2. Periodic behaviour
In the introduction we saw that the values in the list of the iterated points can jump between certain points. On this behaviour we can determine the periodicity of the point and if it belongs to the Mandelbrot set. When drawing the Mandelbrot set in different colors for different periodicities, the result looks as following:
As brighter the color, as higher the periodicity. We can see that the periodicity behaves in certain patterns:
- The larger the 'bud' (the technical term is cardioid), the smaller the periodicity. The largest bud in the middle has the periodicity 1, the second largest has the periodicity 2, and so on.
- All buds touching a larger cardioid increase their periodicity of the size according to the periodicity of the large bud. For example the largest cardioid has the periodicity 1, the next larger has the periodicity 2, the next 3, and so on.
The cardioids on the largest cardioid increase their periodicity by 1, the cardioids on the cardioid with the periodicity 2 increase their periodicity by 2.
When we start on the point (0|0i) and go towards -∞ on the real axis, like the next image, we can see that the periodicity is increasing exponentially. Furthermore, the cardioids are shrinking as we go further into the negative area.
Periodicity of (0|0i) = 1
Periodicity of (-1|0i) = 2
Periodicity of (-1.3|0i) = 4
Periodicity of (-1.4|0i) = 8
Periodicity of (-1.409|0i) = 16
We can also see in which area which periodicity occurs:
Periodicity 1: (0.25|0i) to (-0.75249|0i)
Periodicity 2: (-0.7525|0i) to (-1.253661|0i)
Periodicity 4: (-1.253662|0i) to (-1.394026|0i)
Periodicity 8: (-1.394027|0i) to (-1.407416|0i)
We can also calculate the width of each cardioid on the real axis:
Cardioid with periodicity 1: 1.00249 units
Cardioid with periodicity 2: 0.501161 units
Cardioid with periodicity 4: 0.140364 units
Cardioid with periodicity 8: 0.013389 units
One could further examine the width of the cardioids, as well as the difference in size to other cardioids. But here, I want to focus on the periodic behaviour.
From the point (0|0i) to the point (-1.4074|0i) the periodicity increases exponentially. This changes from the point (-1.4078|0i):
Point (-1.40775|0i): Periodicity 8
Point (-1.4078|0i): Periodicity 40
Point (-1.40785|0i): Periodicity 8
Point (-1.4079|0i): Periodicity 8
Point (-1.40795|0i): Periodicity 16
Point (-1.408|0i): Periodicity 8
Point (-1.4081|0i): Periodicity 16
Point (-1.40815|0i): Periodicity 56
...
The periodicity jumps between very high and low values. This behaviour continues to the point (-1.4167|0i). Between (-1.41675|0i) and (-1.4189|0i) the periodicity is then 12. After that, the periodicity continues leaping. The values are referenced in the next image.
The points with the periodicity 0 in the diagram behave chaotic. The next image shows a detail of the image above.
Between the points (-1.45|0i) and (-2|0i) the periodic behaviour is very chaotic. The most points in this section have the periodicity 0 in the diagram, which means that the periodic behaviour is chaotic or too high.
The borders between the cardioids are very different. Some borders are very clear, for example the end of the Mandelbrot set in the positive area on the real axis.
Exactly from the point (0.25|0i) towards positive infinity the points diverge and don't belong to the Mandelbrot set. In the above image divergent points are marked with periodicity -1.
The border between the cardioid with periodicity 1 and periodicity 2 is also very clear. The border is exactly between the points (-0.752|0i) und (-0.753|0i). The same is the case with the borders between the cardioids with the periodicity 2, 4 as well as between the cardioids with the periodicity 4 and 8.
The border between the cardioids with the periodicity 2 and 4 is between (-1.253661|0i) and (-1.253662|0i).
Between the cardioids with the periodicity 4 and 8 the border is between (-1.394026|0i) and (-1.394027|0i).
As we saw before, from the point (-1.40775|0i) on there is no real border. In this section, the values of the periodicity fluctuate very strong.
The next borders is on the point (-2|0i). This is the end of the Mandelbrot set. The border is very clear like on the point (0,25|0i) on the other side of the set. Before the border, the periodicity behaves chaotic, but exactly on the point (-2|0i) the periodicity is 1.
