open access

https://hdl.handle.net/2115/34776

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

英語

Melnikov method

2008-03-04

Springer New York

Journal of Nonlinear Science

18

485-525

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.

学術雑誌論文

Accepted Manuscript

The original publication is available at www.springerlink.com

0938-8974

1432-1467