Bifurcation Theory

A bifurcation occurs when a small change in a parameter causes a qualitative change in a dynamical system's behavior — new fixed points appear, disappear, or change stability.

Saddle-Node Bifurcation

The normal form is: $\frac{dx}{dt} = \mu + x^2$

Fixed points satisfy $\mu + x^{*2} = 0 \Rightarrow x^* = \pm\sqrt{-\mu}$:

• $\mu < 0$: two fixed points (one stable, one unstable)
• $\mu = 0$: one fixed point (bifurcation point)
• $\mu > 0$: no real fixed points — they annihilate

Transcritical Bifurcation

$\frac{dx}{dt} = \mu x - x^2$

Fixed points: $x^* = 0$ and $x^* = \mu$ (always two). They exchange stability at $\mu = 0$.

Pitchfork Bifurcation (Supercritical)

$\frac{dx}{dt} = \mu x - x^3$

Fixed points:

• $\mu \leq 0$: only $x^* = 0$ (stable)
• $\mu > 0$: $x^* = 0$ (unstable) and $x^* = \pm\sqrt{\mu}$ (stable)

The stable state splits into two — like a pitchfork — as $\mu$ crosses zero.

Period-Doubling in the Logistic Map

$x_{n+1} = r x_n (1 - x_n)$

As $r$ increases, the fixed point loses stability and period-2 cycles appear at $r = 3$ (first period-doubling bifurcation). This cascade continues to chaos at $r \approx 3.569$.

Implementation

import math

def saddle_node_fps(mu):
    # Return list of real fixed points: [sqrt(-mu), -sqrt(-mu)] if mu < 0
    # Return [0.0] if mu == 0, [] if mu > 0
    ...

def transcritical_fps(mu):
    # Always return [0.0, float(mu)]
    ...

def pitchfork_fps(mu):
    # Return [0.0] if mu <= 0
    # Return [0.0, sqrt(mu), -sqrt(mu)] if mu > 0
    ...

def logistic_bifurcation_r():
    # Return the first period-doubling bifurcation value
    ...