  We consider the small-volume dynamics of nonsupersymmetric orbifold and
orientifold field theories defined on a three-torus, in a test of the claimed
planar equivalence between these models and appropriate supersymmetric ``parent
models". We study one-loop effective potentials over the moduli space of flat
connections and find that planar equivalence is preserved for suitable averages
over the moduli space. On the other hand, strong nonlinear effects produce
local violations of planar equivalence at special points of moduli space. In
the case of orbifold models, these effects show that the "twisted" sector
dominates the low-energy dynamics.
