## 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.