# Finding general solution to a non-linear discrete time recurrence relation

I am faced to a non-linear discrete time reccurence relation and I can't find the general solution.

The first question is: Is there a general recipe for finding the general solution to non-linear discrete time recurrence relations?

If not, then how would you find the general solution of the following?

$$p_{t+1} = \frac{sp_t(1-p_t)}{2}+p_t$$

Is there a way to know whether such question is solvable or not without having to try for hours?

Are there several solutions to this equation? If yes, I constructed my model so that I expect $p$ to increase through time iff s>0 and $p$ is expected to decrease through time iff s<0. And indeed looking at the recurence relation, we directly see that this is the way it will behave. I think the general solution should look like a logistic function.

I first Solved for the equilibrium $\hat p$

$$\hat p = \frac{s\hat p(1-\hat p)}{2}+\hat p$$

$$\hat p - \hat p = \frac{s\hat p(1-\hat p)}{2}$$

$$0 = \frac{s\hat p(1-\hat p)}{2}$$

$$0 = s\hat p(1-\hat p)$$

therefore, either $sp=0$, or $1-p = 0$, assuming that $s$ is not $0$, this give the two solutions $p=1$ and $p=0$, which makes sense given the intuition behind this model.

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This is a quadratic recurrence relation. Wolfram says there's no closed form for most of them, and having been thinking about the problem for the past couple of days, I can believe it. –  Jack M Mar 1 at 17:26