The period-doubling route to chaos is universal: any unimodal map with a quadratic maximum undergoes the same cascade, with the same scaling constants.
δ = lim (rₙ − rₙ₋₁)/(rₙ₊₁ − rₙ) ≈ 4.66920
The horizontal distances between successive period-doublings shrink by factor δ. The vertical (orbit) scale shrinks by α ≈ 2.5029.
Feigenbaum (1978) proved these constants are universal — independent of the map. The proof uses renormalization group: the cascade satisfies a functional equation g(x) = −αg(g(x/α)) whose fixed point determines δ and α.
At r∞ (onset of chaos) the attractor is a Cantor set of dimension d ≈ 0.538.