Definition of Butterfly Effect

Question

The Wikipedia definition of the Butterfly Effect seems to imply that linear functions can exhibit the Butterfly Effect. In particular if the state space is $\mathbb{R}$ with the usual metric then if $f(x) = 3x$ then for any $x, y \in \mathbb{R}$ we have $$ \lvert f^\tau(x) - f^\tau(y) \rvert = 3^\tau\lvert x - y\rvert > \mathrm{e}^\tau\lvert x - y\rvert. $$ But the same wikipedia article says that the simplest example of a function exhibiting this effect is a logistic map. Am I missing something?

Answer 1

Yes, the function $f(x)=3x$ on the reals does display "sensitive dependence on initial conditions" (a more precise phrase than "Butterfly effect"), but in an uninteresting way. It's interesting to find sensitive dependence on a bounded domain.

Incidentally, I'd say there are (interesting) examples simpler than the logistic map ($f(x)=4x(1-x)$). For example, the tent map on $[0,1]$ given by $f(x)=2x$ for $0\le x\le1/2$, $f(x)=2-2x$ for $1/2<x\le1$.