= Solution
For the real <logistic map>, $f(x)=x$ factors as
$$
x\bigl(\mu-1-\mu x\bigr)=0.
$$
There is always the <fixed point> $x_0=0$. If $\mu\ne0$ there is also $x_*=1-1/\mu$, distinct from zero unless $\mu=1$. At $\mu=0$ the map is constant and zero is its only <fixed point>. Negative $\mu$ must be retained because the paper specifies $x\in\mathbb R$.
Since $f'(x)=\mu(1-2x)$, the <fixed-point multipliers> are
$$
\lambda_0=\mu,\qquad \lambda_*=2-\mu.
$$
The <fixed point stability for an iteration> criterion gives \b[$x_0$ attracting for $-1<\mu<1$, and $x_*$ attracting for $1<\mu<3$]. The respective <fixed points> are repelling when $|\mu|>1$ and $|2-\mu|>1$. At the unit-modulus boundaries the linear criterion needs a nonlinear supplement.
At $\mu=1$, the two <fixed point> branches meet and exchange stability: this is a <transcritical bifurcation>. Explicitly, with $\delta=\mu-1$,
$$
f(x)-x=\delta x-(1+\delta)x^2,
$$
which has the required nonzero parameter and quadratic terms. At the critical value itself, $f(x)=x-x^2$ attracts small positive points and pushes negative points away, so the merged <fixed point> is not stable from both sides.
At $\mu=-1$ the origin’s <fixed-point multiplier> crosses $-1$, while at $\mu=3$ the nonzero <fixed-point multiplier> crosses $-1$. Both are <period-doubling bifurcations>. To check existence and stability of the new <periodic orbit>, factor the second <iteration of a map>:
$$
f^2(x)-x=-x(\mu x-\mu+1)
\left(\mu^2x^2-\mu(\mu+1)x+\mu+1\right).
$$
The two non-fixed roots are
$$
q_\pm=\frac{\mu+1\pm\sqrt{(\mu-3)(\mu+1)}}{2\mu}.
$$
They form a real <period-two orbit> for $\mu<-1$ or $\mu>3$. Its <periodic-orbit multiplier> is
$$
f'(q_-)f'(q_+)=4+2\mu-\mu^2.
$$
It is less than one in <absolute value> just below $-1$ and just above $3$, respectively. Thus these are <supercritical bifurcations> in the direction where the parent <fixed point> loses stability. At the critical values the <fixed points> themselves are weakly attracting: at $\mu=-1$, $f^2(x)=x-2x^3+x^4$; at $\mu=3$, setting $u=x-2/3$ gives $u\mapsto u-18u^3-27u^4$ after two steps. Both shrink sufficiently small nonzero <absolute values>. The <bifurcation> values for finite <fixed points> are therefore
$$
\boxed{\mu=-1\text{ (period doubling)},\quad1\text{ (transcritical)},\quad3\text{ (period doubling)}.}
$$
The disappearance of the formula for $x_*$ at $\mu=0$ corresponds to that branch going to infinity, not a local <bifurcation> at a finite <fixed point>.
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