I have vaguely known about renormalization since the 1970’s, but had never seriously studied it. Out of curiosity I watched Renormalization and envelopes on YouTube Thursday evening. This was the final lecture of the Asymptotics and perturbation methods course by Prof. Steven Strogatz of Cornell University. I had watched the first two lectures of the course, but none of the others until this one. Fortunately, there were relatively few explicit dependencies on them, so I was able to follow this quite well. Here is the description:

Ideas about renormalization were originally developed in physics, first to remove nagging infinities that kept cropping up in quantum electrodynamics, and later in statistical physics, to explain the power laws and universal exponents seen near second-order phase transitions in everything from magnets to superconductors. Next came their application to nonlinear dynamical systems, most famously in Feigenbaum’s work on the universal aspects of the period-doubling route to chaos. In the 1990s, Chen, Goldenfeld, and Oono used renormalization ideas to find new and improved asymptotic solutions to a wide variety of singularly perturbed ODEs and PDEs. Their renormalization method still hasn’t quite made its way into textbooks on perturbation theory, but it deserves to be discussed at an introductory level.

In this lecture, Prof. Strogatz briefly touches on the basics of renormalization, mainly with the goal to stimulate viewers to learn more! He discusses an example from the work of T. Kunihiro, who showed that renormalization can be understood geometrically by using the concept of an “envelope” to a family of curves. A preprint of the paper by T. Kunihiro is at https://arxiv.org/abs/hep-th/9505166. A PDF of the published version “A geometrical formulation of the renormalization group method for global analysis” is on ResearchGate and SemanticScholar (and also behind paywalls).

I think the ArXiv version of the Chen, Goldenfeld, and Oono paper is Renormalization Group Theory for Global Asymptotic Analysis. I also found The Renormalization Group: A Perturbation Method for the Graduate Curriculum, which is apparently a simplified version of the CGO paper.

As noted above, Kunihiro’s approach is based on the “envelope” of a family of curves. Prof. Strogatz said this used to be a common part of the calculus course sequence, but is not commonly taught now. I found it discussed in my copy of Schaum’s Outline Series Theories and Problems of Advanced Calculus (1963), which I used for my advanced calculus course at Carleton College in the spring of 1970. The result used by Kinihiro and Strogatz is proved on p. 168.

If I am understanding this correctly, the “renormalization” is the replacement of constants A₀ and θ₀ in the naïve perturbation expansion by functions A₀(t₀) and θ₀(t₀). The envelope condition is then used to give t₀ as a function of t, with a convenient assumption (equation(3.9) on page 5) which I do not yet understand, but is very convenient. This may be another of those things in differential equations where a process is justified because it gives good results.