open access

https://doi.org/10.14943/83858

https://hdl.handle.net/2115/69512

High-dimensional heteroclinic and homoclinic manifolds in odd point-vortex ring on sphere with pole vortices

英語

Projection method

2005

Department of Mathematics, Hokkaido University

Hokkaido University Preprint Series in Mathematics

707

1-18

We consider the motion of the $N$-vortex points that are equally spaced along a line of latitude on sphere with fixed pole vortices, called ``$N$-ring''. In particular, we focus on the evolution of the odd unstable $N$-ring. Since the eigenvalues that determine the stability of the odd $N$-ring are double, each of the unstable and stable manifolds corresponding to them is two-dimensional manifold. Accordingly, it is generally difficult to describe the global structure of the manifolds. In this article, based on the linear stability analysis, we propose a projection method to show the structure of the iso-surfaces of the Hamiltonian, in which the orbit of the vortex points exist. Then, applying the projection method to the motion of the $3$-ring and $5$-ring, we discuss the existence of the high-dimensional homoclinic and heteroclinic connections in the phase space, which characterize the evolution of the unstable $N$-ring.

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