Studying discrete-time dynamical systems (IV): generalization of dynamical systems able to produce spirals and clusters of points

Here is an animation of the families mentioned in my previous post: $$S_{D}=\{(x,y, f(z)): (x_{n+1},y_{n+1}) = (Im(f(x_{n} + y_n  i)\cdot\sin{\frac{D}{f}}, Re(f(x_{n} + y_n i))\cdot\cos{\frac{D}{f}})\}$$ Where $f(z)=\frac{1}{(z+1)^t}, t$ from $1$ to $2$ by $0.01$ steps. Each frame is a complete family plot, calculated using $D \in [0,8 \cdot 10^3], n \in [0,8 \cdot 10^3]$ zoom into region $z=[+/-5]+[+/-5]i$. It took me some days to render it:

It is visually astonishing! Still I did not receive any feedback regarding my question at Mathematics Stack Exchange about the dynamical systems that make these beauties. As far as I know it is the first time they are shown.