The Real Line of the Mandelbrot Set
It is interesting to explore the real line of the Mandelbrot Set. Here we can find orbital behaviour of every period. It is also easy to analyse because we only need the real numbers.
So let's have a look at the real line....
The Mandelbrot Set is contained between the values -2 and 0.25 inclusive on the real number line. In the positive direction it actually extends a little further if there is a non-zero imaginary component.
Graphing the Orbits
We can graph the first 12 iterations. The following graph shows the value of z after each iteration. After the first iteration it is the original value of c. You can see that values less than -2 shoot over to the right and escape in the positive direction. Values greater than 0.25 also escape to the right. The other values remain bounded. It seems that those to the right are better behaved, and those to the left are more chaotic.
The main cardioid exists between the value of -0.75 and 0.25. The area is known to be a perfect cardioid.
The orbits in this are are not periodic. We sometimes call this the period 1. Let's see how the orbits in just this are behave.
After 12 iterations it does look like we have some periodic behaviour towards the left. In fact we are getting close to the period 2 area, this is no coincidence. But we see that as long as our orbit starts at a value greater than -0.75 it will eventually settle.
After 100 iterations, we can see that orbits in this region do not remain periodic. All the orbits in the region of the cardioid will eventually converge.
Period 2 Circle
Just to the left of the cardioid is the period 2 circle. It is known to be a perfect circle. It is centred at -1 with a radius of 0.25.
In this circle all of the orbits have a period of 2.
Unlike the cardioid, even when we include many iterations, it remain periodic.
"Circles" to the right of Feigenbaum point
Further to the left we have a series of smaller "circles", doubling in period each time. I'm not sure if these are true circles or not. They get smaller and smaller as they go to the left. However there is a limiting point to this behaviour and it is known as the Feigenbaum point. About −1.4011551890…
The Feigenbaum point marks a boundary between the main Mandelbrot and the area containing mini-Mandelbrots. You will see similar behaviour at the end of all the bulbs on the cardioid.
A closer look...
A closer look...
A closer look...
Looking at the orbit graph in this general area. We can see that there seems to be an overall period of 4. However, you'll notice that it is very roughly contained within a period of 2. Likewise, as we go left the period 8 orbits are kind of within the period 4 orbit (see 2nd graph below).
Selecting a single orbit in this area, you will notice that at first glance it seems to have a period of 4, but in fact if you look closer you will see that it is higher.
Mini-Mandelbrots to the left of the Feigenbaum point.
To the left of the Feigenbaum point are mini-Mandelbrot. In fact, it is my belief that all of this line is made from a collection of mini-Mandelbrot of varying sizes. It seem that no matter where you look on the line, you can find a mini-Mandelbrot close by if you go deep enough.
Orbits in the "Minibrot" zone start to get chaotic. That is because each Mini-Mandelbrot or "Minibrot" has it's own period, and attached circles have an associated period.
With 50 iterations shown.
The Period 3 Mini Mandelbrot
Let's take a look at the biggest mini-Mandelbrot. It's cardioid has a period of 3. The shape is not exactly the same as the main Mandelbrot, there is some minor distortion. So the cardioid and circle aren't exact shapes.
If we graph just the orbits in this area you will see that there is an overall period of 3. The cardioid of this minibrot has a period of 3 However if you look closely you can see that they diverge slightly every second cycle. This is the influence of the orbits in the large circle, which have a period of 6.