The idea began with a practical question in ecology: how does a population change from one season to the next when it grows quickly but the habitat can only feed so many? Multiply this year’s numbers by a growth rate, then hold that growth back in proportion to how full the habitat already is. The result is about as simple as arithmetic gets, and for gentle growth rates it behaves exactly as common sense expects — the population climbs and settles at a comfortable level.
Push the growth rate higher and something strange happens. The steady level splits in two, so the population alternates between a good year and a lean one. Push further and each of those splits again, then again, faster and faster, until the sequence stops repeating altogether and wanders unpredictably forever. Nothing random was ever added — the same starting number always gives the same run — yet two starting values a millionth apart end up completely unrelated within a few dozen steps. That sensitivity is what people mean by chaos.
The biologist Robert May brought this to wide attention in a 1976 paper arguing that very simple rules can produce very complicated behaviour, and that ecologists should not assume messy data means a messy underlying system. Mitchell Feigenbaum then discovered that the splittings crowd together at a fixed ratio of about 4.669, and that the very same number turns up in wholly unrelated systems — dripping taps, oscillating circuits, heated fluids. Hidden inside the chaotic region are narrow windows where order abruptly returns, the widest of them a rhythm of three.