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.