What is chaos?

Through all the tones in Earth’s multi-colored dream,
Sounds one faint note drawn out for the one who listens in secret.

- Friedrich Schlegel

Chaos is bounded aperiodicity.

Not everything in the world is periodic, as we know from experience. There are events, not cycles, that never repeat. There are aspects of things that do repeat, but shot through with a stream of the unique, the singular, the non-repetitive. We might even speak of a complementarity between an event and a cycle, the former unique in the domain of time and the other unique in the domain of frequency.*

If all that a powerful and ubiquitous scientific method like Fourier analysis amounted to was to tell us that everything is, in fact, multiperiodic (repeating with many different frequencies), we might demur: after all, the lower frequencies are longer than a human lifespan or human history. Maybe it all does repeat and we just need to wait for a long time.

But no: there are things that "repeat" once in an infinity of time - that is, they happen only once. They are fallout from what mathematicians and physicists call chaos. One way to define chaos is: bounded aperiodic motion. The first modern scientific work on the subject was meteorologist Edward Lorenz's 1962 classic paper "Deterministic Nonperiodic Flow." Let's unpack that title and the definition: the whole chaos business is there by implication.

• Flow: It's a flow, or a motion, or a change. Lorenz used "flow" to refer to many variables, many functions, changing in time all simultaneously. We could plot these functions in a space of many variables, all functions of time, and have a flow in that space.
  
  
• Deterministic: Really understanding this is a point we'll come back to. But it's enough to say for now that this means that the chaos is causal - it's not random or acausal.
  
  
• Nonperiodic (or aperiodic): A component of the motion that does not repeat, has no frequency. (It's not frequent.) In Fourier analysis, its share of the motion shows up in the zero-frequency bin. If Fourier analysis were always valid, that component would represent a constant, a steady state independent of time.
  
  But this is exactly the situation where Fourier analysis breaks down and its preconditions are violated. Fourier and related techniques are not the right framework for analyzing chaos, a fact known for more than 40 years. Yet techniques valid in other areas of science and engineering, known to break down when applied to chaotic motion, are often lazily applied to chaos anyway, producing meaningless or seriously compromised results. This is a major problem in mathematical modeling of chaotic systems, such as the weather and financial markets. The zero-frequency bin of a Fourier spectrum, in practice, contains the share of the motion that is constant - but it contains the unique events of chaos as well.
  
  
• Bounded: Finally, chaotic motion is bounded. I can always construct nonperiodic motion if I allow, say, two functions of time to diverge from one another to an arbitrary degree. As time goes to infinity, the motions diverge to an infinite degree.
  
  Chaos is aperiodic motion that never repeats, but also remains bounded. Even after an infinite time, the motion never leaves a "box." (We'll get more precise about this "box" later on.) This fact violates a basic intuition embedded in mathematical and scientific thought since ancient times. That wrong intuition says: motion bounded in space should ultimately repeat exactly in time. It's not true, but it is often a major stumbling block in understanding chaos. It means, among other things, that after an infinity of time has passed, the motion in the box contains an infinitely detailed nonrepetitive structure.
  

BTW, ideas about chaos are sometimes incorrectly referred to as "chaos theory." Chaos is just one part of the mathematical study of dynamical systems. It's not a separate theory in its own right.
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* Being related to one another by the Fourier transform, the time domain and the frequency domain are indeed complementary to one another in exactly the same way anyone who knows something about quantum mechanics should recognize. There it's the complementarity of time and energy.