an:03547131
Zbl 0351.92021
Li, Tien-Yien; Yorke, James A.
Period three implies chaos
EN
Am. Math. Mon. 82, 985-992 (1975).
00140141
1975
j
92D25 39A10 54H20 37N25 37C25
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 \(F^3(q)\leq q < F(q) < F^2(q)\) (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 \(q\neq r\) 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.
G.K. Francis