  We use a dynamical systems approach based on the method of orthonormal frames
to study the dynamics of a non-tilted Bianchi Type IX cosmological model with a
bulk and shear viscous fluid source. We begin by completing a detailed
fix-point analysis which give the local sinks, sources and saddles of the
dynamical system. We then analyze the global dynamics by finding the
$\alpha$-and $\omega$-limit sets which give an idea of the past and future
asymptotic behavior of the system. The fixed points were found to be a flat
Friedmann-LeMa\^{i}tre-Robertson-Walker (FLRW) solution, Bianchi Type $II$
solution, Kasner circle, Jacobs disc, Bianchi Type $VII_{0}$ solutions, and
several closed FLRW solutions in addition to the Einstein static universe
solution. Each equilibrium point was described in both its expanding and
contracting epochs. We conclude the paper with some numerical experiments that
shed light on the global dynamics of the system along with its heteroclinic
orbits. With respect to past asymptotic states, we were able to conclude that
the Jacobs disc in the expanding epoch was a source of the system along with
the flat FLRW solution in a contracting epoch. With respect to future
asymptotic states, we were able to show that the flat FLRW solution in an
expanding epoch along with the Jacobs disc in the contracting epoch were sinks
of the system. We were also able to demonstrate a new result with respect to
the Einstein static universe. Namely, we gave certain conditions on the
parameter space such that the Einstein static universe has an associated stable
subspace. We were however, not able to conclusively say anything about whether
a closed FLRW model could be a past or future asymptotic state of the model.
