= Solution
A fixed point is locally attracting when its multiplier satisfies $|F'(x_*)|<1$ and repelling when $|F'(x_*)|>1$. A multiplier crossing $+1$ can create or destroy fixed points in a fold bifurcation, or exchange their stability when branches cross, as in a transcritical bifurcation. A multiplier crossing $-1$ gives a flip, or period-doubling bifurcation, in which a two-cycle can emerge. These are the two multiplier mechanisms; additional <symmetry> can distinguish special cases at $+1$.
For the <logistic map>, the fixed points are
$$
x=0,\qquad x_*=1-\frac1\mu.
$$
The second lies in $[0,1]$ only for $\mu\ge1$. Their multipliers are $\mu$ and $2-\mu$. Thus zero attracts for $0<\mu<1$; the branches meet at $\mu=1$ and exchange stability, with the nonzero branch attracting for $1<\mu<3$. At $\mu=1$, positive orbits decrease to zero since $F(x)=x-x^2$. At $\mu=3$, put $x=2/3+z$. Then $z\mapsto-z-3z^2$ and its second iterate is
$$
z\mapsto z-18z^3-27z^4,
$$
so the fixed point is still weakly attracting at the bifurcation, although linear attraction has disappeared.
The supplied factorization gives the nonfixed two-cycle points for $\mu>3$:
$$
\boxed{x_\pm=\frac{\mu+1\pm\sqrt{(\mu-3)(\mu+1)}}{2\mu}.}
$$
They are interchanged by $F$. Using their sum and product in $F'(x_+)F'(x_-)$ gives
$$
(F^2)'(x_\pm)=4+2\mu-\mu^2.
$$
The two-cycle is attracting exactly for
$$
\boxed{3<\mu<1+\sqrt6.}
$$
It is born from the fixed point at multiplier $+1$ for the second iterate, and at $\mu=1+\sqrt6$ its multiplier reaches $-1$, giving the next period doubling. The boundary cycles are nonhyperbolic; the strict multiplier test is not a proof of attraction there.
Beyond this threshold the <logistic map> undergoes further period doublings, with attracting cycles of periods $4,8,16,\ldots$ and then chaotic parameter ranges, interspersed with periodic windows. It is not correct to say that every parameter above this threshold is chaotic. At $\mu=4$, the substitution $x=\sin^2(\pi\theta)$ gives $F(x)=\sin^2(2\pi\theta)$, illustrating its chaotic doubling-map behavior.
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