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On $a+2b=1$ the coexistence fixed point hasFor $1/9<a<1$ these give a nonreal conjugate pair on the unit circle. Nearby, while the pair remains complex, its modulus is $\sqrt{D}$; increasing $a+2b$ through one changes the modulus from greater than one to less than one. This is the candidate Neimark–Sacker bifurcation, and a nonlinear coefficient is needed to decide its direction.

Here that coefficient can be calculated explicitly. Let $\rho =e^{i\theta }$, $\sigma =\sin \theta >0$, and normalize the right eigenvector and complex projection byFor deviations $(X,Y)=zq+\overline{z}\overline{q}$, the quadratic derivative isThus $f_{20}=p\mathcal{B}(q,q)/2$, $f_{11}=p\mathcal{B}(q,\overline{q})$ and $f_{02}=p\mathcal{B}(\overline{q},\overline{q})/2$ are the quadratic coefficients in $z^{\prime }$. The quadratic coordinate change hasSubstitution into the map and collection of $w^2\overline{w}$ yieldsUsing $b=(1-a)/2$ and ${\sigma }^2=(1-a)(9a-1)/(16a^2)$ simplifies the cubic radial coefficient of a quadratic planar map toConsequently, away from strong resonances of a planar map, the bifurcation is supercritical: on $a+2b>1$ coexistence is attracting, whereas on $a+2b<1$ a small attracting invariant closed curve surrounds the unstable coexistence fixed point. If $\eta =\sqrt{D}-1>0$, its leading radius is $|w{|}^2=-\eta /\ell$, and its radial multiplier is $1-2\eta +\cdots$. Motion on the curve can be quasiperiodic or frequency-locked into periodic motion; stability of the curve alone does not imply an irrational rotation.

Two interior values need qualification: $a=1/7$ is a 1:3 resonance, and $a=1/5$ is a 1:4 resonance. At the former $h_{02}$ is singular and the quadratic ${\overline{w}}^2$ term must be retained. At the latter a cubic ${\overline{w}}^3$ term remains. The resonant terms can be displayed explicitly in the same normalization. At $a=1/7$, $f_{02}=-7/6-7\sqrt{3}i/2\ne 0$. The third iterate, near the resonance and after removing nonresonant quadratics, hasHere $\delta$ is the angular detuning. Its nonzero stationary solutions have $|w|=\sqrt{{\eta }^2+{\delta }^2}/|f_{02}|$ and three phases; they constitute a saddle three-cycle. Indeed the two leading real exponents at these solutions are $3[\eta \pm \sqrt{4{\eta }^2+3{\delta }^2}]$, of opposite signs. Therefore no ordinary attracting-circle conclusion follows arbitrarily close to the exact 1:3 point.

At $a=1/5$, the cubic resonant normal form isThe extra coefficient follows from $g_{03}=f_{11}{h}_{02}+2f_{02}{\overline{h}}_{20}$ and $h=\overline{\rho }{g}_{03}$. Its radial cubic damping is uniform since $\mathrm{Re}g+|h|<0$. The slow angular equation is $\Delta \arg w=\delta +|w{|}^2\mathrm{Im}(g+h{e}^{-4i\arg w})+\cdots$, so fourfold phase locking must be considered. For example, varying $b$ at fixed $a=1/5$ gives $\delta =-\eta /2+\cdots$. The locked branches satisfy $\tan (4\arg w)=6/17$ and $|w{|}^2=-\eta /\mathrm{Re}(g+h{e}^{-4i\arg w})$. For $\eta >0$ they form one attracting and one saddle four-cycle; the two branches alternate in phase. This is a resonant alternative to quasiperiodic motion on an invariant curve. The endpoints also fail the simple-complex-pair assumptions. This distinguishes the generic nearby dynamics from unjustified claims about every point on the printed interval.

At a fixed point $x=y=p$, the quadratic equation is $p^2+(1+b)p-\mu =0$. ConsequentlyThe Jacobian matrix is $\left(\begin{array}{cc}0& 1\\ -b& -2p\end{array}\right)$, 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 whenAt the lower boundary the two fixed points merge with Floquet multipliers $1,b$, a saddle-node bifurcation for $b\ne 1$. 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 givesIt 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 $-1<b<1$ andThus 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 $\mathrm{area}(F(D))=b\mathrm{area}(D)$. 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 $b\ne 1$ 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 $z^{\prime }=e^{i\theta }z(1+i\tau |z{|}^2)+\cdots$ 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.

This map equals an explicit unit-step update of a logistic predator-prey differential equation. Its coexistence fixed point is $(b,(1-a-b)/a)$, requiring $a+b<1$. Its trace and determinant are $T=2-b/a$ and $D=(1-2b)/a$. The Schur stability criterion gives stability precisely when $b>(1-a)/2$ and $b<a+1/3$. The coexistence flip at $b=a+1/3$ is subcritical on $1/9<a<1/3$, with effective cubic coefficient $-9/[(3a+1{)}^2(9a-1)]$ when the $-1$ eigenvector has first component one. The unit-circle crossing $a+2b=1$ is a supercritical Neimark–Sacker bifurcation away from the strong resonances of a planar map $a=1/7,1/5$; in the eigenvector normalization $q_1=1$ its cubic radial coefficient is $-1/[a^2(1-a)]$. The whole nonnegative quadrant is not forward invariant, because $x^{\prime }$ can become negative. Local fixed-point stability is therefore not a global population-dynamics classification.

// Destination: dynamical-systems.bigb

At a strong resonance of a planar map the Floquet multiplier of a fixed point is a root of unity of low order. The usual nonresonant Neimark–Sacker bifurcation requires ${\lambda }^k\ne 1$ for $k=1,2,3,4$. At a 1:3 resonance a quadratic term ${\overline{z}}^2$ cannot be removed from the normal form; at a 1:4 resonance a cubic term ${\overline{z}}^3$ remains. Consequently one cannot infer a quasiperiodic invariant curve from the unit-modulus crossing alone.