Desde que te estoy queriendo

yo no sé lo que me pasa

cualquier vereda que tomo

siempre me lleva a tu casa

(Y mira que mira y mira, Camarón de la Isla)

The verses that head this post are taken from a song of Camarón de la Isla and illustrate very well what is a strange attractor *in the real life*. For non-Spanish speakers a translation is *since I’m loving you, I don’t know what happens to me: any path I take, always ends at your house*. If you don’t know who is Camarón de la Isla, hear his immense and immortal music.

I will not try to give here a formal definition of a strange attractor. Instead of doing it, I will try to describe them *with my own words*. A strange attractor can be defined with a system of equations (I don’t know if all strage attractors can be defined like this). These equations determine the trajectory of some initial point along a number of steps. The location of the point at step i, depends on the location of it at step i-1 so the trajectory is calculated sequentially. These are the equations that define the attractor of this experiment:

As you can see there are two equations, describing the location of each coordinate of the point (therefore it is located in a two dimensional space). These equations are impossible to resolve. In other words, you cannot know where will be the point after some iterations directly from its initial location. The adjective *attractor* comes from the fact of the trajectory of the point tends to be the same independently of its initial location.

Here you have more examples: folds, waterfalls, sand, smoke … images are really appealing:

The code of this experiment is here. You will find there a definition of parameters that produce a nice example image. Some comments:

- Each point depends on the previous one, so iteration is mandatory; since each plot involves 10 million points, a very good option to do it efficiently is to use
`Rcpp`

, which allows you to iterate directly in`C++`

. - Some points are quite isolated and far from the
*crowd of points*. This is why I locate some breakpoints with`quantile`

to remove tails. If not, the plot may be reduced to a*big point*. - The key to obtain a nice plot if to find out a good set of parameters (a
_{1}to a_{14}). I have my own method, wich involves the following steps: generate a random value for each between -4 and 4, simulate a*mini*attractor of only 2000 points and keep it if it doesn’t diverge (i.e. points don’t go to infinite), if x and y are not correlated at all and its kurtosis is bigger than a certain thresold. If the mini attractor overcome these filters, I keep its parameters and generate the big version with 10 million points. - I would have publish this method together with the code but I didn’t. Why? Because this may bring yourself to develop your own since mine one is not ideal. If you are interested in mine, let me know and I will give you more details. If you develop a good method by yourself and don’t mind to share it with me, let me know as well, please.

This post is inspired in this beautiful book from Julien Clinton Sprott. I would love to see your images.

These are really beautiful! Several of them seem reminiscent of Clifford attractors that you wrote about a year or so ago.

Nice to see the reference to Clint Sprott. He is such a landmark figure in chaos and complex systems research.

Yes, that’s true. I love his book. Thank you.