  In this paper we present the cosmological dynamics of a perfect fluid and the
Dark Energy (DE) component of the Universe, where our model of the dark energy
is the string-theoritic Dirac-Born-Infeld (DBI) model. We assume that the
potential of the scalar field and the warp factor of the warped throat region
of the compact space in the extra dimension for the DBI model are both
exponential in nature. In the background of spatially flat
Friedman-Robertson-Walker-Lemaitre Universe, the Einstein field equations for
the DBI dark energy reduce to a system of autonomous dynamical system. We then
perform a dynamical system analysis for this system. Our analysis is motivated
by the invariant manifold approach of the mathematical dynamics. In this
method, it is possible to reach a definite conclusion even when the critical
points of a dynamical system are non-hyperbolic in nature. Since we find the
complete set of critical points for this system, the center manifold analysis
ensures that our investigation of this model leaves no stone unturned. We find
some interesting results such as that for some critical points there are
situations where scaling solutions exist. Finally we present various
topologically different phase planes and stability diagrams and discuss the
corresponding cosmological scenario.
