Playing with Chaos

Rereading Gleick’s “Chaos” and decided to have a look myself at what it looks like.

Redoing this post – and adding 3 more images, with updated program code.

Here is a series of pictures I produced from the simple equation:
xp[next] = r.xp.(1 – xp)

To produce these images I simply ran the equation for each point on the x axis for a set of ranges of values of r (initially between 1 and 4). Starting the population at .1, I ran it for 600 cycles to reach stability if possible, then continued to run it, recording adding one to each y axis pixel corresponding to a result, recording until one pixel reached 3,200 hits. Then I ran a process over it to change those numbers into a visual rainbow – with red being the lowest numbers to violet being the highest (roy g biv). This makes it easy to see the curves of probability distribution of results in the chaotic zone between 3.6 and 4.

A link to the code I wrote is included below the images.

The first image shows the big picture,  as r increases from 1 to 4 the population increases until r gets to 3, when suddenly instead of settling to a single value, it starts to alternate between two values,   then at around 3.4 it changes behaviour again, now alternating between 4 values, then at about 3.55 it starts alternating between 8 values, and by around 3.66, it becomes completely chaotic, not settling into any particular repeating pattern, but over many cycles displaying interesting probability distributions.

big picture – r values 1 – 4

The next image (Key) just shows what I have done with the two images that follow it.

Key showing areas expanded in following images

The image following is the box A stretched to the left, to show more detail.

Box A – r value 3.5 – 4 expanded

The final image is the tiny area shown in Box B – r values between 3.565 and 3.585  and y values between 0.8775 and 0.895.   This massively enlarged section shows the same fractal pattern almost (but not quite) repeating, at ever finer scales.

Box B – r values between 3.565 and 3.585 and y values between 0.8775 and 0.895

Python 2.7 code is on my fishnet site – Teds_Chaos_image44.py