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High-dimensional heteroclinic and homoclinic manifolds in odd point-vortex ring on sphere with pole vortices
| Title: | High-dimensional heteroclinic and homoclinic manifolds in odd point-vortex ring on sphere with pole vortices |
| Authors: | Sakajo, Takashi Browse this author |
| Keywords: | Vortex points | | Flow on a sphere | | Heteroclinic manifold | | Projection method |
| Issue Date: | 2005 |
| Publisher: | Department of Mathematics, Hokkaido University |
| Journal Title: | Hokkaido University Preprint Series in Mathematics |
| Volume: | 707 |
| Start Page: | 1 |
| End Page: | 18 |
| Abstract: | 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. |
| Type: | bulletin (article) |
| URI: | http://hdl.handle.net/2115/69512 |
| Appears in Collections: | 理学院・理学研究院 (Graduate School of Science / Faculty of Science) > Hokkaido University Preprint Series in Mathematics
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Submitter: 数学紀要登録作業用
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