Period three implies chaos. (English) Zbl 0351.92021

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.
Reviewer: G.K. Francis


92D25 Population dynamics (general)
39A10 Additive difference equations
54H20 Topological dynamics (MSC2010)
37N25 Dynamical systems in biology
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics
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