Logistic maps and chaos

This plays around with the basic logistic map, $x_{n+1}=rx_{n}(1-x_{n})$.

The code is not complicated, but that's why chaos is interesting --- a lot of richness and complexity out of what seems like not a lot of equation/code. If you play around with bounds to e.g. zoom into arbitrarily-small sections (as is done below) and even change the map equation to something more complicated (even adding hyperbolic and trig functions, higher polynomials, etc), all the images seem to look basically like this. This is the self-similarity and scale-invariance of chaos coming out --- just like how many kinds of fractals all seem to look Mandelbrot-like, even when similarly more-complicated equations are used beyond $z_{n+1}=z_{n}^2+c$. For logistic maps (probably "traditional" fractals, too), the mathematicians seem to explain this as being a family of "quadratic maps" that all yield the same properties, just scaled differently.

A few outputs

A fully zoomed-out view of the interesting parts. The "period doubling" whose frequency increases at a ratio of the Feigenbaum constant can be seen in the branch from left-to-right.

Zooming in on one of those gaps (note the X axis bounds):

Zooming further into the above image, we start to repeat patterns. A miniature version of the whole "zoomed-out" scene was already visible in the above zoomed image, but scaled differently. ...like when you zoom in a Mandelbrot fractal and see another Mandelbrot-looking shape (and on and on, recursively).

I wonder if the noisiness in the above image is due to using numpy's default floating point (albeit 64-bit) math, rather than an arbitrary-precision library, even though we're not that zoomed-in yet.

Another detailed section:

Market prediction idea

Below was a result from playing with logistic maps in trying to yield curves that looked something like market data. An almost certainly wrong and very qualitative theory for building a quantitative financial market model idea went something like this:

• Markets are chaotic
• Fractals are self-similar
• When zooming in a fractal, you can see in the "distance" what's coming next as you zoom

Perhaps, therefore, if you could find and match up a generated 1D fractal curve against real market data (via cross-correlation), by principle of self-similarity in fractals, perhaps the market's motion too is more constrained than it appears. But, there are an infinite number of curves you could generate to match up against market data --- isn't it a super long shot that you'd find one that matches? To that I say, why do so many sections of the Mandelbrot look the same? As you zoom in and pan around, even when it's different, it often feels familiar. Maybe there are a smaller number of "classes" of curves with variations. And, do you need to have the "right" model? Apparently not, they all yield basically the same result. You get Mandelbrot-like fractals without trying. You get logistic map-like sets without trying. So perhaps, just like how you can "see where you're going" when zooming into a fractal, you could predict e.g. the next few minutes of market data once you've matched one of these classes of fractal curves that you've found.

So? How did it go? I gave up trying to make the curves look like market data, to say nothing of going further with cross-correlation. The ones below are too pointy --- real market data does not look so "Alpine mountain-like" as these curves. Given that I highly doubt this would work in the first place I didn't think it was worth spending much time on. But, at least it made for kind of a cool picture --- to create the curves below I generated a sequence using the logistic map, subtracted the average (if I recall), calculated the cumulative sum, and tried various low-pass filters. The result is a bunch of random walks created with fractals that always starts and ends at 0. Granularity of the curve depends on the length of the logistic sequence.

One more interesting point about these curves: note how they all start out together for a very short time, like a tiny lightning bolt on the left. This is because I varied a parameter in the logistic map only very slightly to generate the different curves. If I varied it even more slightly, they would track together for longer (but not that long, because it really seems to want to go chaotic). This sensitivity to initial conditions is another classic character of chaos. But, this is similar to how you can "see where you're going" when you zoom into a Mandelbrot fractal, even though you can't predict what comes after "the next horizon" and you definitely can't begin to predict even further. Below is a visualization of this fact (it only converges back to zero each time because of how I averaged the sequence).

Worthy of mention: Mandelbrot himself has spent quite a bit of his energy, for decades, looking at fractal behavior in markets (for example, here, in addition to a bunch of academic papers).