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Chaotic Motion of the N-Vortex Problem on a Sphere: I. Saddle-Centers in Two-Degree-of-Freedom Hamiltonians

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Title: Chaotic Motion of the N-Vortex Problem on a Sphere: I. Saddle-Centers in Two-Degree-of-Freedom Hamiltonians
Authors: Sakajo, Takashi Browse this author →KAKEN DB
Yagasaki, Kazuyuki Browse this author
Keywords: Hamiltonian system
Point vortex
Flow on sphere
Chaos
Melnikov method
Issue Date: 4-Mar-2008
Publisher: Springer New York
Journal Title: Journal of Nonlinear Science
Volume: 18
Start Page: 485
End Page: 525
Publisher DOI: 10.1007/s00332-008-9019-9
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.
Rights: The original publication is available at www.springerlink.com
Type: article (author version)
URI: http://hdl.handle.net/2115/34776
Appears in Collections:理学院・理学研究院 (Graduate School of Science / Faculty of Science) > 雑誌発表論文等 (Peer-reviewed Journal Articles, etc)

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