I'll summarize first and then add details. Please forgive me as I'm delving into the field of chaos without much knowledge of it, seeking an expert to help point me in the right direction.
Are there dynamic maps with strange attractors whose coordinates (or defining functions) are almost integers, in the sense that they could be described in the decimal system with very few digits to the right of the decimal point? To make this idea of "almost integers" more precise, I'm looking for systems that essentially require coordinates with $n$ digits to the right of the decimal point, and $\exp(n)$ digits to the left of the decimal point.
For example, in the Mathematica 9 documentation of "RecurrenceTable", they delve into a system of the form:
$$ \begin{align} x_{n+1} &= \alpha x_n + y_n \\ y_{n+1} &= \beta + \left(x_n\right)^2 \\ y_0 &= \delta \\ x_0 &= \gamma \end{align} $$
I'm looking to find dynamic maps like this example that possess at least one strange attractor. The important part is that this strange attractor can be described by large real or complex numbers that are "almost integers".
As for the attractor: no.
The defining property of a strange attractor is that it is fractal-like (i.e., has a non-integer dimension), so you will always have infinitesimal details. This is due to the necessity of pieces of the attractor being folded onto itself by the time evolution (otherwise the dynamics would either be regular or not confined). Therefore every structure of the attractor has to appear in similar shape on one scaling level further down. And if there is no structure to the attractor, it is not strange (or even an attractor).
As for the defining function: Yes. I am not aware of a single chaotic map or differential equation that employs real numbers or an infinite number of rational numbers.