= Generalized flip bifurcation
{title2=$z'=(-1+\beta)z+c\alpha z^3+dz^5+\cdots$}
A generalized flip is a codimension-two period-doubling point with a simple multiplier $-1$, a vanishing cubic coefficient in its scalar centre-manifold map, and a nonzero quintic coefficient. In the odd normal form $z'=(-1+\beta)z+c\alpha z^3+d z^5+\cdots$, symmetric two-cycles solve $\beta+c\alpha z^2+d z^4=0$. The discriminant of this quadratic in $z^2$ produces a fold of two-cycles where its positive roots collide. For $c=3$, $d=-3$, this fold is $\beta=-3\alpha^2/4$, $z^2=\alpha/2$, $\alpha>0$. At $\alpha=0$ the amplitude is proportional to $|\beta|^{1/4}$.
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