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Chaotic motion of the N-vortex problem on a sphere I: Saddle-centers in two-degree-of-freedom

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Please use this identifier to cite or link to this item:http://doi.org/10.14943/84008

Title: Chaotic motion of the N-vortex problem on a sphere I: Saddle-centers in two-degree-of-freedom
Authors: SAKAJO, Takashi Browse this author
YAGASAKI, Kazuyuki Browse this author
Keywords: Hamiltonian system
point vortex
flow on sphere
chaos
Melnikov method
Issue Date: 6-Jun-2007
Journal Title: Hokkaido University Preprint Series in Mathematics
Volume: 858
Start Page: 1
End Page: 34
Abstract: We study the motion of N point vortices with N∈ℕ on a sphere in the presence of fixed pole vortices, which are governed by a Hamiltonian dynamical system with N degrees of freedom. Special attention is paid to the evolution of their polygonal ring configuration called the N-ring, in which they are equally spaced along a line of latitude of the sphere. When the number of the point vortices is N=5n or 6n with n∈ℕ, the system is reduced to a two-degree-of-freedom Hamiltonian with some saddle-center equilibria, one of which corresponds to the unstable N-ring. Using a Melnikov-type method applicable to two-degree-of-freedom Hamiltonian systems with saddle-center equilibria and a numerical method to compute stable and unstable manifolds, we show numerically that there exist transverse homoclinic orbits to unstable periodic orbits in the neighborhood of the saddle-centers and hence chaotic motions occur. Especially, the evolution of the unstable N-ring is shown to be chaotic.
Type: bulletin (article)
URI: http://hdl.handle.net/2115/69667
Appears in Collections:理学院・理学研究院 (Graduate School of Science / Faculty of Science) > Hokkaido University Preprint Series in Mathematics

Submitter: 数学紀要登録作業用

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