Actually, in a chaotic system, any variation of initial conditions, no matter how tiny, eventually propagates to affect the system as a whole.
So it’s not a matter of “sometimes tiny changes have huge effects”, but “every change always eventually affects everything else”.
What is wrong with the example I gave:
It is an empirical fact that known uncertainties in the weather modeling inputs cause total weather unpredictability (at a local level) after X days. Sure, sometimes the model gets it right predicting 10 days in advance, and some times it gets it right only 1 day in advance, but we can say that on average, after X days the predictions reach the baseline of “random chance it will get the weather right”. Adding the fact sperm are so small and numerous that even the most trivial deviation in time of ejaculation, or at any time a tiny deviation in barometric pressure, body movement, or position is enough to change which sperm makes it to the ovum first, it is clear that in general even the most trivial deviation in weather, causing as little as someone to look up at a cloud, is enough to affect who will be born 9 months later. If the entire gene pool is randomized, I think it is without question that history will be significantly altered. And keep in mind, that this is only one specific way in which history could be altered by the tiniest change in weather – I have ignored countless others for the sake of brevity. Furthermore, I have ignored countless other things that become uncertain besides the weather. So even if you argued that we are uncertain about when weather will locally deviate significantly, there are countless other effects, some of which more or less chaotic, each of which serves to strengthen our confidence in some upper bound on how long, at minimum, it would take for macroscopic deviations to become noticeable. At minimum we only need a deviation large enough to cause a few sperms to move one micro meter.
So while I understand your point, I think you underestimate the possibility of setting reasonable upper bounds on the time it would take for history to be meaningfully altered, given a large enough uncertainty.
Is there any type of physical system that would not apply to? Is the universe a chaotic system?
Oh, there are lots of systems that it doesn’t apply to, otherwise we couldn’t model them accurately.
For example, if you’re shooting an artillery shell, small differences in your initial conditions lead to small differences in where the shell lands. Changing the angle of the barrel by a tiny amount will have a negligible effect on the trajectory. Chaotic systems aren’t like that.
Ok, so it’s the small change leading to everything changing, and small change leading to big that characterize chaotic systems. Thanks.
Not exactly; it’s change leading to an unpredictable but not truly random difference that characterizes chaotic systems. A small change leading to a large but precisely calculable difference isn’t chaotic either. A truly random effect like radioactive decay isn’t chaotic either.
Other examples of non-chaotic things would be rocks and pendulums. Rocks just sit there, they are stable not chaotic. And pendulums are periodic, not chaotic. I’ve seen it portrayed as a spectrum of sort; from rigid unchangingness, to periodic behavior, to chaos, to disorder. Rocks, pendulums, life, boiling water. With quantum randomness being in its own special category.
I’d like to find out more about that spectrum if you can recommend resources.
TriPolar – I think Wiki does a decent job of explaining the difference between chaotic behavior and truly random behavior. Chaotic behavior is deterministic but pathologically sensitive to initial conditions. Der Trihs is mostly right that there is a “spectrum”, or rather, three categories of dynamics. They are sort of self-explanatory: deterministic behavior; deterministic behavior that is sensitive to initial conditions (); and non-deterministic behavior (ie random)*. There is no such thing as “rigid unchangingness” btw.
(*) sensitive to initial conditions means that an arbitrarily small change in initial conditions leads to significantly different future behavior.
Thanks. Its the differentiating characteristics of the systems that I’m looking for. That was a good start.
This is the most authoritative explanation of the butterfly effect that I’ve come across, explained by a bona fide expert in time time travel:
This is an interesting display I think.
I took a class in chaotic dynamical systems as an undergraduate, which skimmed the surface (but in a very mathematical manner!) of the area known as “chaos theory”. There was a concise definition of what it means for a function to be “chaotic”. The terms used are “sensitive dependence on initial conditions”, “topologically transitive” and “dense periodic points”. Here are some simple, non-precise, definitions: The first means that at each point there will be a nearby point such that their iterated values will significantly differ eventually. The second roughly means that given two areas, iterating the function on one of the areas will eventually land you in the other. The third is that at each point one is always able to find a nearby point where repeated iterations of the function lead one to return to the same exact spot eventually.
(Aside, I believe during the time between the publishing of the book and the class there was a relatively elementary proof (that is, undergraduates could understand it and it could be explained well within a class period) developed that two of these implied the third, but I was late to class that day and only caught the tail end and thus have no idea which.)
Anyway, why were these three necessary for something to be called chaotic? Well, the first one is the most obvious one we think about in terms of chaos: it means that you can’t do any good numerical analysis of the problem because rounding errors will make your estimates meaningless. The second prevents you from breaking the system down into component parts. The third, about periodic points, means that there is some sense of order, and it’s always close by.
How well physical systems will meet these criteria is not something I have ever seriously considered (I’m a mathematician, not a physicist); there are plenty of abstract functions that display these properties, with f(x) = mx(1-x) for m>4 on the interval [0,1] being the prototype we mostly studied. Proving that something like “the weather” is chaotic as defined above is not practically possible, but it has the same sort of features: small changes in a particle’s initial values can cause large changes in its resultant location, the atmosphere responsible for the weather is continuous and connected, and there are clear regularities to it regardless of where one looks.
I think talking about the weather as a chaotic system and the canonical butterfly effect example are two completely different things. I actually am pretty surprised that the notion of a tiny air disturbance causing long term weather pattern change gets so much serious consideration. Consider that all physical laws we know describe that systems move towards equilibriums of maximal energy distribution. Consider that the time scale for dynamic change in a pocket of air is going to be many times shorter than the scale for the formation of a sizeable weather formation. Basically, the transients caused on such a small scale are going to smooth out, not balloon into some macro-transient orders of magnitude larger.
If atmospheric dynamics actually worked the way suggested in the butterfly wing flapping example, any form of predictability would be out the window. We’d have random wind gusts forming in offices, storms appearing literally out of nowhere, and all sorts of craziness.
I’d agree with DT
(which means this is already a low-probability timeline) 
Technically the only requirement necessary is topological mixing (cite); the other two requirements are redundant. But this is a technicality. The important point to grasp is sensitivity to initial conditions.
You are confusing global phenomena with local phenomena. The ability to predict a local gust of wind or the shape of a wisp of cloud does in fact depend extremely sensitively to initial conditions. What does “smooth out” to some degree (but still ultimately chaotic over the long term) are more global patterns like the trade winds.
The transients don’t “smooth out”. That’s the point of a chaotic system, as **glowacks **pointed out.
We can predict the weather in the short term because it takes time for minor local effects to make their presence felt globally. But over the course of a week or so, minor local effects eventually perturb the entire system. No matter how many weather stations we have gathering data, tiny events between the stations will trigger an inevitable cascade that eventually gives rise to unanticipated large-scale weather patterns.
In addition, systems can exhibit chaotic patterns at one scale of time, size or distance, yet be stable at larger or smaller scales.
The pressure of the gas in a bottle is relatively constant, even though the motion of any individual particle contributing to the pressure may be unpredictable. Weather may exhibit chaotic patterns, while long-term climate may be predictable. Individual movements of stocks or the price of an individual good may move in a seemingly random walk due to sensitivity of initial conditions, but the overall inflation rate or stock market valuation rate may be predictable over scales of years or decades.
I’m not confusing them, the people who suggest a small change in the pressure and velocity of a pocket of air a few centimeters across is sufficient to cause a massive weather formation are.
The reason the atmosphere is so volatile and unpredictable is not because tiny disturbances grow into huge ones, but because ridiculous amounts of energy are being poured into it perpetually by the sun. I feel reasonably confident saying that if the sun went out, weather would become very boring in a fairly short time. And in neither case does doing something like waving my hand cause any measurable difference even seconds later.
You know the definition of a chaotic system, right?
Can anyone chime in who has done empirical research in climate modeling? Even though they typically work with square kilometers rather than centimeters, my understanding is that local weather patterns are a prime example of a chaotic system, and that “a small change in the pressure and velocity of a pocket of air a few centimeters across” is easily enough to significantly affect the local weather weeks down the line.