Surprises in a classic boundary-layer problem

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.
Comment: 23+2 pages, 8 figures. Supplementary material available at https://github.com/duetosymmetry/surprises-in-a-classic-BVP . Version 2: Added 12 references and additional discussion. Comments welcome