  In this paper, we study a cosmological model inspired in the axionic matter
with two canonical scalar fields $\phi_1$ and $\phi_2$ interacting through a
term added to its potential. Introducing novel dynamical variables, and a
dimensionless time variable, the resulting dynamical system is studied. The
main difficulties arising in the standard dynamical systems approach, where
expansion normalized dynamical variables are usually adopted, are due to the
oscillations entering the nonlinear system through the Klein-Gordon (KG)
equations. This motivates the analysis of the oscillations using methods from
the theory of averaging nonlinear dynamical systems. We prove that
time-dependent systems, and their corresponding time-averaged versions, have
the same late-time dynamics. Then, we study the time-averaged system using
standard techniques of dynamical systems. We present numerical simulations as
evidence of such behavior.
