Autonomous Dynamical System Description of de Sitter Evolution in Scalar Assisted $f(R)-\phi$ Gravity

In this letter we will study the cosmological dynamical system of an $f(R)$ gravity in the presence of a canonical scalar field $\phi$ with an exponential potential, by constructing the dynamical system in a way that it is render autonomous. This feature is controlled by a single variable $m$, which when it is constant, the dynamical system is autonomous. We focus on the $m=0$ case which, as we demonstrate by using a numerical analysis approach, leads to an unstable de Sitter attractor, which occurs after $N\sim 60$ $e$-foldings. This instability can be viewed as a graceful exit from inflation, which is inherent to the dynamics of de Sitter attractors.