  The problem of dark energy can be roughly stated as the proposition and
validation of a cosmological model that can explain the phenomenon of the
accelerated expansion of the Universe. This problem is an open discussion topic
in modern physics. One of the most common approaches is that of the "Dark
Energy" (DE), a matter component still unknown, with repulsive character (to
explain the accelerated expansion), which fills about 2/3 of the total content
of the Universe. In this thesis are investigated two cosmological models, a
non-minimally coupled quintessence field, based on a Scalar-Tensor Theory of
gravity, formulated in the Einstein's frame, and a quintom dark energy model,
based on General Relativity. A normalization and parametrization procedure is
introduced for each model, in order to investigate the flow properties of an
associated autonomous system of ordinary differential equations. In our study
are combined topological, analytical and numerical techniques. We are mainly
interested in the past dynamics. However, some results concerning the
intermediate and future dynamics are discussed. The mathematical results
obtained have an immediate interpretation in the cosmological context.
