Kasner and mixmaster behavior in universes with equation of state $w>~1$

We consider cosmological models with a scalar field with equation of state $w>~1$ that contract towards a big crunch singularity, as in recent cyclic and ekpyrotic scenarios. We show that chaotic mixmaster oscillations due to anisotropy and curvature are suppressed, and the contraction is described by a homogeneous and isotropic Friedmann equation if $w>\mathrm{}1.\mathrm{}$ We generalize the results to theories where the scalar field couples to p forms and show that there exists a finite value of w, depending on the p-forms, such that chaotic oscillations are suppressed. We show that $Z_2$ orbifold compactification also contributes to suppressing chaotic behavior. In particular, chaos is avoided in contracting heterotic M-theory models if $w>\mathrm{}1\mathrm{}$ at the crunch.