Q: What are chaos and chaos theory? How can you talk about chaos?

Physicist: Chaos theory, despite what Jurassic Park may lead you to believe, has almost nothing to do with making actual predictions, and is instead concerned with trying to figure out how much we can expect our predictions to suck.

“Pure chaos” is the sort of thing you might want to argue about in some kind of philosophy class, but there are no examples of it in reality.  For example, even if you restrict your idea of chaos to some method of picking a number at random, you find that you can’t get a completely unpredictable number.  There will always be some numbers that are more likely or less likely.  Point is; “pure chaos” and “completely random” aren’t real things.

“Chaos” means something very particular to its isolated and pimply practitioners.  Things like dice rolls or coin flips may be random, but they’re not chaotic.

Dr. Ian Malcolm, shown here distracting a T-rex with a road flare, is one of the few completely accurate depictions of a chaos theorist in modern media.

A chaotic system is one that has well understood dynamics, but that also has strong dependence on its initial conditions.  For example, if you roll a stone down a rocky hill, there’s no mystery behind why it rolls or how it bounces off of things in its path (well established dynamics).  Given enough time, coffee, information, and slide rulers, any physicist would be able to calculate how each impact will affect the course of the tumbling stone.

Although the exact physics may be known, tiny errors compound and trajectories that start similarly end differently.  Chaos!

But there’s the problem: no one can have access to all the information, there’s a diminishing return for putting more time into calculations, and slide rulers haven’t really been in production since the late 70’s.

So, when the stone hits another object on it’s ill-fated roll down the hill, there’s always some error with respect to how/where it hits.  That small error means that the next time it hits the ground there’s even more error, then even more, …  This effect; initially small errors turning into huge errors eventually, is called “the butterfly effect“.  Pretty soon there’s nothing that can meaningfully be said about the stone.  A world stone-chucking expert would have no more to say than “it’ll end up at the bottom of the hill”.

Chaotic systems have known dynamics (we understand the physics), but have a strong dependence on initial conditions.  So, rolling a stone down a hill is chaotic because changing the initial position of the stone can have dramatic consequences for where it lands, and how it gets there.  If you roll two stones side by side they could end up in very different places.

Putting things in orbit around the Earth is not chaotic, because if you were to put things in orbit right next to each other, they’d stay in orbit right next to each other.  Slight differences in initial conditions result in slight differences later on.

The position of the planets, being non-chaotic, can be predicted accurately millennia in advance, and getting more accurate information makes our predictions substantially more accurate.  But, because weather is chaotic, a couple of days is the best we can reasonably do (ever).  Doubling the quality of our information or simulations doesn’t come close to doubling how far into the future we can predict.  Few hours maybe?

Answer gravy: When you’re modeling a system in time you talk about its “trajectory”.  That trajectory can be through a real space, like in the case a stone rolling down a hill or a leaf on a river, or the space can be abstract, like “the space of stock-market prices” or “the space of every possible weather pattern”.  With the rock rolling down the hill you just need to keep track of things like: how fast and in which direction it’s moving, its position, how it’s rotating, that sort of thing.  So, at least 6 variables (3 for position, and 3 for velocity).  As the rock falls down the hill it spits out different values for all 6 variables and traces out a trajectory (if you restrict your attention to just it’s position, it’s easy to actually draw a picture like above).  For something like weather you’d need to keep track of a hell of a lot more.  A good weather simulator can keep track of pressure, temperature, humidity, wind speed and direction, for a dozen layers of atmosphere over every 10 mile square patch of ground on the planet.  So, at least 100,000,000 variables.  You can think of changing weather patterns around the world as slowly tracing out a path through the 100 million dimensional “weather space”.

Chaos theory attempts to describe how quickly trajectories diverge using “Lyapunov exponents“.  Exponents are used because, in general, trajectories diverge exponentially fast (so there’s that).  In a very hand-wavy way, if things are a distance D apart, then in a time-step they’ll be Dh apart.  In another time-step they’ll be (Dh)h = Dh² apart.  Then Dh³, then Dh⁴, and so on.  Exponential!

Because mathematicians love the number e so damn much that they want to marry it and have dozens of smaller e’s (_eeeeeeee), they write the distance between trajectories as , where D(t) is the separation between (very nearby) trajectories at time t, and D₀ is the initial separation.  is the Lyapunov exponent.  Generally, at about the same time that trajectories are no longer diverging exponentially (which is when Lyapunov exponents become useless) the predicting power of the model goes to crap, and it doesn’t matter anyway.

Notice that if is negative, then the separation between trajectories will actually decrease.  This is another pretty good definition of chaos: a positive Lyapunov exponent.

λ < 0. Pick two points that are close together, then run time forward. They get closer.

A beautiful, and more importantly fairly simple, example of chaos is the “Logistic map”.  You start with the function , pick any initial point between 0 and 1, then feed that into f(x).  Then take what you get out, and feed it back in.  That is; x₁ = f(x₀), x₂ = f(x₁), …  This is written “recursively” as ““.  The reason this is a good model to toy around with is that you can change r and get wildly different behaviors.

For 0<r<1, x_n converges to 0, regardless of the initial value, x₀.  So, nearby initial conditions just get closer together, and .

For 1<r<3, x_n converges to one point (specifically,), regardless of x₀.  So, again, .

For 3<r<3.57, x_n oscillates between several values.  But still, regardless of the initial condition, the values of x_n still settle into the same set or values (same “trajectory”).

But, for 3.57<r, x_n just bounces around and never converges to any particular value.  When you pick two initial values close to each other, they stay close for a while, but soon end up bouncing between completely unrelated values (their trajectories diverge, exponentially).  The is the “chaotic regime”.  .

(bottom) Many iterations of the logistic map for r = 3.30 (2 value region) and r = 3.9 (chaotic region).  (middle) The values that Xn converge to, regardless of Xo, for various values of r.  The blue lines correspond to the examples.  (top) Lyapunov exponent of the Logistic map for various values of r. Notice that the value is always zero or negative until the system becomes chaotic.

Long story short, chaos theory is the study of how long you should trust what your computer simulations say before you should start ignoring them.