  Scalar field cosmologies with a generalized harmonic potential and a matter
fluid with a barotropic Equation of State (EoS) with barotropic index $\gamma$
for Locally Rotationally Symmetric (LRS) Bianchi III metric and open
Friedmann-Lema\^itre-Robertson-Walker (FLRW) metric are investigated. Methods
from the theory of averaging of nonlinear dynamical systems are used 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 behavior. Depending on values of barotropic index
$\gamma$ late-time attractors of physical interests for LRS Bianchi III metric
are Bianchi III flat spacetime, matter dominated FLRW universe (mimicking de
Sitter, quintessence or zero acceleration solutions) and matter-curvature
scaling solution. For open FLRW metric late-time attractors are a matter
dominated FLRW universe and Milne solution. With this approach, oscillations
entering nonlinear system through Klein-Gordon (KG) equation can be controlled
and smoothed out as the Hubble factor $H$ - acting as a time-dependent
perturbation parameter - tends monotonically to zero. Numerical simulations are
presented as evidence of such behaviour.
