Distinguished Limits and Drifts: between Nonuniqueness and Universality

TL;DR

This paper clarifies the two-timing method for oscillatory systems, revealing only two distinguished limits and their universal structures, with practical guidance and biological examples.

Contribution

It identifies and explains the two fundamental distinguished limits in the two-timing method, providing accessible instructions and illustrating with biological examples.

Findings

Only two distinguished limits exist in the two-timing method.

Universal structures of averaged equations are demonstrated.

Explicit solutions for different oscillation classes are provided.

Abstract

This paper deals with a version of the two-timing method which describes various `slow' effects caused by externally imposed `fast' oscillations. Such small oscillations are often called \emph{vibrations} and the research area can be referred as \emph{vibrodynamics}. The governing equations represent a generic system of first-order ODEs containing a prescribed oscillating velocity u, given in a general form. Two basic small parameters stand in for the inverse frequency and the ratio of two time-scales; they appear in equations as regular perturbations. The proper connections between these parameters yield the \emph{distinguished limits}, leading to the existence of closed systems of asymptotic equations. The aim of this paper is twofold: (i) to clarify (or to demystify) the choices of a slow variable, and (ii) to give a coherent exposition which is accessible for practical users in applied mathematics, sciences and engineering. We focus our study on the usually hidden aspects of the two-timing method such as the \emph{uniqueness or multiplicity of distinguished limits} and \emph{universal structures of averaged equations}. The main result is the demonstration that there are two (and only two) different distinguished limits. The explicit instruction for practically solving ODEs for different classes of u is presented. The key roles of drift velocity and the qualitatively new appearance of the linearized equations are discussed. To illustrate the broadness of our approach, two examples from mathematical biology are shown.