γ 1 + Γ 2 + Γ 3 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 quasiperiodic or periodic motions. Moreover, it is shown that the motion become chaotic as the inter‐ring distances are increased relative to the mean radius.