Dept. Math, Hokkaido Univ. EPrints Server

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

Preprint Series # 707
Sakajo, Takashi High-dimensional heteroclinic and homoclinic manifolds in odd point-vortex ring on sphere with pole vortices. (2005);

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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.

Item Type:Preprint
Additional Information:30 Copies
Uncontrolled Keywords:Vortex points; Flow on sphere; Heteroclinic manifold; Hamiltonian systems;
Subjects:76-xx FLUID MECHANICS
37-xx DYNAMICAL SYSTEMS AND ERGODIC THEORY
ID Code:890