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I am trying to solve the recurrence relation $a_{n}=\alpha a_{n-1}^2+\beta a_{n-1}$ where $\alpha$ and $\beta$ are constants. I have been trying to find specific techniques for solving this equation in closed form. I have not come across any such work, yet. I may not be looking into the proper literature as I guess this problem to be a well studied one. Any pointer towards the solution and/or the direct solution will be really helpful.

Thanks.

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Your recurrence equation is called a Quadratic map,most quadratic maps don't have a closed form solution.

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You can try to get an approximation. If $\alpha a_n^2$ dominates, a first cut is $a_{n + 1} = \alpha a_n^2$, which you can solve by taking logarithms and solving the resulting equation. If $a_n \to 0$, the second term eventually dominates and a first cut is $a_{n + 1} = \beta a_n$, also easy to solve.

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