  Bouncing cosmologies are often proposed as alternatives to standard inflation
for the explanation of the homogeneity and flatness of the universe. In such
scenarios, the present cosmological expansion is preceded by a contraction
phase. However, during the contraction, in general the anisotropy of the
universe grows and eventually leads to a chaotic mixmaster behavior. This would
either be hard to reconcile with observations or even lead to a singularity
instead of the bounce. In order to preserve a smooth and isotropic bounce, the
source for the contraction must have a super-stiff equation of state with
$P/\rho=w>1$. In this letter we propose a new mechanism to solve the anisotropy
problem for any low-energy value of $w$ by arguing that high energy physics
leads to a modification of the equation of state, with the introduction of
non-linear terms. In such a scenario, the anisotropy is strongly suppressed
during the high energy phase, allowing for a graceful isotropic bounce, even
when the low-energy value of $w$ is smaller than unity.
