I stand corrected.
Stranger
The error would be about 19 feet. Now imagine that the error after one mile were 19 feet, but after two miles were 10 miles and after three it was 5 light years. That is the real significance of non-linear dynamics, it is non-linear and really small changes can have outlandish effects. That is why weather forecasting for more than about five days is a crap shoot and after ten is meaningless. But, oddly enough, long term trends are clear from phenomenons like el nino.
The title of the 1952 Ray Bradbury story is A Sound of Thunder.
It made my day that the great Hari Seldon came along to add his voice to a discussion of predicting the future…
Although, Bradbury’s story was about sensitive dependence on initial conditions. You could say that it was ahead of its time. What Bradbury intuitively grasped is that history, like the weather, is sensitively dependent on initial conditions.
Games shows invented a very effective chaos demonstrator a while ago. They are those tall, vertical boards with pegs that you drop a chip in and the chip drops and bounces through the pegs until it lands in a prize slot. If those are built correctly, you can’t predict what slot the chip will land in even if you trigger the drop precisely and mechanically. The chaos impact builds as it goes.
Imagine if on the bottom, there were buttons to nuke either Russia or Iran. Any tiny effect could have a huge impact on the history of the world.
It’s aforementioned because biqu mentioned it in post #9.
Not at all. The point of Bradbury’s story is that any change - however small - would have a multiplicatively huge effect on all future events.
This is an essentially linear view of starting conditions, exactly the point of view that chaos theory overturned. It reflected its time, rather than being ahead of it.
I guess you’re right, the text of the story does suggest a more linear dependency. I would have thought that the received view would be that killing the butterfly wouldn’t have mattered at all – that the small difference would be insignificant.
Ain’t pseudo science wonderful?
IIRC the great Nikola Tesla put Colorado Springs in the dark more that once in attempts to control the weather at the antipode!
Real hard facts are hard to come by.
BS abounds by the buckets, bushels, and barrels in full measure, pressed down, heaped up, and running over!
Sounds like the “Conneticut Yankee in King Arthur’s Court” circa 1940’s version hearing sounds of the distant past on a modern (at that time) radio, prior to a concussion and a halucination about the Kings Court.
Theoretical non-sense, practical impossibilities. 
The continental divide, perhaps?
The excerpt quoted by Exapno Mapcase might actually overstate Bradbury’s limited understanding, especially when you consider the story’s conclusion.
The time travelers return to the 20th century after their T-Rex hunt goes south, and the changes they see are very superficial. Even the same presidential candidates are running in the altered present; the only difference is who won the election (presumably further evidence of a dumber population, which the returning T-Rex hunters already suspected from the wording of the signs in the time travel office).
Of course, a more drastic conclusion would probably be too much for Bradbury’s intended audience, so one could argue that he chose to portray decidedly sublinear dependence on initial conditions, eschewing scientific accuracy for the sake of a good story.
Yes, Bradbury certainly minimized the cascading effects of the change because it would kill the dramatic tension to catalog thousands of differences. However, you missed a critical subtlety in the ending.
[spoiler]"He did not move. Eyes shut, he waited, shivering. He heard Travis breathe loud in the room; he heard Travis shift his rifle, click the safety catch, and raise the weapon.
“There was a sound of thunder.”
IOW, the T. Rex and presumably all the dinosaurs have survived. That’s the big change in the world.[/spoiler]
Not to take away from your point, but the error after 1 mile would be 1.1876 feet (assuming a nautical mile, just because it’s what I’m used to working with, the error after travelling one statute mile would be even less.)
1.1876 feet, quite insignificant, which is what the scientist in question would have expected.
Those are two different ways of saying the same thing.
No. There’s a difference to highlighting the sensitivity to initial conditions–indicating that the final result, while falling within well-defined bounds, is nonetheless unpredictable in detail effects–and asserting that a system will become dynamically unstable and progressively out of control.
Let me offer an example: we’ve all seen flocks of birds moving together in a harmonic, coordinated fashion. These flocks, as Colibri says, don’t have a specific leader; their motion is due to what is termed “emergent behavior,” i.e. that the system acts as a distributed network despite the individual inputs and lack of hierarchical structure. Because even slight inputs are propagated rapidly through the system–more rapidly, indeed, than would be suggested by normal decision-making or hierarchical command progression–one can’t predict the motion of the flock for more than a few seconds at a time, yet the flock remains in a coherent, fluid group. Unpredictable but not unstable.
Stranger
Hmm, I came to a different conclusion:
that Travis had shot the guy who stepped on the butterfly. Certainly a reasonable reaction since the guy was negligent.
Was there something I missed?
That’s a conventional interpretation, but I’ve always been dissatisfied with it.
[spoiler]Shooting Eckels would make Travis a murderer for the sake of revenge. Yes, he’s shown to be a hothead and the worst possible person to put in charge of such an operation - Eckels can’t possibly be the only person to react this way - but it’s an overreaction to a change in President and it’s unlike Bradbury to write characters as simplistic as this.
Second, and more important to my mind, that interpretation makes “the sound of thunder” the firing of the gun. But the phrase has already been used once in the story and it makes specific reference to T. rex:
"Silence.
"A sound of thunder.
“Out of the mist, one hundred yards away, came Tyrannosaurus rex.”
Are we to think that Travis or his popgun have turned into T. rex? They kill, yes, but that would be a very ordinary sort of killing. And one hardly worthy of being the title as well.
I also find foreshadowing in the lines a page earlier:
And there was a feel. His flesh twitched. His hands twitched."
What happens the very next line after Eckels sees the T. rex?
“‘My God!’ Eckels twitched his mouth.”
Could Bradbury be so careless a writer as to use that same word inadvertently? I don’t think so.
There’s more to the ending than a fit of pique. Dinosaurs roam the earth. All for the life of a butterfly.
Italics in original. Grade the essay as you will 
[/spoiler]
Ah. The Plinko effect.
Plinko is what they call it on The Price Is Right, I think, but I’m sure it’s modeled on Pachinko, which apparently has been around since just after WWII
Using various approximations for the length of a statute mile and the value of pi, it looks like Jinx intended the resultant error to be 1 foot.
As others have said, that would be a linear dependence (as the distance doubles, the error doubles), but chaos is nonlinear and the errors grow much more rapidly.
To put it in terms of significant figures, since that’s how the OP expressed it: In a non-chaotic system (say, a two-body problem in Newtonian gravity), if you put in numbers which are good to about 2 sigfigs, you’ll expect to get numbers out which are good to about 2 sigfigs. If you put in numbers good to 20 sigfigs, you’ll expect to get 20 sigfigs in your output. In a chaotic system, by contrast, you might start off with 20 significant figures, but only have 1 significant figure in your output, or none at all, if the physically possible range of outcomes is large enough. There can sometimes be emergent patterns, but there doesn’t need to be, and if such patterns exist, they’ll apply to the statistical bulk, not to the individual elements.
Continental divides, incidentally, are not chaotic systems. I can find huge regions of South America where you know for darned sure where the raindrop is going to end up. Drop it anywhere in Brazil, and it’s going to reach the mouth of the Amazon. And even if you do drop a raindrop very close to the Divide, it’s going to get further away from it. So even if it wasn’t immediately clear where the drop was going to go, after a few moments you’ve lost all that uncertainty.