  We investigate Bianchi type IX ''Mixmaster'' universes within the framework
of the low-energy tree-level effective action for string theory, which (when
the ''stringy'' 2-form axion potential vanishes) is formally the same as the
Brans-Dicke action with $\omega =-1$. We show that, unlike the case of general
relativity in vacuum, there is no Mixmaster chaos in these string cosmologies.
In the Einstein frame an infinite sequence of chaotic oscillations of the scale
factors on approach to the initial singularity is impossible, as it was in
general relativistic Mixmaster universes in the presence of stiff -fluid matter
(or a massless scalar field). A finite sequence of oscillations of the scale
factors approximated by Kasner metrics is possible, but it always ceases when
all expansion rates become positive. In the string frame the evolution through
Kasner epochs changes to a new form which reflects the duality symmetry of the
theory. Again, we show that chaotic oscillations must end after a finite time.
The need for duality symmetry appears to be incompatible with the presence of
chaotic behaviour as $t\to 0$. We obtain our results using the Hamiltonian
qualitative cosmological picture for Mixmaster models. We mention possible
relations of this picture to diffeomorphism-independent methods of measuring
chaos in general relativity. Finally, we discuss the effect of inhomogeneities
and higher dimensions on the possible emergence of chaos within the string
cosmology.
