  Starting from an anisotropic flat cosmological model(Bianchi type $I$), we
show that conditions leading to isotropisation fall into 3 classes,
respectively 1, 2, 3. We look for necessary conditions such that a Bianchi type
$I$ model reaches a stable isotropic state due to the presence of several
massive scalar fields minimally coupled to the metric with a perfect fluid for
class 1 isotropisation. The conditions are written in terms of some functions
$\ell$ of the scalar fields. Two types of theories are studied. The first one
deals with scalar tensor theories resulting from extra-dimensions
compactification, where the Brans-Dicke coupling functions only depend on their
associated scalar fields. The second one is related to the presence of complex
scalar fields. We give the metric and potential asymptotical behaviours
originating from class 1 isotropisation. The results depend on the domination
of the scalar field potential compared to the perfect fluid energy density. We
give explicit examples showing that some hybrid inflation theories do not lead
to isotropy contrary to some high-order theories, whereas the most common forms
of complex scalar fields undergo a class 3 isotropisation, characterised by
strong oscillations of the $\ell$ functions.
