let us consider the following quadratic map:
$$s_{n}=s_{n-1}²+c$$ $$(*)$$
There is several papers disscuting the dynamics of (*). I want to know the behavior of this map for $c=-2$ and I am asking if this map is equivalent to the logistic map $$s_{n}=rs_{n-1}(1-s_{n-1})$$ for some $0≤r≤4$.
$rs_n=rs_{n-1}(r-rs_{n-1})$, let $t_n=-rs_n$,
so $t_n+r/2=(t_{n-1}+r/2)^2-r^2/4+r/2$, let $u_n=t_n+r/2$
and $u_n=u_{n-1}^2-(r^2-2r)/4$.
So $r^2-2r=8$,$(r-1)^2=9$,$r=4$
$u_n=2-4s_n$, $u_n=u_{n-1}^2-2$,$s_n=4s_{n-1}(1-s_{n-1})$