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

Order and Chaos in the $SU(2)$ Matrix Model: Ergodicity and Classical Phases