  We investigate the Bianchi I cosmology with the homogeneous SU(2) Yang-Mills
field governed by the non-Abelian Born-Infeld action. Similar system with the
standard Einstein-Yang-Mills (EYM) action is known to exhibit chaotic behavior
induced by the Yang-Mills field. When the action is replaced by the
Born-Infeld-type non-Abelian action (NBI), the chaos-order transition is
observed in the high energy region. This is interpreted as a smothering effect
due to (non-perturbative in $alpha'$) string corrections to the classical EYM
action. We give a numerical evidence for the chaos-order transition, and
present an analytical proof of regularity of color oscillations in the limit of
strong Born-Infeld non-linearity. We also perform some general analysis of the
Bianchi I NBI cosmology and derive an exact solution in the case when only the
U(1) component of the Yang-Mills field is excited. Our new exact solution
generalizes the Rosen solution to the Bianchi I Einstein-Maxwell cosmology to
the U(1) Einstein-Born-Infeld theory.
