It's an irrational, nonlinear world out there

Chaos is part of a larger phenomenon in the natural world, nonlinearity, a disproportion between cause and effect. There are two types:

• Cause leads to disproportionately small effect. This is saturation, something familiar from overloaded audio circuits (where adding more power to your speakers doesn't lead to much louder sound, for example) and in economics, as diminishing returns. This type of nonlinearity doesn't lead to chaos and actually constitutes a kind of "anti-chaos."
  
  
• Cause leads to disproportionately large effect. This is the property needed (although by itself not enough) for chaos. The nonlinearity (the disproportion) has to not only be present, but large enough.

The nonlinearity can arise from internal or external sources. To exhibit chaotic behavior, a closed system must have at least two degrees of freedom (independent ways of changing). For system open to external influences - able to both receive and output flows of energy and other "stuff" - only one degree of freedom is needed. In both cases, it's the coupling between two subsystems - either within the system, or between the inside and the outside of the system - that's in play. That coupling has to be strong enough to change the Fourier spectrum of the system's evolution, the way its motion is broken down into independent periods.

As the nonlinearity, this coupling between parts, increases from zero, the different parts exchange energy with one another. Whatever periodic or multiperiodic behavior the subsystems had without the nonlinearity present, the nonlinearity "pulses" the exchange of energy or other "stuff" at periods longer than the original ones; periods that are actually multiples of the originals. In Fourier language, the different degrees of freedom resonate at frequencies lower than the originals, and in fact, at rational fractions of the original frequencies - like 1/2, 1/3, or 2/3. These below-fundamental frequencies are called subharmonics. They're very different from the harmonic overtones we met earlier. Subharmonics occur only when nonlinearities are present.

If the nonlinearity is made stronger, the number and strength of these subharmonics grow, so much that in fact, the original fundamental frequencies are better viewed as themselves being harmonics of the subharmonics. The subharmonics become more and more finely spaced and extend down closer and closer to zero frequency.

The remarkable thing about nonlinearity is that at some finite nonlinearity, the discrete spectrum of subharmonics breaks down completely into a continuum of frequencies extending all the way down to zero. That transition is the signal of chaos.

It's the existence of the continuum of irrational numbers that ultimately makes this possible. If all frequencies were rational numbers, there would be a countable infinity of them, but never extending to zero. The subharmonics that pile up near zero frequency, mix and subdivide; because they're irrational numbers and can be arbitrarily close to one another, the resulting full spectrum of subharmonics faces nothing keeping it from going all the way to zero. The resulting component of motion at zero frequency is (apart from any constant-in-time contribution) the chaotic piece.