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James Yorke: fifty years of chaos (theory)

Almost fifty years ago, when my student T. Y. Li and I wrote a math paper titled “Period 3 Implies Chaos”, I could not predict the effect that title would have. Chaos. It would go on to have a life of its own, far beyond the mathematical proof contained in our short paper.

Since then, the word has risen and fallen in popularity, while others, like complexity, have emerged. But the principles are the same: there are limits to what we can accurately predict. Many systems are sensitive to initial conditions. And above all: a fully deterministic system can still be unpredictable.

This reflection could have easily been titled “50,000 Years of Chaos.” Humans have always known that slight differences can have dramatic consequences: a boulder lands a meter away from a man’s sleeping head—a meter that contains an entire world. Mathematicians merely put numbers to a principle that needed no introduction.

When novelists use Heisenberg’s uncertainty principle as an explanation for divorce, they are speaking by analogy. Chaos, on the other hand, is both a formal mathematical discipline, and a fact about the contingency of our daily lives—we buy health and life insurance in order to manage it. A world that is shorn of chaos doesn’t look anything like ours.

But chaos is also a phenomenon that applies to precise numerical quantities, whether those quantities are water molecules or animal populations. The reason why it took so long to mathematically formalize, is because people wanted to find clean linear solutions to differential equations. But the vast majority of such equations are not easily solvable, being nonlinear and chaotic. As Stanislaw Ulam famously quipped, “to call the study of chaos ‘nonlinear science’ was like calling zoology ‘the study of non elephant animals’.”

James Yorke – 50 Years of Chaos in The Latecomer


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