  We study inflationary universes with an SU(3) gauge field coupled to an
inflaton through a gauge kinetic function. Although the SU(3) gauge field grows
at the initial stage of inflation due to the interaction with the inflaton,
nonlinear self-couplings in the kinetic term of the gauge field become
significant and cause nontrivial dynamics after sufficient growth. We
investigate the evolution of the SU(3) gauge field numerically and reveal
attractor solutions in the Bianchi type I spacetime. In general cases where all
the components of the SU(3) gauge field have the same magnitude initially, they
all tend to decay eventually because of the nonlinear self-couplings.
Therefore, the cosmic no-hair conjecture generically holds in a mathematical
sense. Practically, however, the anisotropy can be generated transiently in the
early universe, even for an isotropic initial condition. Moreover, we find
particular cases for which several components of the SU(3) gauge field survive
against the nonlinear self-couplings. It occurs due to flat directions in the
potential of a gauge field for Lie groups whose rank is higher than one. Thus,
an SU(2) gauge field has a specialty among general non-Abelian gauge fields.
