Averaging with a time-dependent perturbation parameter

TL;DR

This paper presents a theorem on the long-term behavior of solutions to systems with oscillatory dynamics, motivated by cosmological models, extending averaging methods in nonlinear dynamical systems.

Contribution

It introduces a new theorem applicable to oscillatory systems in cosmology, enhancing the mathematical understanding of their asymptotic behavior.

Findings

Theorem characterizes large-time behavior of oscillatory systems.

Applicable to spatially homogeneous cosmological models.

Extends averaging theory in nonlinear dynamics.

Abstract

Motivated by recent problems in mathematical cosmology, in which temporal averaging methods are applied in order to analyze the future asymptotics of models which exhibit oscillatory behavior, we provide a theorem concerning the large-time behavior for solutions of a general class of systems. We thus propose our result to be applicable to a wide range of problems in spatially homogenous cosmology with oscillatory behavior. Mathematically the theorem builds upon the standard theory of averaging in non-linear dynamical systems.