  We study the classical non-linear dynamics of the $SU(2)$ Yang-Mills matrix
model introduced in [1] as a low-energy approximation to two-color QCD.
Restricting to the spin-0 sector of the model, we unearth an unexpected
tetrahedral symmetry, which endows the dynamics with an extraordinarily rich
structure. Amongst other things, we find that the spin-0 sector contains
co-existing chaotic sub-sectors as well as nested chaotic basins, and displays
alternation between regular and chaotic dynamics as energy is varied. The
symmetries also grant us a considerable amount of analytic control which allows
us to make several quantitative observations. Next, by noting that several
features of the model have natural thermodynamic interpretations, we switch
from our original chaos-theoretic viewpoint to a more statistical perspective.
By so doing, we see that the classical spin-0 sector has a rich phase
structure, arising from ergodicity breaking, which we investigate in depth.
Surprisingly, we find that many of these classical phases display numerous
similarities to previously discovered quantum phases of the spin-0 sector [2],
and we explore these similarities in a heuristic fashion.
