# Chaos: If the horseshoe fits

This is a commentary on Chapter 2 of the book Chaos by James Gleick which I discussed in my previous post. The chapter is titled “Revolution” and looks at the 60’s and 70’s when mathematicians reversed their position about unpredicatibility and began resolving a longstanding estrangement between physics and mathematics. Gleick discusses the importance of Steve Smale, a mathematician who developed a method for thinking about chaos that would become known as the “Smale horseshoe”. For an overview on the horseshoe, try either Wikipedia or Scholarpedia. Here I’ll mention some trivia about the horseshoe, followed by two big points that hit me.

First, let me offer some trivia. Smale’s life story is interesting in itself. There’s even a biography available by Steve Batterson. Smale actively protested the USA’s involvement in the Vietnam War and supposedly was controversial enough to lose his NSF funding as a result. His “horseshoe” concept is also a colorful story that took place on the Copacabana beach in Rio de Janeiro (see here or here for his own telling of the story). He commented once that the horseshoe shape itself was suggested by Lee Neuwirth in 1960 after seeing Smale’s less recognizable figures. Another tidbit I like is Smale’s own use of coin flips as an example of chaos (again see here or here). It’s a simple but accessible example that I haven’t seen used to demonstrate sensitivity to initial conditions, and Smale relates it to the horseshoe. More on that later.

Now I’ll describe two aspects of the horseshoe that struck me as I dug deeper. The first is a point emphasized by Gleick concerning Smale’s own personal revolution that mirrored the larger transformation that was occurring within mathematics. Regarding an earlier paper on dynamics, Smale says (see earlier links), “I was delighted with a conjecture in that paper which had as a consequence (in modern terminology) ‘chaos doesn’t exist’!” Gleick mentions that someone wrote to Smale to prove him wrong, citing a system with chaotic properties known as the van der Pol oscillator. Gleick doesn’t tell us who the “somone” was, but Smale explained that Norman Levinson wrote the letter that was to prove so cataclysmic for Smale.

I find this story significant because of my own failed attempt to discover where Smale originally supposed that systems with chaotic properties could not exist. Later I’ll give some details on my failed search. For now, I’ll just say that hours and hours of reading Smale’s papers gave me no clue about his claim. My point is that the future Fields Medalist had to be told he was completely wrong. I find this significant because it seems to be a hallmark of the classic Kuhn paradigm shift.

There’s a second major point that has impacted me about Smale’s horseshoe. It’s important to me because I’m still a neophyte in my understanding of chaos theory, and it highlights an aspect of the horseshoe that probably confuses quite a few neophytes like me. The issue is what happens at the ends of the horseshoe. They’re called “caps” in Wikipedia and “semi-discs” in Scholarpedia. Whatever you call them, their importance is not just unclear in such articles, but it seems to be almost completely ignored. This brings me to my personal hero of dynamical systems, Steven Strogatz, and his book Nonlinear Dynamics and Chaos. In this book, Strogatz actually spares the reader from confronting the Smale horseshoe head-on. Instead he saves it as an exercise for the reason I am about to reveal.

Strogatz first presents a version without the ends (or caps or semi-discs) and describes it as a “pastry map” where everything is stretched and squished and nothing is left out. This uses the same concept as the squished putty in the Wikipedia figure. He later explains that the horseshoe ends actually account for what he calls “transient chaos”, and I have not seen an accessible discussion of this concept anywhere else (yet). In transient chaos, the behavior is still sensitive to initial conditions, but the system eventually escapes the aperiodic behavior. Remember the coin-flip example I cited from Smale? That’s actually a form of transient chaos in that it’s quite unpredictable but eventually settles to an equilibrium. Strogatz uses a rolling die as an example and also points out a regime in the Lorenz equations for transient chaos. This is a major point for me because it resolves one frustration in attempting to understand the Smale horseshoe. It’s equally important because it bridges the gap between standard examples of chaos that oscillate forever and other unpredictable cases like a coin-flip or a rolling die.

This ends the main points I had about Gleick’s chapter on the revolution. For posterity, I will close with some details about my failed search for proof of Smale’s personal revolution. I’ll also give a couple references for his early horseshoe publications. Smale posted his personal bibliography here. I consulted a few colleagues, and one of them believes that Smale made his initial false conjecture here: Morse inequalities for a dynamical system, *Bulletin of the AMS*, 66 (1960), pp. 43–49. However, in this and other closely-dated publications, I am not able to relate Smale’s theorems about diffeomorphisms and structural stability to what I understand about chaotic systems. As for the horseshoe itself, that same colleague believes it may have first appeared in a 1961 conference paper, but I could not confirm that. The horseshoe transformation, in the form of equations, seems to be described here: Diffeomorphisms with many periodic points, *Differential and Combinatorial Topology (A symposium in honor of Marston Morse)*, Princeton University Press (1965), pp. 63—80. This can also be found in *The Collected Papers of Stephen Smale: Volume 2*. However, the first graphical depiction of the horseshoe seems to be in this 1963 paper (in Russian no less!): A structurally stable differentiable homeomorphism with an infinite number of periodic points, *Report on the Symposium on Non Linear Oscillations*, Kiev Mathematics Institute (1963), pp. 365–366. The first English publication with a graphic of the horseshoe seems to be this: Differentiable dynamical systems, *Bulletin of the AMS*, 73 (1967), pp. 747–817. Along with the classic horseshoe, it also displays Smale’s more complicated geometries that apparently preceded the horseshoe idea he received from Neuwirth.