= Solution
For the quadratic <unimodal map>, an attracting <fixed point> lies on the positive branch
$$
x_*(\mu)=\frac{2}{1+\sqrt{1+4\mu}},\qquad
f_\mu'(x_*)=1-\sqrt{1+4\mu}.
$$
Its multiplier has modulus less than one for $0\leq\mu<3/4$ and reaches $-1$ at the first <period-doubling bifurcation> $b_1=3/4$. The resulting two-cycle has points
$$
x_\pm=\frac{1\pm\sqrt{4\mu-3}}{2\mu},\qquad
f_\mu(x_+)=x_-,\quad f_\mu(x_-)=x_+.
$$
Its <periodic-orbit multiplier> is $4\mu^2x_+x_-=4(1-\mu)$, so it is attracting for $3/4<\mu<5/4$ and doubles at $b_2=5/4$. Successive attracting periods are $4,8,16,\ldots$, with their parameter intervals shrinking toward $s_\infty\simeq1.401155189$.
A <superstable periodic orbit> contains the <critical point> zero, so its <derivative> product vanishes. Along the first <period-doubling cascade>, $s_n$ is the first-cascade parameter with a critical orbit of exact period $2^n$, satisfying $f_{s_n}^{2^n}(0)=0$. For example, $s_1=1$, $s_2\simeq1.310702641$, and $s_3\simeq1.381547484$. These are interior points of stability intervals, distinct from the bifurcation parameters $b_n$.
The following <bifurcation diagram> shows attracting orbit values, including the first bifurcations and superstable points. At the accumulation parameter the limiting critical orbit is not a finite attracting cycle; its closure is the period-doubling limit set.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2001/iii/paper-53-bifurcation.png]
{title=Attracting quadratic-map orbits and the first period-doubling cascade}
The <Feigenbaum constants> use the positive spatial-scaling convention. Their parameter definition is
$$
\boxed{\delta=\lim_{n\to\infty}\frac{s_n-s_{n-1}}{s_{n+1}-s_n}\simeq4.669201609.}
$$
The same limit is obtained from successive bifurcation parameters. To define spatial scaling without ambiguity about signs, let
$$
d_n=f_{s_n}^{2^{n-1}}(0),\qquad n\geq1.
$$
This is the signed central return separation in the superstable $2^n$ orbit; its sign alternates between successive levels of the primary cascade. Then
$$
\boxed{\alpha=-\lim_{n\to\infty}\frac{d_n}{d_{n+1}}\simeq2.502907875.}
$$
Equivalently, the ratio of the corresponding unsigned separations tends to $\alpha$. Some conventions call the signed ratio itself $\alpha$; here the spatial scale of the <Feigenbaum renormalization fixed point> is $a=-1/\alpha<0$, matching the orientation reversal in the <doubling operator>.
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