 Solution

For $a,b<0$, ${\mu }_s<1$. The local attracting nonzero segments have $x<-b/(2a)$ or $x>0$, again subject to $M>-1$; the segment between the saddle-node and zero repels. The crossings at ${\mu }_s$ and one are saddle-node and transcritical. The quadratic-cubic flip criticality at the origin requires additional information: writing $\mu =-1+ϵ$ givesIt is supercritical if $a+b^2>0$ and subcritical if $a+b^2<0$, with a degenerate flip if $a+b^2=0$. The sign assumptions in case (iii) do not determine this criticality. The original “no further bifurcations in the region sketched” qualification is local; it does not remove this possible degeneracy or the additional $M=-1$ crossings farther along fixed branches.