= Solution
The nonzero <fixed point> of the <logistic map> is $x_*=(\mu-1)/\mu$, with multiplier $F'(x_*)=2-\mu$. It is stable for $1<\mu<3$, and its multiplier crosses $-1$ at $\mu=3$. To identify the emerging orbit explicitly, factor
$$
F^2(x)-x=-x(\mu x-\mu+1)
\left[\mu^2x^2-\mu(\mu+1)x+\mu+1\right].
$$
The remaining two roots, interchanged by $F$, are
$$
\boxed{x_{1,2}=\frac{\mu+1\pm\sqrt{(\mu+1)(\mu-3)}}{2\mu}}.
$$
They coincide with $x_*$ at three and are distinct for $\mu>3$. Their cycle multiplier is
$$
\Lambda_2=F'(x_1)F'(x_2)=\mu^2(1-2x_1)(1-2x_2)
=-\mu^2+2\mu+4.
$$
It lies between $-1$ and $1$ for $3<\mu<1+\sqrt6$. This establishes the first <period-doubling bifurcation> and the stable bifurcating two-cycle.
At $\mu_2=1+\sqrt6$, $\Lambda_2=-1$ and $d\Lambda_2/d\mu=-2\sqrt6\ne0$. The second flip is nondegenerate: for $H=F^2$ at either cycle point, the cubic flip coefficient is $c=(H'''/6)+(H''/2)^2=-\mathcal S H/6$ when $H'=-1$. Here $\mathcal S$ denotes the <Schwarzian derivative>. Since $\mathcal SF=-6/(1-2x)^2<0$ and $\mathcal S(F^2)=(\mathcal SF\circ F)(F')^2+\mathcal SF$, this coefficient is positive. Locally, after following the <fixed point> of $H$, its second iterate has the form $H^2(y)-y=2a(\mu-\mu_2)y-2cy^3+\cdots$, with $a=2\sqrt6>0$. Nonzero roots therefore appear with $y^2\sim a(\mu-\mu_2)/c$ on the increasing-parameter side. These are a four-cycle of $F$, proving \b[the second <period-doubling bifurcation> occurs at $\mu=1+\sqrt6$].
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