I've run across the following recursion which at times seems very steady and predictable and at other times seems very chaotic.

Let $c_1, \dots c_k, b_0, m \in \mathbb{Z}$ with $b_0>m\ge 3$ and $b_i = 0$ for $i \le 0$. For $n \ge 1$, define $$b_n = \sum_{i=1}^k c_i\left \lfloor \frac{b_{n-i}}{m} \right \rfloor .$$

Here's two examples where the sequence behaves very differently:

In both, take $k=2$, $m=5$, $b_0=26$.

**Example 1:** For $c_1=2$, $c_2=4$, the sequence of $b_n$ eventually repeats: $$26, 10, 24, 16, 22, 20, 24, 24, 24, 24, \dots$$ where $b_i =24$ for $i \ge 6$.

**Example 2:** For $c_1=3$, $c_2=4$, the sequence of $b_n$ diverges: $$26, 15, 29, 27, 35, 41, 52, 62, 76, 93, 114, 138, \dots$$ where the sequence is strictly increasing for $i \ge 4$.

Is this a well-known recursion and/or dynamical system that has been studied before?

The main question I'm curious about is: given $m$ and $b_0$, under what conditions for $c_i$ is it eventually periodic vs. divergent?