  The classical solutions to higher dimensional Yang--Mills (YM) systems, which
are integral parts of higher dimensional Einstein--YM (EYM) systems, are
studied. These are the gravity decoupling limits of the fully gravitating EYM
solutions. In odd spacetime dimensions, depending on the choice of gauge group,
these are either topologically stable or unstable. Both cases are analysed, the
latter numerically only. In even spacetime dimensions they are always unstable,
describing saddle points of the energy, and can be described as {\it
sphalerons}. This instability is analysed by constructing the noncontractible
loops and calculating the Chern--Simons (CS) charges, and also perturbatively
by numerically constructing the negative modes. This study is restricted to the
simplest YM system in spacetime dimensions $d=6,7,8$, which is amply
illustrative of the generic case.
