Logistic and Quadratic map

Question

I am trying to understand the relation between a logistic map and a quadratic map. For example, how can you modify a logistic map for the quadratic map, i.e., modifying the logistic map $x_{n+1}=rx_n(1-x_n)$ to the quadratic $x_{n+1}=x_n^2+c.$

Answer 1

$rx_n(1-x_n)=rx_n-rx_n^2=-(\sqrt{r})^2(x_n-\frac{1}{2})^2 + \frac{r}{4}$.

Now make the Substitution $(x_n- \frac{1}{2}) \sqrt{r} = y_n$ from which follows $y_{n+1}=(x_{n+1}- \frac{1}{2}) \sqrt{r}$. When you replace $x_n$ by the $y_n$ you will have the quadratic map.