Why does iterating $a(b,n)$ and highlighting loops produce these patterns?

Let $a(b,n)$ be the number of integer tuples $(x_1, x_2, ..., x_{k+1})$ where $0 \leq x_i \leq b-1$, such that $|x_i - x_{i+1}| = d_i$ for all $i$, where $(d_1, d_2, ..., d_k)$ are digits of $n$ in base $b$.

Related patterns in this specific sequence are discussed here and here.

Now consider the iterative definition $a(b_m,n) = b_{m+1}$, with starting value $(b_0,n)$. For any given starting value the sequence of terms $a(b_0,n),a(b_1,n),a(b_2,n),...$ will either loop or shoot off to infinity.

This can be visualised on a 2d grid by taking the initial values $(b_0,n)$ as the coordinate of the cells which we'd colour black if the sequence explodes and white if the sequence falls in a loop.

Surprisingly it has the following pattern:

Ofcourse we can explore this idea further by changing the definition of $a(b,n)$. Say if $a(b,n) = (b \oplus n) + |b-n|$, then the following pattern emerges:

where color is assigned based on how quickly it explodes or loops.

I had the idea to build an algorithm to explore possible rules for interesting patterns and I encountered some pretty interesting patterns:

The rules for these are pretty complex in comparison. How do these patterns emerge? Has it or something similar been researched on/found before?