  We derive a set of equations monitoring the evolution of covariant and
gauge-invariant linear scalar perturbations of
Friedman-Lema\^itre-Robertson-Walker models with multiple interacting
non-linear scalar fields. We use a dynamical systems' approach in order to
perform a stability analysis for some classes of scalar field potentials. In
particular, using a recent approximation for the inflationary dynamics of the
background solution, we derive conditions under which homogenization occurs for
chaotic (quadratic and quartic potentials) and new inflation. We also prove a
cosmic no-hair result for power-law inflation and its generalisation for two
scalar fields with independent exponential potentials (assisted power-law
inflation).
