 Orbit equations for a triangular cubic map

For $F(x,y)=((a+1)x-x^3+y,(b-1)y-(b-1)x^3+y^3)$, the origin multipliers are $a+1$, $b-1$. A nonzero fixed point has $S=x^2>0$, $y=x(S-a)$, and $S[1-(S-a{)}^3]=a(2-b)$. Since the map is odd, a symmetric two-cycle has $F(x,y)=-(x,y)$; it satisfies $y=x(S-a-2)$ and $S[1+(S-a-2{)}^3]=b(a+2)$. Eliminating $y$ from the two map equations gives these identities directly. They locate the side of each pitchfork or flip without relying on a potentially misleading cubic sign in the original coordinates. At $(a,b)=(-1,0)$ the second identity becomes $3S^2-3S^3+S^4=b$, giving the quartic amplitude scaling of a generalized flip bifurcation.