*See here for the first part of this article.*

### Shifty dynamics continued

The point *p* is the fixed point with a sequence consisting of all 1s. The point *s* is the the fixed point with a sequence consisting of all 0s.

### The butterfly effect

The butterfly effect got its name from the idea that the bat of a butterfly's wing in one place on the Earth can cause a tornado elsewhere. That's sensitive dependence on initial conditions.

### Poincaré

Henri Poincaré. He was shocked by his discovery of homoclinic tangles.

What does all this have to do with the mess Poincaré found himself in 1889? In 1960 Stephen 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.

Everything hinges on the existence of a particular point. In the case of the horseshoe map, an example is the point $q$ with the sequence $...11111.01111...$ with only $1$s at the left and right end. Points in the forward orbit of $q$ get closer and closer to the fixed point $p$, whose sequence consists entirely of $1$s. That's a result of the fact that $f$ squeezes things in the horizontal direction: $q$ itself might be far away from $p$, but as you apply $f$ things get squeezed so the forward orbit of $q$ moves closer to $p$. The backward orbit of $q$ also gets closer and closer to $p$ — that's a result of the fact that $f$ stretches things in the vertical direction (if $f$ stretches things, then $f$ run backwards squeezes them). The existence of the point $q$ somehow embodies the squeezing and the stretching of $f$, whose interaction appears to be behind the chaos. The point $q$ is called a \emph{homoclinic point} of $f$.

Loosely speaking, if a dynamical system has such a \emph{homoclinic point}, which behaves just like $q$ does for the horseshoe map, then the dynamical system contains the full dynamics of the horseshoe map. (This is a very loose formulation of Smale's theorem. For a more technical version, see here.)

The dynamical systems Poincaré considered when thinking about the three-body system can contain such homoclinic points. The mess that results from the dynamics is called a \emph{homoclinic tangle} and trying to draw it is a bit like trying to draw the forward and backward orbits of our region $D$, which include all those 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.*"

There's another interesting connection between Smale and Poincaré we should mention. Poincaré made an important conjecture, not in the area of dynamical systems, but in the area of topology. The conjecture now bears Poincaré's name, and wasn't proved until the early 2000s. In 1961 however, Smale proved a result very much related to the Poincaré conjecture (he proved it for dimensions greater than 5). Rather satisfyingly his prove made use of dynamical techniques. It earned him the 1966 Fields Medal, one of the highest honours in maths.

### About this article

Marianne Freiberger is editor of *Plus*.
She talked to Stephen Smale at the Heidelberg Laureate forum 2017.