A General View on Double Limits in Differential Equations

TL;DR

This paper reviews singularly perturbed differential equations with multiple small parameters, proposing a three-step framework to unify and analyze various double-limit problems across different contexts.

Contribution

It introduces a general conceptual framework with a three-step process for comparing and contrasting double-limit problems in differential equations.

Findings

The three-step process effectively unifies diverse double-limit problems.

Double-limit parametric diagrams serve as a powerful unifying tool.

The methodology can be transferred to various classes of differential equations.

Abstract

In this paper, we review several results from singularly perturbed differential equations with multiple small parameters. In addition, we develop a general conceptual framework to compare and contrast the different results by proposing a three-step process. First, one specifies the setting and restrictions of the differential equation problem to be studied and identifies the relevant small parameters. Second, one defines a notion of equivalence via a property/observable for partitioning the parameter space into suitable regions near the singular limit. Third, one studies the possible asymptotic singular limit problems as well as perturbation results to complete the diagrammatic subdivision process. We illustrate this approach for two simple problems from algebra and analysis. Then we proceed to the review of several modern double-limit problems including multiple time scales, stochastic dynamics, spatial patterns, and network coupling. For each example, we illustrate the previously mentioned three-step process and show that already double-limit parametric diagrams provide an excellent unifying theme. After this review, we compare and contrast the common features among the different examples. We conclude with a brief outlook, how our methodology can help to systematize the field better, and how it can be transferred to a wide variety of other classes of differential equations.