Take an organism that ordinarily migrates south, put it in an artificial environment, and reset its circadian clock by $x$ hours (by gradually shifting the hours of daylight). Release it into the wild, and instead of migrating south, it will migrate in a new direction $y$. Thinking of $x$ and $y$ as elements of $S^1$ and $y$ as a function of $x$, this gives a map $S^1\rightarrow S^1$. Apparently the winding number of this map is characteristic of the species (e.g. $0$ for the pond skater, $1$ for the sunfish). (Source: Arthur Winfree's Geometry of Biological Time.)