  In this paper we shall analyze the $f(\mathcal{G})$ gravity phase space, in
the case that the corresponding dynamical system is autonomous. In order to
make the dynamical system autonomous, we shall appropriately choose the
independent variables, and we shall analyze the evolution of the variables
numerically, emphasizing on the inflationary attractors. As we demonstrate, the
dynamical system has only one de Sitter fixed point, which is unstable, with
the instability being traced in one of the independent variables. This result
holds true both in the presence and in the absence of matter and radiation
perfect fluids. We argue that this instability could loosely be viewed as an
indication of graceful exit in the $f(\mathcal{G})$ theory of gravity.
