  We perform a detailed dynamical system analysis for the behaviour of a
Dirac-Born-Infeld (DBI) field in a spatially closed
Friedmann-Lema\^itre-Robertson-Walker (FLRW) cosmology. The DBI field is
characterised by a potential and brane tension. We study power-law or
exponential functions for the potential and tension. We find that in a
spatially closed FLRW cosmology, a DBI field in the ultra-relativistic limit
allows for a broader range of initial conditions resulting in a bouncing
universe than in the non-relativistic limit. We further note that the range of
initial conditions allowing for a bounce is larger if we consider power-law
functions for the potential and tension, compared to the exponential case. Our
dynamical analysis shows that a DBI field does not exhibit stable cyclical
behaviour, including the case in which a negative cosmological constant is
present.
