Autonomous dynamical system description of de Sitter evolution in scalar assisted f(R) - φ gravity

In this paper 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 rendered 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 similar to 60 e-foldings. This instability can be viewed as a graceful exit from inflation, which is inherent to the dynamics of de Sitter attractors.