Smoothing out oscillations of a coupled Axion model

In this paper we study an axion model given by two canonical scalar fields $\phi_1,\,\phi_2$ coupled via a given potential $V(\phi_{1},\phi_{2})$. We introduce novel dynamical variables and dimensionless time variables, which have not been used in analyzing these cosmological dynamics where expansion normalized dynamical variables are usually adopted. In the uncoupled case the equilibrium points can be fully identified and characterized. However, in the coupled case we need to solve transcendental equations and a little progress is made. The main difficulties that arise using standard dynamical systems approaches are due to the oscillations entering the nonlinear system through the Klein-Gordon (KG) equations. This motivates the analysis of the oscillations using averaging techniques. Methods from the theory of averaging nonlinear dynamical systems allow us to prove that time-dependent systems and their corresponding time-averaged versions have the same late-time dynamics. Therefore, simple time-averaged systems determine the future asymptotic behaviour. Then, we study the time-averaged system using standard techniques of dynamical systems. Therefore, with this approach, oscillations entering a nonlinear system through the KG equation can be controlled and smoothed out as the Hubble factor $H$ - acting as a time-dependent perturbation parameter - tends monotonically to zero. We present numerical simulations as evidence of such behaviour.