Consider the one-dimensional map $f: \mathbb{R} \rightarrow \mathbb{R}$, where $f(x)=\mu x^{2}(1-x)$ with $\mu$ a real parameter. Find the range of values of $\mu$ for which the open interval $(0,1)$ is mapped into itself and contains at least one fixed point. Describe the bifurcation at $\mu=4$ and find the parameter value for which there is a period-doubling bifurcation. Determine whether the fixed point is an attractor at this bifurcation point.