Surprises in a classic boundary-layer problem

TL;DR

This paper explores a classic boundary-layer problem revealing overlooked phenomena such as bifurcations and tiny terms, providing educational insights for courses in perturbation methods and dynamical systems.

Contribution

It uncovers new phenomena in a well-known problem, including bifurcations and small terms, enriching the understanding of singular perturbation problems.

Findings

Discovery of a pitchfork bifurcation in solutions.

Identification of transcendentally small initial terms.

Potential as an educational tool for perturbation and dynamical systems courses.

Abstract

We revisit a textbook example of a singularly perturbed nonlinear boundary-value problem. Unexpectedly, it shows a wealth of phenomena that seem to have been overlooked previously, including a pitchfork bifurcation in the number of solutions as one varies the small parameter, and transcendentally small terms in the initial conditions that can be calculated by elementary means. Based on our own classroom experience, we believe this problem could provide an enjoyable workout for students in courses on perturbation methods, applied dynamical systems, or numerical analysis.