Consider the map $x_{n+1}=\lambda x_{n}\left(1-x_{n}^{2}\right)$ for $-1 \leqslant x_{n} \leqslant 1$. What is the maximum value, $\lambda_{\max }$, for which the interval $[-1,1]$ is mapped into itself?

Analyse the first two bifurcations that occur as $\lambda$ increases from 0 towards $\lambda_{\max }$, including an identification of the values of $\lambda$ at which the bifurcation occurs and the type of bifurcation.

What type of bifurcation do you expect as the third bifurcation? Briefly give your reasoning.