Averaged dynamics in anisotropic interacting cosmologies

We apply the theory of averaging and standard dynamical system techniques to obtain relevant information from the field equations of an anisotropic cosmological model based on the locally rotational Bianchi I spacetime, incorporating a scalar field, a non-zero cosmological interaction term and a scalar field potential function that generalizes the usual harmonic potential. The gravitational field equations form a highly nonlinear dynamical system, thus the averaging approach allows us to extract information regarding the effective large-scale and late-time behavior of the solutions. By considering a generalized potential with a quadratic term and cosine-like corrections, the averaging methods become necessary to smooth out oscillations and to obtain simplified dynamical equations that have the same asymptotic behavior as the original system. We perform a detailed qualitative analysis of the gravitational field equations to obtain valuable information on the structure of the solution space for the anisotropic scalar field model. The interaction between the scalar field and matter is described by a general expression that depends on the Hubble parameter, the time derivative of the scalar field, and the energy densities of cold dark matter and dark energy. This formulation involves real parameters that modulate the interaction, as well as a coupling constant with the dimensions of the Hubble parameter. We show that the Hubble parameter serves as a time-dependent perturbation parameter, controlling the discrepancy between the original system and its time-averaged counterpart. As this parameter decreases, both systems converge to the same asymptotic behavior, which makes the averaging approach important for the study of the dynamics at late times. This enables the suppression of oscillatory effects, which simplifies the dynamical analysis. Finally, we identify conditions on the parameters of the model that ensure the presence of equilibrium points that are not present in the Bianchi I model without considering the interacting term.