Solving logistic map $u_{n+1}=4u_{n}(1-u_{n})$ for $u_{0} >1$:
It is straight forwards to show via induction that if $0 \leq u_0 \leq 1$ we can let $u_0 = \sin^2 \theta $ for some real $\theta$ then $u_n = \sin^2 (2^n \theta) $.
Clearly this cant hold in the $u_{0} >1$ case due to the range of sin, so I would think there would be some hyperbolic substitution I could do instead, however the basic sinh, cosh attempts dont seem to work.
Is there such an expression for $u_n$ in this case?
If $u_0>1$ then $u_1<0$ and $u_n\to-\infty$. Let $u_n=-\sinh^2\theta_n$ for $n\ge1$. Then $\theta_{n+1}=2\,\theta_n$.