  Scalar field cosmologies with a generalized harmonic potential are
investigated in flat and negatively curved
Friedmann-Lema\^itre-Robertson-Walker and Bianchi I metrics. An interaction
between the scalar field and matter is considered. Asymptotic methods and
averaging theory are used to obtain relevant information about the solution
space. In this approach, the Hubble parameter plays the role of a
time-dependent perturbation parameter which controls the magnitude of the error
between full-system and time-averaged solutions as it decreases. Our approach
is used to show that full and time-averaged systems have the same asymptotic
behavior. Numerical simulations are presented as evidence of such behavior.
Moreover, the asymptotic behavior of the solutions is independent of the
coupling function.
