Autonomous Dynamical System Approach for Inflationary Gauss-Bonnet Modified Gravity

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.