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

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 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 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