  We present two calculations for a class of robust homoclinic cycles with
symmetry Z_n x Z_2^n, for which the sufficient conditions for asymptotic
stability given by Krupa and Melbourne are not optimal.
  Firstly, we compute optimal conditions for asymptotic stability using
transition matrix techniques which make explicit use of the geometry of the
group action.
  Secondly, through an explicit computation of the global parts of the Poincare
map near the cycle we show that, generically, the resonance bifurcations from
the cycles are supercritical: a unique branch of asymptotically stable period
orbits emerges from the resonance bifurcation and exists for coefficient values
where the cycle has lost stability. This calculation is the first to explicitly
compute the criticality of a resonance bifurcation, and answers a conjecture of
Field and Swift in a particular limiting case. Moreover, we are able to obtain
an asymptotically-correct analytic expression for the period of the bifurcating
orbit, with no adjustable parameters, which has not proved possible previously.
We show that the asymptotic analysis compares very favourably with numerical
results.
