We prove that when a multiplier of a fixed point passes through $-1$, a period-2 orbit bifurcates. This is the fundamental mechanism of the period-doubling route to chaos.

Setup: Consider a one-parameter family of maps $f_\mu: \mathbb{R} \to \mathbb{R}$ with a fixed point $x^*(\mu)$ satisfying $f_\mu(x^*(\mu)) = x^*(\mu)$. Assume the multiplier $\lambda(\mu) = f'_\mu(x^*(\mu))$ passes through $-1$ transversely: $\lambda(0) = -1$ and $\lambda'(0) \neq 0$.

Step 1: Period-2 orbits as fixed points of $f^2$

Period-2 points satisfy $f_\mu^2(x) = x$ but $f_\mu(x) \neq x$. Near the bifurcation, we analyze $g_\mu(x) = f_\mu^2(x)$ to find its fixed points. At $\mu = 0$, $x^* = x^*(0)$ is a fixed point of $g_0$ with multiplier:

$g'_0(x^*) = f'_0(f_0(x^*)) \cdot f'_0(x^*) = [f'_0(x^*)]^2 = (-1)^2 = 1$

This is a non-hyperbolic situation requiring higher-order analysis.

Step 2: Taylor expansion of $g_\mu$ near $(x^*, 0)$

Expand $g_\mu(x)$ around $x = x^*$, $\mu = 0$:

$g_\mu(x) = x^* + g'_0(x^*)(x - x^*) + \frac{1}{2}g''_0(x^*)(x - x^*)^2 + a\mu + b\mu(x - x^*) + O((x-x^*)^3, \mu^2)$

where $g'_0(x^*) = 1$. Computing $g''_0(x^*)$ involves derivatives of the composition $f^2$. For generic maps, $g''_0(x^*) \neq 0$ (non-degeneracy).

Step 3: Change of variables

Let $y = x - x^*$ and study $h_\mu(y) = g_\mu(x^* + y) - x^*$. Fixed points of $g_\mu$ satisfy $h_\mu(y) = y$, or equivalently:

$0 = \frac{1}{2}g''_0(x^*) y^2 + a\mu + b\mu y + O(y^3, \mu^2)$

For small $\mu, y$, this simplifies to:

$y = -\frac{a\mu}{by} + O(y^2, \mu^2)$

Rearranging: $y^2 \approx -\frac{a\mu}{c}$ where $c = g''_0(x^*)/2 + b/2$ (combining coefficients).

Step 4: Bifurcation condition

For period-2 orbits to exist, we need $y^2 > 0$, requiring $-a\mu/c > 0$. Assuming $a, c \neq 0$ (genericity), period-2 points emerge when $\mu$ and $-a/c$ have the same sign. Specifically:

$y = \pm\sqrt{-\frac{a\mu}{c}}$

For $\mu > 0$ (if $-a/c > 0$), two new fixed points of $g_\mu$ appear, corresponding to a period-2 orbit $\{p, q\}$ with $f_\mu(p) = q$ and $f_\mu(q) = p$.

Step 5: Stability of the period-2 orbit

The multiplier of the period-2 orbit is $\lambda_2 = g'_\mu(y)$. Computing:

$\lambda_2 = 1 + g''_0(x^*) y + O(y^2, \mu) = 1 + g''_0(x^*) \sqrt{-\frac{a\mu}{c}} + O(\mu)$

For small $\mu > 0$, this is approximately $1 + \text{const} \cdot \sqrt{\mu}$. The sign determines stability. For supercritical bifurcations (common case), the period-2 orbit is stable immediately after bifurcation when the fixed point $x^*$ becomes unstable.

Conclusion: At $\mu = 0$ where the multiplier of $x^*$ crosses $-1$, a pair of period-2 points bifurcates. For $\mu > 0$ (or $< 0$ depending on parameters), a stable period-2 orbit exists, while the original fixed point becomes unstable. This is the period-doubling (flip) bifurcation.