Ah, there he is.
The dice are not a good example of a linear system. A better linear system is a nail, partially embedded in wood, and a hammer to hit it with. Hit it with a gentle tap and it goes in a little farther. Smack it with a more formidable wallop and it goes in quite a bit farther. You could graph the force of impact against the distance of nail-travel and most likely see a smooth line or curve.
You would be very surprised if you increased the impact a bit more than the previous hit and instead of traveling a bit further than it had the previous time, the nail retreated from the wood a quarter inch. Or suddenly exploded into a mushroom of nail fragments. Or smashed right through the wood, head and all, flew through the air, and embedded itself 1/4 inch in the concrete floor.
So “Chaos” doesn’t mean “contains Strange Attractor?”
OK, then we’re back to the OP question. What’s a simple definition of “Chaos?” What criterion determines when Chaos is present and when it is not?
Throwing dice is an example of Chaos. So is flipping a coin.
Say you flip a coin and it spins 15 times before bouncing the first time. If you’d flipped it slightly harder, it would still flip 15 times before bouncing, and it’s future path after the bounce wouldn’t be much different. But if you flip it slightly harder than that, then it flips 16 times before bouncing. Then it takes a totally different path than it would have with only the 15 flips.
It takes only a tiny bit of extra energy to make the edge of the spinning coin miss grazing the floor and convert the 15-flip pathway into the 16-flip version.
Systems with chaos contain strings of decisions. They’re sensitive to initial conditions because the smallest change will eventually affect one of these decisions, and once you’ve changed one decision, everything from then on follows a totally different pathway.
Simple definition of Chaos Theory:
A linear system is one in which all variables are understood. When a system reaches a certain scale it will eventually become “non-linear” due to our inability to calculate all the possible variables percisely enough. In any system that is non-linear, we as humans are incapable of understanding or measuring all the possible variables that will impact the outcome of an action. As such, the outcome will appear as random to us. However, should this system be repeatedly tested enough times in a similar fasion a pattern will begin to become apparent.
At least that is how it has been explained to me by my betters.
Unfortunately that definition also would include random noise with a signal added: it looks random at first, but if you average it long enough you’ll start to see a pattern. Chaos doesn’t just contain a pattern, instead it contains a fractal pattern (called a strange attractor.) The classic experiment involves plotting the timing of dripping water as points on a graph. If you wait long enough, the randomly scattered points start filling in to form a loopy fractal.
If it’s Chaos, it’s not random. In other words, the current events totally control what happens next, with no fuzzy random stuff being injected from outside. The apparent randomness happens naturally; it boils up from within.
If it’s Chaos, it never repeats no matter how long you wait.
If it’s Chaos, it appears random at first glance, but actually it’s pseudo-random. It’s a horifficlly complex detailed pattern.
If it’s Chaos, and if you change it into a signal and look at it’s frequency spectrum, you see an overall 1/freq slope made of large low frequency signals with smaller higher freq signals riding on them, with even smaller even higher freq on those. If you play it through loudspeakers, it doesn’t sound like a white-noise hiss, instead it sounds like a jet engine added to a motorcycle.
Also, JS Princeton makes a good point: we’re talking about IDEAL chaos here. We can define Ideal Chaos, but it doesn’t exist in the real world because quantum noise interferes. Since a chaotic process amplifies small changes, that means quantum noise is amplified too… and that means that given enough time, a chaotic process will show true randomness (composed of hugely amplified quantum randomness.) Or in other words, if you zoom in on the fractal pattern hidden inside the motions, eventually the tiniest fractal details fuzz out because of Heisenberg Uncertainty. With true chaos the pattern is fractal down to infinity. With true chaos if you change the initial conditions by even an infinitely small amount, the chaotic motion pattern totally changes in the long run.
What then is real-world chaos? How much true randomness is allowed before the randomness overwhelms the fractally-chaos pattern? I don’t know. I would suspect that the hidden fractal would have to have lots of levels. If it didn’t, then you’d get repetitions in the original happenings (the process would do some random-looking stuff for awhile, but then it would start over and repeat that exact sequence.)