  In general relativity, for fluids with a linear equation of state (EoS) or
scalar fields, the high isotropy of the universe requires special initial
conditions, and singularities are anisotropic in general. In the brane world
scenario anisotropy at the singularity is suppressed by an effective quadratic
equation of state. There is no reason why the effective EoS of matter should be
linear at the highest energies, and a non-linear EoS may describe dark energy
or unified dark matter (Paper I, astro-ph/0512224). In view of this, here we
study the effects of a quadratic EoS in homogenous and inhomogeneous
cosmological models in general relativity, in order to understand if in this
context the quadratic EoS can isotropize the universe at early times. With
respect to Paper I, here we use the simplified EoS P=alpha rho + rho^2/rho_c,
which still allows for an effective cosmological constant and phantom behavior,
and is general enough to analyze the dynamics at high energies. We first study
anisotropic Bianchi I and V models, focusing on singularities. Using dynamical
systems methods, we find the fixed points of the system and study their
stability. We find that models with standard non-phantom behavior are in
general asymptotic in the past to an isotropic fixed point IS, i.e. in these
models even an arbitrarily large anisotropy is suppressed in the past: the
singularity is matter dominated. Using covariant and gauge invariant variables,
we then study linear perturbations about the homogenous and isotropic spatially
flat models with a quadratic EoS. We find that, in the large scale limit, all
perturbations decay asymptotically in the past, indicating that the isotropic
fixed point IS is the general asymptotic past attractor for non phantom
inhomogeneous models with a quadratic EoS. (Abridged)
