I need to obtain the z-transform of difference equations that are as follows:
My problem however is multivariate and looks like this:
$$ \begin{align} x_{k+1}&=ay_{k}+ x_{k}^2y_{k}\tag{1} \\ y_{k+1}&=b+(1-a)y_{k} - x_{k}^2y_{k}\tag{2}\\ \end{align} $$
where $a$,$b$ are positive constants.
What would be the z-transform of the above equations? In particular how does one treat the $x_{k}^2y_{k}$ term when taking the z transform? I know the z transform of linear difference equations is simple, for instance $x_{k+1}=2x_{k}$ is transformed as $zX(z)-zx_{0}=2X(z)$
But how do I take a z-transform of equations (1) and (2)? If you could guide me to any relevant literature that addresses this issue as well I would be grateful.
Regards
One way of doing this (as I can think of) is to use the basic summation formula of z-transform in terms of the iterates and obtain the z-transform. My intention of asking the question was, is there a way to obtain an expression (like zX^2(z).Y(z)) for the solution instead of the summation?