Averaging Theory and Dynamical Systems in Cosmology: A Qualitative Study of Oscillatory Scalar-Field Models

We study cosmological models using dynamical systems and averaging methods, encompassing flat and open FLRW geometries as well as the LRS Bianchi types I, III, and V. Under mild regularity and frequency-scaling assumptions, we obtain a near-identity conjugacy between the oscillatory flow and an averaged slow flow, with $\mathrm{\parallel }\mathbf{x}(t)-\bar{\mathbf{x}}(t)\mathrm{\parallel }=\mathcal{O}(H(t))$. The effective systems preserve the original asymptotics and yield geometry-dependent late-time attractor classifications. A corollary addresses the case in which the leading averaged vector field vanishes, so the system exhibits no autonomous drift at order $H^0$.