I was working on a problem in economics and came across a system of non-linear difference equations of the form:

$$x_{n}=x_{n-1}+(y_{n-2}-y_{n-1})\left(1-\frac{2}{b}y_{n-1}\right)$$ $$y_n = a-\sqrt{a^2+bx_{n}}$$

I tried substituting $y_n$ into $x_n$ and vice versa to reduce it to just one difference equation, but it ends up 2nd order, non-linear with no obvious closed form solution. I'm aware that chances are pretty big that there is none. If there is a way to solve the system I would greatly appreciate any tips. However, I'm not interested in solving the system per say. It would be sufficient for me to understand what happens to

$$\lim_{n\to \infty}{x_n} \text{ and } \lim_{n\to \infty}{y_n} \text{ for different } a \text{ and } b.$$

Would it be sufficient to examine fixed points if I wanted to observe the behavior of limits at infinity? (analogous to system of ODEs and a phase diagram?) If so, how should I go about it? I can't rewrite it to vector form and use the usual textbook approach (i.e. drop the subscripts and solve the normal equation). I had trouble finding any good material on how to find and classify them in case like this.

Thank you very much.


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