Find general solution of $u_{n+1}=4u_n(1-u_n)$

Question

Find the general term of the sequence $u_{n+1}=4u_n(1-u_n)$

I know that the solution can be found for $|u_0|\leq 1$ by using a trig substitution, but what about for $u_0>1$?

It seems the characteristic equation method won't work because the degree of the polynomial is too high and guessing seems difficult.

Answer 1

The solution pointed out by @Mittens in the Comments is indeed the solution you are seeking. However, it is assumed there that $u_0\in[0,1]$, contrary to your expressed $u_0>1$. Now, I have found the same solution as the logistic maps in WolframAlpha, which can be expressed as follows

$$u_n=\sin^2(C\cdot2^n)$$

so that $C=\sin^{-1}(\sqrt{u_0})$

Now, you can choose $u_0>1$, in which case the $\arcsin$ is complex, but the resulting $u_n$ is real. However, note that these solutions grow exponentially fast with both $u_0$ and $n$. I have verified these results by numerical calculation of the recurrence relation and the analytic solution for random values of $|u_0|>1$. The only caveat is that there is (numerically) an infinitesimal residual complex component.