# Period 3 Implies Chaos

Robert P. Munafo, 2023 Jun 20.

The title of a famous paper in chaos theory establishing the ubiquitous nature of so-called "chaotic oscillation", in which the iterates of a recurrence relation often follow patterns that have no period, repeating pattern or other traditionally "well-defined" behaviour.

Li, Tien-Yien, and James A. Yorke.

Period three implies chaos.

The theory of chaotic attractors. Springer, New York, NY, 2004. pp. 77-84.

If f(x) is a (scalar) real-valued continuous function of a real argument, the authors show that if there is any value of x for which

f^{3}(x) = f(f(f(x))) = x

(a "period-3 point"), then there exist points of all positive integer periods (1, 2, 3, ...), and furthermore that there are uncountably many values of x for which the iterates of x have no period. They also show that such x values come arbitrarily close to the same-indexed iterates of other non-periodic values x':

given any two parameters p and q that have non-periodic iterates

and given any e>0

there is some n for which f^{n}(p)-f^{n}(q) < e

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2024.

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