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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.

Display scanned Zentralblatt-MATH page with this review.
[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

Cited in: Zbl 1208.37002 Zbl 1202.93049 Zbl 1196.92019 Zbl 1153.37017 Zbl 1175.37039 Zbl 1153.39004 Zbl 1136.54025 Zbl 1096.39019 Zbl 1076.37024 Zbl 1072.92038 Zbl 1114.37002 Zbl 1049.37026 Zbl 1121.37304 Zbl 1059.65117 Zbl 1057.39021 Zbl 1054.92002 Zbl 1035.37028 Zbl 0991.37010 Zbl 0978.37003 Zbl 0986.37022 Zbl 0930.37010 Zbl 0874.11017 Zbl 0873.54023 Zbl 0813.54027 Zbl 0805.54038 Zbl 0768.54031 Zbl 0741.39002 Zbl 0716.90008 Zbl 0722.58030 Zbl 0628.58027 Zbl 0654.26004 Zbl 0639.54029 Zbl 0637.58009 Zbl 0558.54031 Zbl 0543.90017 Zbl 0541.26002 Zbl 0532.92014 Zbl 0525.93046 Zbl 0521.58040 Zbl 0519.39005 Zbl 0523.93038 Zbl 0511.28013 Zbl 0502.58023 Zbl 0501.28010 Zbl 0519.54028 Zbl 0511.58025 Zbl 0424.58019 Zbl 0542.54038 Zbl 0473.92018 Zbl 0434.39003 Zbl 0455.58022 Zbl 0383.58010

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