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The classical solutions to higher dimensional Yang--Mills (YM) systems, whichare integral parts of higher dimensional Einstein--YM (EYM) systems, arestudied. These are the gravity decoupling limits of the fully gravitating EYMsolutions. In odd spacetime dimensions, depending on the choice of gauge group,these are either topologically stable or unstable. Both cases are analysed, thelatter numerically only. In even spacetime dimensions they are always unstable,describing saddle points of the energy, and can be described as {\itsphalerons}. This instability is analysed by constructing the noncontractibleloops and calculating the Chern--Simons (CS) charges, and also perturbativelyby numerically constructing the negative modes. This study is restricted to thesimplest YM system in spacetime dimensions $d=6,7,8$, which is amplyillustrative of the generic case.Comment: 16 pages, 3 figures ; comments added, to appear in J. Phys.

Yang–Mills sphalerons in all even spacetime dimensionsd= 2k,k> 2:k= 3, 4