  We analyze the dynamics of a Dirac-Born-Infeld (DBI) field in a cosmological
set-up which includes a perfect fluid. Introducing convenient dynamical
variables, we show the evolution equations form an autonomous system when the
potential and the brane tension of the DBI field are arbitrary power-law or
exponential functions of the DBI field. In particular we find scaling solutions
can exist when powers of the field in the potential and warp-factor satisfy
specific relations. A new class of fixed-point solutions are obtained
corresponding to points which initially appear singular in the evolution
equations, but on closer inspection are actually well defined. In all cases, we
perform a phase-space analysis and obtain the late-time attractor structure of
the system. Of particular note when considering cosmological perturbations in
DBI inflation is a fixed-point solution where the Lorentz factor is a finite
large constant and the equation of state parameter of the DBI field is $w=-1$.
Since in this case the speed of sound $c_s$ becomes constant, the solution can
be thought to serve as a good background to perturb about.
