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- W412470314 abstract "We study the motion of $N$ point vortices with $NinNset$ 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 textit{$N$-ring}, in which they are equally spaced along a line of latitude of the sphere when it is unstable. When the number of the point vortices is $N=5n$ or $6n$ with $n in Nset$, 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. Utilizing 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 prove 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." @default.
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- W412470314 date "2007-06-06" @default.
- W412470314 modified "2023-09-26" @default.
- W412470314 title "Chaotic motion of the N-vortex problem on a sphere I: Saddle-centers in two-degree-of-freedom" @default.
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