Basically a chaotic system is such that it tends to expand differences in path instead of shrink them.

An example of a non-chaotic system is a cannonball fired at X1 speed and X2 angle in a gravitational field, where you measure the distance Y being how far it flies.

You fire the cannon, the ball lands a little short of the target, so you know you can increase the firing velocity a bit and probably hit that target.

If you overshoot, you shoot slower next time. If you undershoot, you shoot faster. (I’m not playing with the angle here because the angle is slightly weirder, too low and angle and you undershoot, but too high an angle and you also undershoot).

So the relationship between the position of the muzzle speed knob, and the final position of the cannonball, follows a “linear” relation. If turning the knob 100 mph results in the cannonball landing 100 m away, and turning the knob 300 mph cresults in the cannonball being 500 m away, then because it’s a linear system you can reason that turning the knob 200 mph results in the cannonball being somewhere between 100 m and 500 m away.

In short, given a function mapping points in one space onto points in another space, a linear function ensures that two points close on the input space will be close to two points in the output space.

A chaotic function doesn’t preserve this. It scrambles the relationships between input and output. The lines between input points and output points cross each other; if they were hair they’d no longer be combed.

Say you replace your empty sky with a giant 3D pinball machine, and then you fire cannonballs into that.

You set the dial to 100 mph: cannonball lands 100 m away.
You set the dial to 300 mph: cannonball lands 500 m away. you set the dial to 200 mph: where will it now?

Because you’ve introduced the pinball machine to the sky, you can no longer predict that the cannonball is going to land between 100 and 500 meters away.

Maybe the 100 mph cannonball went under a bumper, the 300 mph cannonball went over it, but the 200 mph cannonball hits it, and bounces back over your head and it lands behind you.

Now you’ve got this table of inputs to outputs:

| muzzle speed | landing position | | 100 mph | 100 m | | 200 mph | -750 m | | 300 mph | 500 m |

(The choice to mix imperial and SI units is deliberate, by the way, because in real life these variables might have totally unrelated units. Like “value of a gram of gold in yen” “number of red blood cells passing into the brain per hour”. Nonlinearity isn’t a property specifically of space so I didn’t want it to look like it was an equation about space or distance)

Anyway, I’m probably wrong about 20% of that but it’s my understanding of chaos.

Now to your question:

How [would it be unpredictable]?

It relies on my assumption that there is an infinite amount of information in the positions and velocities of the particles.

(This is probably false but I’m carrying the argument through anyway to see where it goes)

In other words, that if you wrote the position or velocity of the particle it would take an infinite number of written digits to capture the position, or velocity, precisely.

The chaos comes from this: when there’s no longer a relationship forcing points “between” each other to be “between” each other in the output, it happens at every level. (for math people: linear isn’t literally a line, but any complex polynomial with real number coefficients. I think? Maybe it’s no compound terms that makes it linear? Something like that.It’s been 20 years and I lost my old diff eq book). But it can be a very squiggly line and still be mathematically a “linear function”.

Meaning that each input number is unpredictably related to the number next to it: 1’s relationship to 2 isn’t known. Whether 2 falls between 1 and 3 isn’t known.

But surely 1.5 is between 1 and 2 right? Nope it happens in the first decimal place too.

You might be able to predict the probability, because maybe the input can’t travel too far from the output. But after the system cycles, the positions are at least a little shuffled. And as time goes on, just the random shuffling will tend to move input lines further from each other.

The end result is that tiny deviations in input become huge deviations in output, and 1.11342 might map to 100 while 1.11343 maps to 975, and 1.11344 maps to 42, and 1.113421 maps to 4,350.

It’s like zooming into a fractal. Each tiny detail gets larger and larger and can move the entire thing.

And it can even happen with finite input (though it does become literally predictable by simulation, but is still computationally irreducible).

Shit I’m really bad at keeping things short.

It’s like pointing a camera at a live feed of itself. If you haven’t seen it, try it or look it up on youtube.

TL;DR: Tiny differences in input become huge differences in output in a chaotic system, meaning in a continuous universe imperfections too small to measure doom the prediction

So, at what point is your personal decision making controlling the divergence so that it reflects your will? That is what I am asking.