## Abstract

We study the motion of $N$ point vortices with $N\in\Nset$
on a sphere in the presence of fixed pole vortices, which are
governed by a Hamiltonian dynamical system with $N$ degrees of freedom.
Special attention is paid to the evolution of their polygonal ring
configuration called the \textit{$N$-ring}, in which they are equally
spaced along a line of latitude of the sphere when it is unstable.
When the number of the point vortices is $N=5n$ or $6n$ with $n \in \Nset$,
the system is reduced to a two-degree-of-freedom Hamiltonian
with some
saddle-center equilibria, one of which corresponds to the unstable $N$-ring.
Utilizing a Melnikov-type method applicable to two-degree-of-freedom
Hamiltonian systems with saddle-center equilibria and a numerical
method to compute stable and unstable manifolds,
we prove numerically that there exist transverse homoclinic orbits
to unstable periodic orbits in the neighborhood of the saddle-centers
and hence chaotic motions occur. Especially, the evolution of the
unstable $N$-ring is shown to be chaotic.