  We consider a spherically symmetric (purely magnetic) SU(2) Yang-Mills field
propagating on an ultrastatic spacetime with two asymptotically hyperbolic
regions connected by a throat of radius $\alpha$. Static solutions in this
model are shown to exhibit an interesting bifurcation pattern in the parameter
$\alpha$. We relate this pattern to the Morse index of the static solution with
maximal energy. Using a hyperboloidal approach to the initial value problem, we
describe the relaxation to the ground state solution for generic initial data
and unstable static solutions for initial data of codimension one, two, and
three.
