Fixed-point and two-cycle thresholds of the Hénon map (${\mu }_{\mathrm{SN}}=-(1+b{)}^2/4,{\mu }_{\mathrm{flip}}=3(1+b{)}^2/4$)

For the Hénon map $(x,y)\mapsto (y,\mu -bx-y^2)$ with $b>-1$, fixed points exist at $y_{\pm }=-(1+b)/2\pm \sqrt{\mu +(1+b{)}^2/4}$. Their multipliers satisfy ${\lambda }^2+2y\lambda +b=0$. The Jury stability criterion gives asymptotic stability only for the upper branch, with $-1<b<1$ and $-(1+b{)}^2/4<\mu <3(1+b{)}^2/4$. The two-cycle has alternating values $y,z$ satisfying $y+z=1+b$ and $(y-z{)}^2/4=\mu -3(1+b{)}^2/4$. Its second-iterate trace is $4[(1+b{)}^2-\mu ]-2b$ and determinant $b^2$, so it is attracting for $-1<b<1$ until $\mu =(5b^2+6b+5)/4$. The fixed-point fold and first flip are nondegenerate away from endpoints and persist under small smooth perturbations.