Deterministic Chaos

Uniform stability of fⁿ(x) fixed points

for an n-cycle of f(x)

The idea of the proof is captured by considering a 2-cycle {x₁, x₂} of f(x).

First, note f(x₁) = x₂ and f(x₂) = x₁, so x₁ and x₂ are fixed points of f²(x):

f²(x₁) = f(f(x₁)) = f(x₂) = x₁

and

f²(x₂) = f(f(x₂)) = f(x₁) = x₂

Now to test the stability of x₁ as a fixed point of f²(x), we compute the derivative (f²) '(x₁)).

By the chain rule

(f²) '(x₁) = f '(f(x₁)) f '(x₁) = f '(x₂) f '(x₁)

Similarly,

(f²) '(x₂) = f '(f(x₂)) f '(x₂) = f '(x₁) f '(x₂)

That is, (f²) '(x₁) = (f²) '(x₂), so x₁ is a stable fixed point of f²(x) if and only if x₂ is.

Return to stability of cycles.