A general linear iteative system can be represented as a matrix:
$(x,y)\mapsto(ax+by,cx+dy)$
is essentially the same as
$\left[\begin{array}{cc} a&b\\ c&d\\ \end{array}\right] \left[\begin{array}{c} x\\ y\\ \end{array} \right]$
which is useful because it can be iterated quickly (matrix exponentiation) and enables various matrix techniques for determining asymptotic behavior and the like. (Of course the number of variables can be increased as needed.)
Is there a similar tool for quadratic iterative systems like
$(x,y)\mapsto(ax^2+bxy+cy^2,dx^2+exy+fy^2)$ ? I'm interested in computing the $n$th iterate ($n$ not too small), finding asymptotic behavior, and any other interesting things that can be determined for a given collection of constants $a,b,\ldots$.
My immediate interest (genetics, oddly enough) does not use any of the diagonal terms $x^2,y^2$ so a treatment that ignores them would be fine (though I suspect including is more natural).