= Solution
A <fixed point> $z_*$ of a map $F:\mathbb R^2\to\mathbb R^2$ satisfies $F(z_*)=z_*$. Its linear perturbations evolve by the <Jacobian matrix> $DF(z_*)$: both <eigenvalues> must lie strictly inside the <unit disk> for linear <asymptotic stability>. One inside and one outside gives a saddle; both outside give a repeller. Unit-modulus <eigenvalues> require nonlinear analysis, unlike the continuous-time test using signs of real parts.
Here a <fixed point> has $x=y=p$ and $2p=a(p-p^3/3)$. Therefore
$$
\boxed{p=0\text{ for all }a>0,\qquad p=\pm\sqrt{3-6/a}\text{ for }a>2.}
$$
At zero the characteristic equation is $\lambda^2+\lambda-a=0$. Its negative root is less than $-1$ for every $a>0$, while its positive root crosses $+1$ at $a=2$. Thus zero is a saddle for $0<a<2$ and a real repeller for $a>2$, with a pitchfork of fixed-point branches at $a=2$.
At either nonzero point the characteristic equation is $\lambda^2+\lambda+2a-6=0$, whose <discriminant> is $25-8a$. For $2<a<3$ the <fixed points> are saddles. At $a=3$ their <periodic-orbit multipliers> are $0,-1$, giving a flip threshold. For $3<a<25/8$ they are attracting real nodes; at $25/8$ repeated-eigenvalue <degenerate nodes>; and for $25/8<a<7/2$ attracting foci. The <Jury stability criterion> conditions reduce to $a>3$ and $a<7/2$, confirming the full stable interval.
At $a=7/2$ the <periodic-orbit multipliers> are $e^{\pm2\pi i/3}$. For larger $a$ they form a repelling <focus equilibrium>. This is a unit-circle crossing at a <strong resonance of a planar map> with ratio $1:3$; <linearization> alone does not justify asserting a generic nonresonant Neimark-Sacker bifurcation. The period-two branch at the flip threshold is derived next, including an additional family absent from the source's proposed quadratic.
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