Simple Mathematical Models with Very Complicated Dynamics
A one-line population equation, fully deterministic, can settle to a steady level, swing in endless cycles, or erupt into chaos — proving simple rules need not have simple behaviour.
One of the simplest equations you could write down — a single rule for next year's population — turns out to behave so wildly that its future becomes impossible to predict, even though nothing about it is random.
The idea, unpacked
May studied a one-line model of a population: each year's size depends on last year's by a simple rule with a single dial, the growth rate. Turn the dial low and the population settles to a steady number. Turn it up a little and it stops settling — it flips between a high year and a low year, forever: boom, bust, boom, bust. Turn it up more and the cycle doubles to four years, then eight, faster and faster.
Past a certain setting, the doubling never finishes: the population's path becomes chaotic — it never repeats, and two populations starting a hair apart soon diverge completely. The startling part is that the rule is perfectly fixed and contains no chance at all. Complicated, random-looking behaviour can come from a dead-simple deterministic cause.
Where it came from
By the mid-1970s Robert May, an Australian physicist turned ecologist at Princeton, was studying how animal populations rise and fall year to year. Models like his were meant to predict steady levels or gentle cycles. Iterating them, he kept hitting numbers that jumped around with no pattern — and realised the mess was not a bug in the model but a genuine feature of the equation.
Others had glimpsed pieces of this — mathematicians had noticed the strange behaviour of such maps, and Edward Lorenz had found unpredictability in weather equations in 1963 — but in 1976 May wrote a short, forceful review in Nature that pulled it together and rang the alarm. Soon after, Mitchell Feigenbaum showed the cascade of doublings follows a universal numerical pattern, turning a curiosity into a law.
Why it mattered
It overturned a deep, comfortable assumption: that irregular, unpredictable behaviour must have irregular, complicated, or random causes. May showed the opposite is common — the plainest deterministic rule can be unpredictable in principle. That changed how scientists read data: a wildly fluctuating record of fish stocks, insect outbreaks, or epidemics might be deterministic chaos, not noise. And it taught a humbling lesson that runs through weather, ecology and economics alike — being able to write the exact rule does not mean you can forecast the outcome.
An analogy
Imagine a photocopier set to copy a copy of a copy, with one knob for contrast. At a low setting the image settles down and stops changing. Nudge the knob and it starts flickering between two images; nudge again and between four; nudge past a threshold and every copy differs from the last, wandering with no pattern — even though the machine is doing the exact same deterministic thing each time. The widget below is that machine: slide the growth rate and watch the population go from a single steady level, to a tidy cycle, to chaos.
Where it sits
This is the Library's simplest gateway into chaos. Edward Lorenz (1963) found the same sensitivity in the smooth equations of weather; Benoit Mandelbrot (1967) mapped the jagged fractal geometry such systems trace out; the Lotka–Volterra equations (1926) gave the older, tamer picture of predator and prey cycling smoothly. May's logistic map distilled the whole drama into one line of arithmetic, and made chaos something a schoolchild can compute by hand.
First-order difference equations arise in many contexts in the biological, economic and social sciences. Such equations, even though simple and deterministic, can exhibit a surprising array of dynamical behaviour, from stable points, to a bifurcating hierarchy of stable cycles, to apparently random fluctuations.