Fractal identification

Solution 1:

This is not an actual answer, but it's too long for a comment.

Here is a similar construction:

Here is a Matlab code to plot this family of fractals:

N = 10000; % Number of points
R = 1; % Radius of outer circle
K = 4; % Parameter K
i = 0; % Initial point
f = 1; % Final point
d = f-i;
k = abs(K);
Q = round(log(N*R*(k-1)/d)/log(k)); % Number of iterations
t = 2*pi*(i:d/N:f-d/N);
[a,N] = size(t);
x = zeros(1,N);
y = zeros(1,N);
for q=1:Q
    x = x - (R*(k-1)/k^q)*sin(t*K^q/k);
    y = y + (R*(k-1)/k^q)*cos(t*K^q/k);
end
plot(x,y)
axis equal

The fractal in the link above is for $K = -3$. In your case, you used $K = 4$:

I was particularly interested in the case $K = -2$ and created this animated gif: