Complex natural systems defy standard mathematical analysis, so one ecologist is throwing out the equations.

The roots of empirical dynamic modeling go back more than 30 years. In the late 1970s, the Dutch mathematician Floris Takens was studying chaos theory, which had begun to emerge in the 1960s as scientists recognized that many of nature’s complex phenomena seem to defy prediction. In chaotic systems, small perturbations can have large and seemingly unpredictable effects, as in the archetypical example of a butterfly’s flapping wings influencing the weather thousands of miles away.

Takens helped find order in the chaos. Along with the physicist David Ruelle, he developed the notion of a “strange attractor” — a set of points in a coordinate system made of the variables that influence a system, around which the system’s state, plotted over time, swirls like a ball of yarn.

In many natural systems, however, the number of relevant variables that make up the coordinate system is immense. The factors that determine the weather in a certain place at a certain time are almost limitless, and some of these can be very hard to measure — the air pressure three miles above the North Pole, for example.

But let’s say you could consistently and accurately measure one variable, such as the temperature in New York City. Takens found a way to use present and past measurements of that one variable to capture all the information in the system. The method involves creating an alternate coordinate system from those past measurements; in other words, one coordinate axis might be the temperature in Times Square today, a second axis might be the temperature yesterday, a third the temperature two days ago, and so on. Takens showed that the full state of a chaotic system can, in theory at least, be embedded in a time series of a single variable. He published his “embedding theorem” in 1981.

The theorem “caused a big hullabaloo,” said

Timothy Sauer, a mathematician at George Mason University in Fairfax, Va., who has extended the original theorem so it can be applied more generally.

Read more here:

https://www.quantamagazine.org/20151013-chaos-theory-and-ecology/