Let $f_\mu(x)=\mu+x^2$. What bifurcation occurs for $\mu=-1$?
Pretty straight forward, but I'm having a hard time with this entire section in my book. It's not making any sort of sense and the theorems they want me to use are completely confusing. Can anyone please break this problem down into the simplest form possible?
There is no bifurcation at $\mu=-1$! This corresponds to the fact that $\mu=-1$ lies squarely in a stable zone of the bifurcation diagram.
Algebraically, the critical orbit of zero is, in fact, a member of a super-attractive orbit of period 2 here. If you change $\mu$ a little bit, you don't change the fact that zero is in a stable orbit.