Quadratic-cubic flip criticality
= Quadratic-cubic flip criticality
{title2=$a+b^2$}
For $F_\mu(x)=\mu x+bx^2+ax^3$ near $\mu=-1$, write $\mu=-1+\epsilon$. Then $F_\mu^2(x)-x=-2\epsilon x-2(a+b^2)x^3+O(\epsilon^2x,\epsilon x^2,x^4)$. The small two-cycle is attracting on $\mu<-1$ if $a+b^2>0$ and repelling on $\mu>-1$ if $a+b^2<0$. Vanishing of $a+b^2$ gives a degenerate flip requiring higher terms.