Consider a point on the $xy$-plane whose position is updated in iterations. In each iteration, the point undergoes, with equal probability, either an $A$- or a $B$-update, defined as follows:
$A$-update: $\cases{x_{i+1} = \sqrt{x_i^2-k}\\ y_{i+1} = y_i + \delta(x_i) \\ },\qquad$ $B$-update: $\cases{x_{i+1} = x_i + \delta(y_i) \\ y_{i+1} = \sqrt{y_i^2-k}},$
where $\delta(\cdot) = \frac{\alpha}{\sqrt{(\cdot)^2 - \beta}}$, $k,\alpha> 0$, and $0\leq \beta \leq k.$
The sequence terminates when either $x_i$ or $y_i$ $\leq \sqrt{k+1}$.
The dynamics are interesting because, in an $A$-update, $x$ decreases while $y$ increases, and $y$ increases more as $x$ gets smaller (and vice-versa for a $B$-update).
Questions:
For which values of $k$, $\alpha$ and $\beta$ is the sequence guaranteed to terminate after a finite number of iterations?
How many iterations does it take for the sequence to terminate in the worst case scenario, given $x_0$ and $y_0$?
PS: I had asked an earlier version of this question, which was too convoluted, and remained without a satisfactory answer. I have presented a much simplified version here.
For the process to terminate in a finite number of steps, say $N$, at step $N-1$ it must be certain that it will do so irrespective of the update. So
$$\sqrt{x_{N-1}^2-k}\le\sqrt{k+1}\text{ and }\sqrt{y_{N-1}^2-k}\le\sqrt{k+1}$$
or (knowing that all square roots are positive)
$$\sqrt{k+1}\lt x_{N-1},y_{N-1}\le\sqrt{2k+1}$$
At $N-2$
$$\begin{align} \sqrt{k+1}&\lt x_{N-1}=\sqrt{x_{N-2}^2-k}&\le\sqrt{2k}\\ \sqrt{2k+1}&\lt x_{N-2}\le\sqrt{3k+1} \end{align}$$
and
$$\begin{align} \sqrt{k+1}&\lt y_{N-1}=y_{N-2}+\frac{\alpha}{\sqrt{x_{N-2}^2-\beta}}\le\sqrt{2k+1}\\ \sqrt{k+1}-\frac{\alpha}{\sqrt{x_{N-2}^2-\beta}}&\lt y_{N-2}\le\sqrt{2k+1}-\frac{\alpha}{\sqrt{x_{N-2}^2-\beta}}\\ \end{align}$$
By symmetry, this must also be true if the other option was selected. So since we know that the $\alpha$ term is positive, it is clear that the lower limit is defined by the first equation and the upper limit by the second, giving
$$\begin{align} \sqrt{2k+1}&\lt x_{N-2},y_{N-2}\le\sqrt{2k+1}-\frac{\alpha}{\sqrt{x_{N-2}^2-\beta}}\\ \end{align}$$
This inequality cannot be satisfied given the constraints.
Unless I have made a mistake it is not certain that the function will terminate in a finite number of steps.
That said, it is quite likely that it will.