Consider the discrete Logistic model $P_{n+1}=r P_n\left(1-P_n\right)=g_r\left(P_n\right), 0 \leq r \leq 4$. Find the fixed point and analysis its stability. For what value of $r$, do the non-hyperbolic fixed points occur? Show that $g$ has periodic points of prime period two if $r>3$.
Fixed points are, $$ \begin{align} P^*&=rP^*(1-P^*)\\ P^*(r-rP^*-1)&=0\\ P^*=0&\textrm{ or }P^*=\frac{r-1}{r} \end{align} $$ Now, I suspect the range of $r$ needs to be split into many cases. Is $r=1,<1,\geq1$ enough? How could I know that more splits are necessary? And I couldn't figure out "For what value of $r$, do the non-hyperbolic fixed points occur? Show that $g$ has periodic points of prime period two if $r>3$."
non-hyperbolic for continuous version is $\frac{dx}{dt}=0$. From where I guess the discrete case will be $P_{n+1}-P_n=\nabla P_n=1$. but the discrete model seems not compatible with my guess. Any help will be appreciated.