Averaging of ordinary differential equations with slowly varying averages (Q1956536)

The averaging method asserts that a good approximation to solutions of a time varying ordinary differential equation with small amplitude \[ \frac{dx}{dt}=\varepsilon f(x,t) \] are solutions of the averaged equation \[ \frac{dx}{dt}=\varepsilon f^0(x), \] where \[ f^0(x)=\lim_{t\to\infty}\frac{1}{T}\int_0^Tf(x,t)\,dt. \] The error between solutions with the same initial conditions is maintained small on a long time interval. In the paper, a similar result is established allowing the averaged equation to vary in time thus allowing slowly varying averages of the original equation. Both the modeling aspect and a discussion including a comparison of the model with a model from the literature and possible modifications of other available forms of the equation are presented. Examples that illustrate various possibilities of slowly varying averages and the application of the method established in the paper are given.