Hénon map

The Hénon map is a two-parameter quadratic planar map, conventionally $(X,Y)\mapsto (1-a{X}^2+Y,b_{H}X)$. The alternative family $(x,y)\mapsto (y,\mu -bx-y^2)$ has constant Jacobian determinant $b$. When $\mu \ne 0$, the change $X=y/\mu$, $Y=-bx/\mu$ gives the conventional parameters $a=\mu$, $b_H=-b$. Fixed points in the alternative normalization satisfy $x=y$ and $y^2+(1+b)y-\mu =0$. Its invertibility requires $b\ne 0$; at $b=1$ it preserves area.

At $b=1$ the Hénon map has unit Jacobian determinant. The upper fixed point is linearly elliptic for $-1<\mu <3$, and its newborn two-cycle is elliptic for $3<\mu <4$. These are neutral multiplier statements, not attraction. The endpoints have double multipliers $+1$ and $-1$. Constant-area scaling also excludes a simple invariant closed curve enclosing positive area when $b>0$ and $b\ne 1$: invariance of its interior would require both $\mathrm{area}(F(D))=b\mathrm{area}(D)$ and $F(D)=D$. Thus the $b=1$ elliptic stability boundary is a conservative degeneracy, not a generic dissipative Neimark-Sacker bifurcation. Arbitrary non-area-preserving perturbations can change this behavior even though generic fold and flip curves persist.

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.