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Scalar field cosmologies with a generalized harmonic potential and a matterfluid with a barotropic Equation of State (EoS) with barotropic index $\gamma$for Locally Rotationally Symmetric (LRS) Bianchi III metric and openFriedmann-Lema\^itre-Robertson-Walker (FLRW) metric are investigated. Methodsfrom the theory of averaging of nonlinear dynamical systems are used to provethat time-dependent systems and their corresponding time-averaged versions havethe same late-time dynamics. Therefore, simple time-averaged systems determinethe future asymptotic behavior. Depending on values of barotropic index$\gamma$ late-time attractors of physical interests for LRS Bianchi III metricare Bianchi III flat spacetime, matter dominated FLRW universe (mimicking deSitter, quintessence or zero acceleration solutions) and matter-curvaturescaling solution. For open FLRW metric late-time attractors are a matterdominated FLRW universe and Milne solution. With this approach, oscillationsentering nonlinear system through Klein-Gordon (KG) equation can be controlledand smoothed out as the Hubble factor $H$ - acting as a time-dependentperturbation parameter - tends monotonically to zero. Numerical simulations arepresented as evidence of such behaviour.

Averaging Generalized Scalar Field Cosmologies I: Locally Rotationally Symmetric Bianchi III and open Friedmann-Lema\^itre-Robertson-Walker models