Addicted to Chaos

Still reading Chaos by Gleick. One can use Excel to demonstrate chaos as follows.

Take the Malthusian growth model, which is a logistic equation to represent population growth:

x next year = rx (1-x)

where x represents the proportion of surviving individuals at initial conditions, x next year is the proportion of surviving individuals in the next time period, and r is the rate of growth.

Here we have two plots, both with the same rate of growth, but with one starting at 90% survival rate and the other at 10%. Not surprisingly, after about 7-10 years, both are at the same spot, 44.44%.

Here's the same thing, but upping the rate of growth to 2.7:

Both still equalize after a few years at 62.96%, after bouncing on both sides for a bit. It gets weird when the growth rate gets near 3, though:

These graphs never stabilze, but always bounce back and forth between 67.94% and 65.35% indefinitely. And if we up the growth rate to 4:

Or 5, for that matter:

The whole thing falls apart if the population grows too quickly, it outstrips resources. I know, so what? Well, look back at the graphs with a growth rate of 3. They have no solution in that there is no single number that the population tends toward.

It gets better if you look at a graph like this, which I did not create in Excel:

This is a graph of the growth rate on the x-axis and the final population on the y-axis. As the rate nears 3, the possible final populations bifurcate, then quadfurcate, then octofurcate, until there's no solution. Simple enough, this is a product of math.

What's really cool are those bands of solution in the graph, like that big one right after 3.8. Interesting shit is happening in there. Those spots reduce to a trifurcated solution for a moment, each of which functions like its own population, birfurcating, quadfurcating and octofurcating. This is a zoom of that big spot right after 3.8:

That's Chaos, order masquerading as disorder. There is a solution, there is a pattern, but it's based on being slightly different that the previous pattern, but still a pattern nonetheless. It's like that Lorentz Loop from my previous post, there is always a figure eight pattern, but it's never the same size as before.