= Solution
At a <fixed point> $x=y=p$, the <quadratic equation> is $p^2+(1+b)p-\mu=0$. Consequently
$$
\boxed{p_\pm=-\frac{1+b}{2}\pm\sqrt{\mu+\frac{(1+b)^2}{4}},\qquad
\mu\ge-\frac{(1+b)^2}{4}.}
$$
The <Jacobian matrix> is $\begin{pmatrix}0&1\\-b&-2p\end{pmatrix}$, and its <Floquet multipliers> solve $\lambda^2+2p\lambda+b=0$. The <Jury stability criterion> requires $|b|<1$ and $1+2p+b>0$, $1-2p+b>0$. With $b>-1$, this becomes $-1<b<1$, $-(1+b)/2<p<(1+b)/2$. The lower branch never satisfies it; the upper branch is attracting precisely when
$$
\boxed{-1<b<1,\qquad-\frac{(1+b)^2}{4}<\mu<\frac{3(1+b)^2}{4}.}
$$
At the lower boundary the two <fixed points> merge with <Floquet multipliers> $1,b$, a <saddle-node bifurcation> for $b\ne1$. At the upper boundary the upper point has <Floquet multipliers> $-1,-b$, giving a generic <period-doubling bifurcation> for $-1<b<1$.
Its direction can be established explicitly. A distinct <two-cycle> alternates values $y,z$ satisfying $(1+b)z=\mu-y^2$, $(1+b)y=\mu-z^2$. Subtracting gives $y+z=1+b$, and substitution gives
$$
\boxed{y,z=\frac{1+b}{2}\pm\sqrt{\mu-\frac{3(1+b)^2}{4}}.}
$$
It exists beyond the fixed-point <period-doubling bifurcation>. The second-iterate <Jacobian matrix> has <determinant> $b^2$ and <trace> $4yz-2b=4[(1+b)^2-\mu]-2b$. Applying the same stability inequalities shows attraction for $-1<b<1$ and
$$
\frac{3(1+b)^2}{4}<\mu<\frac{5b^2+6b+5}{4}.
$$
Thus the first <period-doubling bifurcation> is supercritical; the next upper boundary has a <Floquet multiplier> $-1$ for the <two-cycle>. These are the <fixed-point and two-cycle thresholds of the Hénon map>.
At $b=1$ the map preserves <phase-space area>, so attraction is impossible. The fixed-point fold is at $\mu=-1$, with double <Floquet multiplier> $+1$, and is a conservative saddle-centre bifurcation. The upper branch $p_+=-1+\sqrt{1+\mu}$ is linearly elliptic for $-1<\mu<3$; the lower branch is a <saddle fixed point of a map>. At $\mu=3$, the upper branch has double <Floquet multiplier> $-1$ and becomes a <saddle fixed point of a map> for larger $\mu$. The <two-cycle> $y,z=1\pm\sqrt{\mu-3}$ is elliptic for $3<\mu<4$, then reaches double <Floquet multiplier> $-1$ for the second iterate at $\mu=4$. The partial sketch distinguishes elliptic, neutrally stable <Floquet multipliers> from attracting states. Resonances may require additional nonlinear analysis; unit <Floquet multiplier> moduli alone do not prove nonlinear <Lyapunov stability>.
The $b=1$ segment is a <conservative Hénon stability boundary>, not an ordinary nondegenerate dissipative <Neimark–Sacker bifurcation>. Indeed for $b>0$ constant <phase-space area> scaling is $\operatorname{area}(F(D))=b\operatorname{area}(D)$. A simple invariant closed curve would enclose a bounded <invariant set> $D$ of positive <phase-space area>, forcing $b=1$. Thus no such closed invariant curve can bifurcate into $b\ne1$ within this constant-determinant family.
Generic folds and flips away from these degeneracies persist under sufficiently small smooth <perturbations>: their simple unit <Floquet multiplier>, transverse parameter crossing and nonzero leading nonlinear coefficient persist, while the other <Floquet multiplier> stays away from the <unit circle>. The <implicit function theorem> then continues the bifurcation curves and their criticality. The unqualified <perturbation> claim needs this restriction. Arbitrary <perturbations> need not preserve the conservative $b=1$ behavior. For example, in local complex coordinates at a nonresonant elliptic <fixed point>, the <area-preserving map> property gives a cubic <normal form> $z'=e^{i\theta}z(1+i\tau|z|^2)+\cdots$ with zero cubic radial damping. Composing with the arbitrarily small radial <perturbation> $z\mapsto(1-\eta|z|^2)z$ introduces radial coefficient $-\eta$. Varying the modulus through one can now give a genuine <Neimark–Sacker bifurcation>. Conservative degeneracies and resonances are therefore not covered by a blanket structural-stability assertion.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2006/iii/paper-85-maps.png]
{title=Hénon fixed points and two-cycles at unit area determinant, and local branches of the triangular cubic map}
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