Generalized flip bifurcation ($z^{\prime }=(-1+\beta )z+c\alpha z^3+d{z}^5+\cdots$)

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 $z^{\prime }=(-1+\beta )z+c\alpha z^3+d{z}^5+\cdots$, 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 $|\beta {|}^{1/4}$.