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Zbl 0351.92021
Li, Tien-Yien; Yorke, James A.
Period three implies chaos.
(English)
[J] Am. Math. Mon. 82, 985-992 (1975). ISSN 0002-9890

Let \$F\$ be a continuous function of an interval \$J\$ into itself. The period of a point in \$J\$ is the least integer \$k>1\$ for which \$F^k(p) = p\$. If \$p\$ has period 3 then the relation \$F^3(q)\le q < F(q) < F^2(q)\$ (or its reverse) is satisfied for \$q\$ one of the points \$p\$, \$F(p)\$, or \$F^2(p)\$. The title of the paper derives from the theorem that if some point \$q\$ in \$J\$ has this Sysiphusian feature, ``two steps forward, one giant step back", then \$F\$ has periodic points of every period \$K=1,2,3,\dots\$. Moreover, \$J\$ contains an uncountable subset \$S\$ devoid of asymptotically periodic points, such that \$\$ 0=\liminf|F^n(q)-F^n(r)| < \limsup|F^n(q)-F^n(r)|\$\$ for all \$q\ne r\$ in \$S\$. (a point is asymptotically periodic if \$\lim|F^n(p) - F^n(q)| = 0\$ for some periodic point \$p\$.) The proof is eminently accessible to the nonspecialist and is therefore of interest to anyone modeling the evolution of a single population parameter by a first order difference equation. The authors compare the logistic \$x_{n+1} = F(x_n) = rx_n(1-x_n/K)\$ with a model of which, by contrast, \$|dF(x)/dx|>1\$ wherever the derivative exists. For such a system no periodic point is stable, in the sense that \$|F^k(q)-p| < |q-p|\$ for all \$q\$ in a neigborhood of a periodic point \$p\$ of \$k\$. A brief survey of a theorem motivated by ergodic theory completes this fascinating paper.

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[G.K. Francis]
MSC 2000:
*92D25 Population dynamics
39A10 Difference equations
54H20 Topological dynamics
37N25 Dynamical systems in biology
37C25 Fixed points, periodic points, fixed-point index theory

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