Plus Magazine Monthly Column – November 2017 – Part I: Smale’s chaotic horseshoe

By on 08/12/2017

Plus is a free online magazine about mathematics aimed at a general audience. It is part of the Millennium Mathematics Project, based at the University of Cambridge and our aim is to open a door onto the world of maths for everyone. We run articles, videos and podcasts on all aspect of mathematics, from pure maths and theoretical physics to mathematical aspects of art, medicine, cosmology, sport and more. Plus has a news section, covering news from the world of maths as well as the maths behind the mainstream news, reviews of books, plays and films, as well as puzzles for you to sharpen your wits.


Smale’s chaotic horseshoe

It’s not often you get to meet one of your all time mathematical heroes (especially since mathematical heroes tend to be dead), but back in September we had the great pleasure of meeting Stephen Smale, 1966 Fields Medallist honoured for his proof of the generalised Poincaré Conjecture in dimension greater than 5. Apart from his work on topology Smale is famous for contributions to (the not unrelated) theory of dynamical systems. In particular, he is famous for the horseshoe map, which beautifully encapsulates mathematical chaos.

Fittingly, it was none other than Poincaré who first encountered the dynamics the horseshoe map embodies in the late 19th century. He had been awarded a mathematical prize, offered by King Oscar II of Sweden, for his work on the three-body problem. To Poincaré’s horror the work turned out to contain a mistake. He did the right thing. He owned up and even paid for the copies of his work that had already been printed. He then revised his competition entry, still got the prize, and later wrote more extensively about what the mistake had revealed to him. What it had revealed was the existence of chaos.

Fast forward to 1960 and the beaches of Rio, where Stephen Smale was thinking about a problem inspired by the maths of radio waves. The complexity of the problem worried him, especially since he had previously conjectured that there was no such thing as chaos. “I made some pretty bad predictions,” he told us. “[Norman] Levinson at MIT pointed out that there were these old papers by a pair of English mathematicians [Mary Cartwright and J. L. Littlewood]. They had interesting results which contradicted what I had predicted, and I wanted to understand what they did. So I put what I did in a very geometric context so I could understand it.”

The result was Smale’s horseshoe map. It goes straight to the heart of the problems Poincaré had encountered when thinking about his three-body problem. “Poincaré got a big mess, and [the horseshoe] put order in the mess.”

The map is quite easy to define. Look at the region D shown below, made up of a square S with two rounded-off ends stuck on at the top and at the bottom. Imagine picking this region up, squeezing it in the horizontal direction and stretching it into the vertical direction so that it becomes long and thin, and then bending it round to the right to form a horseshoe. Now place it back down into the original region D as shown.

The horseshoe map.

The horseshoe map.

Things become quite complex when you apply the map to the region D repeatedly. The horseshoe then turns into a folded horseshoe, which in turn becomes a twice folded horseshoe, and so on.


Applying the horseshoe map twice to the region D gives a folded horseshoe.

In one of mathematics’ most beautiful uses of symbolic dynamics Smale showed that a particularly interesting set of points in D can be represented exactly by infinite sequences of 0s and 1s, so that an application of the map corresponds to shifting a sequence along to the right (find out more here.) With this conjugacy to the shift map, it immediately becomes obvious that the horseshoe map behaves chaotically on that interesting set of points.

In 1960 Smale (building on work by David Birkhoff) proved that the horseshoe is in some sense universal. A large class of dynamical systems, defined by mathematical equations, contain the dynamics of the horseshoe map and therefore also contain its chaos. The dynamical systems Poincaré considered when thinking about the three-body system are an example. The mess that results from the dynamics is called a homoclinic tangle and trying to draw it is a bit like trying to draw the forward and backward orbits of our region , which include increasingly complex multi-folded horseshoes. As Poincaré wrote:

“When one tries to imagine the figure formed by these two curves and their infinitely many intersections […] these intersections form a kind of lattice, web or network with infinitely tight loops […] One is struck by the complexity of this figure which I am not even attempting to draw.

You can find out more about the horseshoe map in Smale’s chaotic horseshoe.

to be continued…

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