Fractal Flow

How fractals flow as their parameters change.

Hello, my name is Michael Ostroff, a PhD Physics student at FAU, and this post is about a fractal phenomenon I discovered called fractal flow. If one has a set of parameters, a function which depends on said parameters, and they slightly vary said parameters, they’ll notice that the outputs of the function over a domain will shift around. For our purposes today, we will say this function is the ∞th iteration of complex function f[z,c]. This motion is how much the input z must change in order for the final output to remain constant as c changes over time. The velocity field ż is known as the fractal flow, and it’s a function of z, c, and ċ, the rate of change of the parameter c. Generally, ż has the same domain as z. Despite the ∞th iteration of f[z,c] diverging to infinity for values outside the corresponding Julia set, the fractal flow is a finite-valued complex function in this region. As you can see below, this complex function has a fractal structure.

For the case of a general Julia fractal where z_(n+1)=f[z_n,c], the fractal flow of a given point z₀ is defined by the iterative process which starts with

m=-1;s=0;z=z₀;

and then the following is applied repeatedly

m/=f10[z,c];s+=m f01[z,c];z=f[z,c];

where f10=∂f/∂z and f01=∂f/∂c. Outside the Julia set, z will diverge to complex infinity, at which point m will tend towards 0, causing s to converge. Inside the Julia set though, s seems to always diverge to complex infinity. The fractal flow is ż/ċ=s.

Every value of z where f10[z,c]=0 is known as a critical point. These critical points z_c are a function of c, and a given function f[z,c] can have several. A Mandelbrot is created by iterating f[z,c] with a starting value of z_c and seeing how many iterations it takes to diverge to infinity, if it ever does. If it diverges to infinity, then that value of c is outside the corresponding Mandelbrot set. If a value of c is outside all Mandelbrot sets, the corresponding Julia set, and by extension its fractal flow, will be disconnected. In the image below, you can see the fractal flow(bottom half) for a value of c outside all three Mandelbrot sets(top half). As you can see, random Mandelbrot sets tend to have weird bloby appearances. You can also see that the fractal flow is disconnected.

In contrast, if the value of c is in some Mandelbrot sets, but not all of them, you get a connected dust Julia set.

If a value of c is inside all Mandelbrot sets, you get a fully connected Julia set.

For the simple case of f[z,c]=z²+c, I’ve made a video of how the Julia set flows as one moves the parameter c around the Mandelbrot set: