The main 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 $latex z^2+c$, 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 $latex z^2+c$ 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, $latex bug(x+iy) = x + i* sin(y)$ 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.