= Fixed-point and two-cycle thresholds of the Hénon map
{title2=$\mu_{\rm SN}=-(1+b)^2/4,\quad\mu_{\rm flip}=3(1+b)^2/4$}
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.
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