characteristic of the bugs fractal series is the “web” network
surrounding and connecting the areas of stability. That will be the focus for
this series. However, they also often contain regions of chaotic behavior. So
today we take a digression to look at the chaos.
First the TL;DR version. Fractal images based on the ubiquitous , are clean and orderly. Complex yes, but also clean. Fractal based on non-holomorphic functions, (and many rational functions) are messy with dust and brush strokes, like today’s image.
In a dynamical systems, orbits either (see Stability of orbits.)
- escape to infinity
- converge to single point or a finite periodic cycle
- are non-escaping, non-periodic (chaotic)
For a dynamical system based on polynomials, almost all orbits fall into the first two buckets. In the usual fractals, the orbits of the third type lie on the thin one dimension boundary between escaping and periodic regions, and the one dimensional filaments. Much too small to see.
But when you get
away from well behaved polynomials, two dimensional regions with chaotic orbits
of the third type appear.
While the bug formula, seems simple, it is not differentiable as a complex function. So adding this to the fractal calculation produces a non-holomorphic formula with potential for two dimensional chaos.
In general, fractals
with two dimension chaos is a worthy topic to study in its own right. I am sure
I will return to it from time to time. But tomorrow, focus
returns to the webs in the cleaner areas of the bugs fractals.
Here is the same image as yesterday, but with “black inside” turned off.
Traditionally for fractals only the escaping points are colored, and the color indicates how quickly the point escapes. The non-escaping, or “inside” points are colored black. For traditional fractals, the insides are boring. The orbits of the points in the non-escaping region flow towards a single fixed point, or an attracting cycle. A solid inside color helps shifts the viewer’s focus to the swirls surrounding the fractal body.
With non-traditional fractals, especially fractals based on non-holomorphic functions, the inside area is more interesting. The orbits can bounce around in chaotic fashion, never settling into a fixed point or cycle. It is not easy to find a meaningful way to color such regions. But that makes it all the more rewarding when you discover a way to visualize the area.
Time for some nerd talk.
First, let’s meet the bug formula: .
Typical fractals iterate the formula . Here the basic fractal formula is combined with the bug formula. There are two ways to combine them.
Mathematically the order matters, but for pictures the order makes very little difference. I am not consistent in choosing one or the other. This image happens to use the “bug first” variation.
The image is the “Mandelbrot set” for the combined formula. Technically speaking, if you are not using z^2+c, then it is not the Mandelbrot set. A proper description would be the “parameter space” view. But just this once I will indulge in sloppy language since the process is exactly like creating a Mandelbrot set image, but with the slight change in the formula. Compare to basic fractal.
The image is centered at (-1,0) with width 3.0 and height set to maintain a 1:1 mathematical aspect aspect ratio. (Meaning each pixel represents a square in the complex plane.)
I played with these bug fractals a lot in the 90s. Many images on the old site were bug fractals. I return to the formula often, especially when nothing else seems to be working. Two months ago I started playing with them again. There are several images on my hard drive, that need to be sorted and organized. I plan on doing that over the next few days. The first two are presented without explanation as a visual introduction to the series.
This one starts the
I will provide all
the nerdy details on how I create these in a later post. For now, just enjoy
the image without thinking about how or why.