An example of an obviously chaotic system.

I have written before about chaos but was thinking of it again today and came up with a more obviously chaotic system than the bouncing balls on an oscillating plate mentioned last time. It’s a system you can simulate yourself.

Pick a real number, any real number you want. We’ll call that number the “‍state‍” of the system. Then repeatedly do the following:

1. Throw away the integer part of the state (e.g., 3.1419… becomes 0.1419…).

2. Multiply the state by ten (e.g., 0.1419… becomes 1.419…).

That’s it. Deterministic chaos can be very simple.

Let’s work an example. Suppose you start with state 2.45618435047…. Knowing this, I know your next state will be 4.5618435047…, then 5.618435047…, then 6.18435047…, 1.8435047…, 8.435047…, 4.35047…, 3.5047…, 5.047…, 0.47…, 4.7…, etc. Still, no matter how accurately I measure your initial state it doesn’t take very many steps before I have no idea what your state will be.

Now suppose we add a third step,

3. Add to your state inputs from the outside world.

Now we have access to the so-called “‍butterfly effect‍”.

Let’s suppose I have you add in to the state the relative impact of the motion of a single butterfly on the world’s air currents. It’s an absolutely minuscule number. There’s about 5,​000,​000,​000,​000,​000,​000 kilograms of air on the planet moving at an average of 7 meters per second for something on the order of a zettajoule of kinetic energy in the air. Even a really heavy butterfly isn’t going to be burning more than a few calories, which won’t make more than a 0.000,​000,​000,​000,​000,​000,​01% impact on the air even if it is perfectly applied. But give the chaos long enough and even that tiny addition still becomes significant.

The weather isn’t believed to be as rapidly chaotic as my example; it may take years before a butterfly’s impact even shows up in wind patterns. But if weather really is chaotic, which is seems to be, then eventually it does show up. No one has any way of predicting what that impact will be, but we believe it exists nonetheless.