Deterministic Chaos

Stability of the Logistic Map Fixed Points

The 45 degree blue bowtie condition is just a visual statement of the fact that at the fixed point the absolute value of the derivative is less than 1.

For the logistic map

L(x) = s*x*(1 - x),

the derivative is

L'(x) = s - 2*s*x.

For the fixed point x_f = 0 we have

L'(x_f) = L'(0) = s.

Consequently, this fixed point is stable for 0 <= s < 1.

For the fixed point x_f = (s - 1)/s we have

L'(x_f) = L'((s - 1)/s) = s - 2*s*(s - 1)/s = 2 - s.

Consequently, this fixed point is stable for |2 - s| < 1, hence 1 < s < 3, agreeing with our observations.

Return to Fixed Points of the Logistic Map.