open access

https://doi.org/10.14943/84008

https://hdl.handle.net/2115/69667

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

英語

Melnikov method

2007-06-06

Department of Mathematics, Hokkaido University

Hokkaido University Preprint Series in Mathematics

858

1-34

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.

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