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
f3(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 fn(p)-fn(q) < e
From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2024.
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