Can we create a system of recurrences that exhibits chaotic behavior, with the restriction that we are only allowed to use the integers?
For example, we could create a system:
$$ \begin{align} a(t) &= a(t-1)^2 +b(t-2) + 3c(t-1) \\ b(t) &= 3 + a(t-1) \\ c(t) &= b(t-1)b(t-2)^2 \\ & \vdots \end{align} $$
I'm of course looking for a system that stays within some boundaries for certain intial conditions. In other words, maybe:
$$ \begin{align} -20 \le a(t) \le 100 \\ 15 \le b(t) \le 50 \\ -100 \le c(t) \le 100 \\ & \vdots \end{align} $$
Of course, all of the functions I mentioned are just examples. The main problem is that the functions must exhibit what I will call "totally integer functions". That is to say that they must be functions from integers to integers. That disallows division, since we could get a non-integer result. It also disallows floors, ceilings, and modulo functions, for example. I also cannot use function composition, unless the total depth of recursion is less than $\log{(\log{(n)})}$ for a function with all values between $-n$ and $n$.
Digital computers are integer systems. Can we create chaos in an digital computer? Does a system of NAND gates pass your requirements?
Also look at cellular automata in Wolfram's New Kind of Science.