I am studying a sequence based on the following recurrence: $$X[t] = \sqrt{\alpha X[t-1]^2+(X[t-1]^2-\alpha X[t-2]^2)\frac{(2-X[t-1])^2}{X[t-1]^2}}$$ $$X[0]=0$$ $$X[1]>0$$ $$\alpha \in (0,1)$$ I would like to prove that under the above conditions, the sequence is convergent.

For particular initialization, such that $\exists k, X_k=2$ it is straightforward to show that the sequence converges to 0. However, simulations shows that random initialization make this sequence oscillate around the value of 1, and asymptotically converge to it (for example: https://i.stack.imgur.com/dK3Lc.png).

Now, I've managed to prove some intermediary results, such that the elements will eventually be "trapped" in the interval (0,2); the sequence will certainly maintain its monotony until it passes the value of 1. Ideally I would like to find the general solution, but all my attempts in this direction failed (including generating functions).

In order to prove convergence without knowing the general term, I tried bounding the sequence between two other sequences convergent to 1. I also attempted to show that the "turning points" (change of monotony above and bellow 1) gets closer and closer to 1. Sadly, none succeeded.

I would be really grateful for any hint or idea of other ways I can approach this problem.


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