Let f_a:R to R be defined by f_a(x)=-x³+2x²+ax, where a is a real parameter. Prove that the difference equation y(n+1)=f_a(y(n)), n>=0 undergoes a bifurcation at a=1. What kind of bifurcation occurs at this bifurcation point? Explain your answer.

I have found the equillibrium solutions y₁=0, y₂=1-1/2 root(4a) andy₃=1+1/2 root(4a).

So looking at the derivative of f_a(x), we can see that y₁ is asymptotically stable if a<1 and unstable if a>1..... but I don't really understands what happens fory₂ andy_3, is it a period doubling? transcritical? And why? Thanks for the help!