Period-doubling bifurcation

Remarks

Under these conditions, for $\mu$ near $\mu_0$ on one side of $\mu_0$ there is exactly one orbit of period $2$ near $x_0$. On the other side there is none, and the period-$2$ orbit tends to $x_0$ as $\mu \to \mu_0$. The sign of the quantity in clause 4 decides whether the bifurcation is supercritical, with an attracting period-$2$ orbit, or subcritical, with a repelling one.

Example: the logistic family $f_r(x) = r x (1 - x)$ at $r_0 = 3$ and $x_0 = 2/3$. Clause 1: $3 \cdot \frac{2}{3} \cdot \frac{1}{3} = \frac{2}{3}$. Clause 2: $f_x = r(1 - 2x) = 3 \cdot (-\frac{1}{3}) = -1$. Clause 3: on the branch $x(r) = 1 - 1/r$ one has $f_x = 2 - r$, with derivative $-1$ in $r$. Clause 4: $f_{xx} = -2r = -6$ and $f_{xxx} = 0$, so the quantity is $18 \neq 0$. For $r > 3$ the period-$2$ orbit is $\frac{r + 1 \pm \sqrt{(r - 3)(r + 1)}}{2r}$.

Non-example: the family $f_\mu(x) = \mu - x$ satisfies clauses 1 and 2 at $(x_0, \mu_0) = (0, 0)$, but $f_x = -1$ for every $\mu$, so clause 3 fails. Here every point other than the fixed point $\mu/2$ has period $2$ for every $\mu$, and no isolated orbit is born.

A fixed point with derivative $+1$ instead of $-1$ gives a saddle-node bifurcation or one of its degenerate relatives.