  We discuss the coupled Einstein-Klein-Gordon equations for a complex scalar
field with and without a quartic self-interaction in a zero curvature
Friedman-Lema\^{\i}tre Universe. The complex scalar field, as well as the
metric, is decomposed in a homogeneous, isotropic part (the background) and in
first order gauge invariant scalar perturbation terms. The background equations
can be written as a set of four coupled first order non-linear differential
equations. These equations are analyzed using modern theory of dynamical
system. It is shown that, in all singular points where inflation occurs, the
phase of the complex scalar field is asymptotically constant. The analysis of
the first order equations is done for the inflationary phase. For the short
wavelength regime the first order perturbation term of the complex scalar field
is smeared out and the Bardeen potential oscillates around a nearly constant
mean value. Whereas for the long wavelength regime the first order perturbed
quantities increase.
