Butterfly effect

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The butterfly effect is a much misunderstood observation of complexity theory - that the behaviour of a complex system is highly susceptible to its initial configuration, and small differences in that initial state — in an ecosystem, the flapping of a butterfly’s wing — may mean the system behaves in vastly — and quite unpredictably — different ways.

This is not the same as saying, as people are prone to, that “a butterfly flapping its wings in the Amazon causes a hurricane in China”.

People who not only should, but do know better, can fall into this trap. “To show what a difference an initial condition can make, consider the double-jointed pendulum”.

Set off two double-jointed pendulums from an apparently identical condition and quickly their trajectories will wildly diverge, it is true. But this to vergence is not accountable solely for atomic differences in the initial configuration, but for ongoing atomic differences throughout the pendulums cycle. The systems are path-dependent, not initial-condition-dependent. The longer the the system continues the more dependent the system will be on on the infinity if subsequently intervening causes. Unless the pendulum is somehow powered, the laws of thermodynamics predict that it will in a short time period come to rest. Over time, then, even insoluble mathematical operations converge. We can see this path dependency to be noise. For an unpowered jointed pendulum, the signal is is, over time, clear. However you start it, it will end up in entropic rest.

For a complexity theorist, the butterfly’s wing metaphor makes the point not that hurricanes can be reduced to their infinitesimal operating causes, but that they cannot. These systems are so complex — so ontologically indeterminate — that it is theoretically impossible to predict how they will behave.

Butterfly wing-flaps are discrete independent events. Unless you are prepared to hypothesise some kind of spooky quantum butterfly entanglement, one butterfly flapping its wings will not make more or less likely another butterfly’s decision to do the same, let alone any of the other environmental factors that might cause a tropical storm.

Furthermore, there are millions of butterflies in the Amazon, all discretely — even if not discreetly — wing-flapping, and Gaussian nature of these events will largely cancel each other out, putting butterfly wing-flapping in “Mediocristan” and not “Extremistan”.

If individual-butterfly-wing-flapping-in-Brazil status is a material part of the complex system that generates weather systems in China, then how many other factors are as or more material?

Trillions. Have fun deconstructing the causal chain to see how much of a role your butterfly actually had. Just ten generations dilutes your ancestor’s genetic contribution to your DNA to 1/1024, and in the same as way there is no meaningful sense in which a flighty Brazilian butterfly causes Chinese hurricanes.

See also

• Complexity