  We investigate the evolution of non vacuum
Friedmann-Lema\^itre-Robertson-Walker (FLRW) with any spatial curvature in the
context of Gauss-Bonnet gravity. The analysis employs a new method which
enables us to explore the phase space of any specific theory of this class. We
consider several examples, discussing the transition from a decelerating into
an acceleration universe within these theories. We also deduce from the
dynamical equations some general conditions on the form of the action which
guarantee the presence of specific behaviours like the the emergence of
accelerated expansion. As in $f(R)$ gravity, our analysis shows that there is a
set of initial conditions for which these models have a finite time singularity
which can be an attractor. The presence of this instability also in the
Gauss-Bonnet gravity is to be ascribed to the fourth-order derivative in the
field equations, i.e., is the direct consequence of the higher order of the
equations.
