Butterfly effect

Remarks

The constant $\delta$ is the same for all $x$; the point $y$ and the time $n$ depend on $x$ and $\varepsilon$. For a flow $(\varphi_t)$ one asks instead for a time $t \ge 0$ with $d(\varphi_t(x), \varphi_t(y)) > \delta$. Sensitive dependence is one of the conditions in Devaney's definition of chaos.

Example: the doubling map $f(x) = 2x \bmod 1$ on the circle $\R / \Z$, with the distance $d$ to the nearest integer of the difference. Take $\delta = 1/8$. Given $x$ and $\varepsilon > 0$, choose $n$ with $2^{-n} < \varepsilon$ and let $y = x + 2^{-n-2}$. Then $d(x, y) < \varepsilon$, and $f^n(y) - f^n(x) = 1/4 \bmod 1$, so $d(f^n(x), f^n(y)) = 1/4 > \delta$.

Non-example: the rotation $f(x) = x + \alpha \bmod 1$ on $\R / \Z$ is an isometry, so $d(f^n(x), f^n(y)) = d(x, y)$ for all $n$. Points closer than $\delta$ stay closer than $\delta$ forever, so no $\delta > 0$ works.