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5.1.3 Chaos

Like non-linearity, chaos is best understood as a dynamical concept—a feature of how systems changed over time that is represented by certain conditions on the DyST models of those systems. Chaos has played an increasingly central role in a number of sciences since the coinage of the term “butterfly effect” in the mid 20th century as a response to Lorenz (1963)[1]. Indeed, the evocative idea of the butterfly effect—that idea that the flapping of a butterfly’s wings on one side of the world can lead to a hurricane on the other side of the world days later—has percolated so thoroughly into popular culture that the broad strokes of the concept are familiar even to many laypeople. Still, the specifics of the concept are often misunderstood, even by many philosophers of science. In particular, chaotic systems are sometimes thought to be indeterministic, a mistake which has the potential to create a great deal of confusion. Let’s think things through slowly, and add on the formalism as we get a better handle on the concept.

Let’s start here: suppose that it is in fact true that the flapping of a butterfly’s wings in Portugal can spawn a hurricane off the coast of Mexico days later. Here’s a question that should immediately jump out at us: under what conditions does something like this happen? Clearly, it cannot be the case that every butterfly’s flapping has this sort of catastrophic effect, as there are far more butterfly flaps than there are hurricanes. That is, just saying that a tiny change (like a flap) can cause a big change (like a hurricane) doesn’t tell us that it will, or give us any information about what the preconditions are for such a thing to happen. This point is worth

  1. Lorenz (1963) never employs this poetic description of the effect, and the precise origin of the phrase is somewhat murky. In 1972, Lorenz delivered an address to the American Association for the Advancement of Science using the title “Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas?” The resemblance between the Lorenz system’s state space graph (Figure 2) and a butterfly’s wings is likely not coincidental.