Defining chaos

July 29, 2011 in Math

I've been doing a lot of reading on chaos, in particular on the nature of chaotic systems. I was recently trying to explain to a friend why a dynamic system, which can be perfectly captured by a "deterministic" equation, can nonetheless exhibit chaotic behavior. His refusal at first to accept that fact reminded me of my initial skepticism upon being told that statisticians study and characterize randomness -- it just doesn't seem to make sense. It's easy to conflate "randomness" or "chaos" with "unpredictable" (or with each other!) when that's not necessarily the case.

What inspired me to write this, however, was a succinct definition that I came across in a paper which described a form of chaos simply as the following property: In a chaotic system, two different sets of initial conditions which are separated by an arbitrarily small distance will grow exponentially farther apart as the system evolves.

This will be recognizable to some readers as a characterization of the Lyapunov exponent, and to others as an overly mathematized version of the butterfly effect. In any case, despite my familiarity with the subject I found this relatively simple definition to be quite illuminating in its clarity. There's something to be said for the literature majors who have thankfully applied their talents to mathematics.

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