= 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 $\mu=-1+\epsilon$ gives
$$
F^2(x)-x=-2\epsilon x-2(a+b^2)x^3+O(\epsilon^2x,\epsilon x^2,x^4).
$$
It is supercritical if $a+b^2>0$ and subcritical if $a+b^2<0$, with a degenerate flip if $a+b^2=0$. \b[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.
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