A General Averaging Theorem for Slowly-Decaying Oscillatory Systems

We prove a general averaging theorem for non-autonomous slow-fastsystems coupled to a positive function that decays at a rate proportionalto its own square. The driving field of the fast system is assumed toadmit a splitting into a continuously differentiable averaged vector fieldand an almost-periodic oscillatory remainder with time-average equal to zero. We work underthree abstract hypotheses: regularity of the averaged vector field, thetrigonometric structure of the oscillation, and the existence of a smoothnear-identity corrector obtained by quadrature. Under these hypotheses weshow that the orbit of the original system converges to the orbit of theaveraged system as time goes to infinity, with an explicit error bound oneach interval of an adaptive bootstrap sequence whose lengths shrink tozero. As an application, we specialize the theorem to a Bianchi Icosmology with an axion-like scalar potential and a barotropic perfectfluid coupled through a non-gravitational interaction term. The averaged system is autonomous,two-dimensional, and free of the fast oscillation of the scalar field, soits attractor structure transfers to the original oscillatory dynamics inthe late-time limit.