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Why is the butterfly effect so surprising? What is the key element which makes scientist wonder when they read its statement?

It seems obvious and almost intuitive that a butterfly flapping it wings is a change which can lead to a corresponding change in any output of any magnitude.

So why is everyone so surprised by it? Am I missing some key insight?

I would like to clarify here that while I have written obvious please don't take it as a sign of my obnoxiousness.

I ask this question not to be rebuked, but seek a explanation wherein I can realize and fill in the gaps in my understanding.

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    $\begingroup$ It might help to understand something about the historical philosophy of the discipline and in particular how the notion of determinism was understood. Consequently this might be better on History of Science and Mathematics. $\endgroup$ – dmckee --- ex-moderator kitten Feb 23 '17 at 18:31
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    $\begingroup$ You say it's obvious. Ok, write down a mathematical model of any physical system such that a tiny change in initial conditions leads to an enormous change the result. $\endgroup$ – DanielSank Feb 23 '17 at 18:58
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    $\begingroup$ @DanielSank: Surely this is too easy. If I drop you right next to the edge of the Grand Canyon, I'll get one result; if I drop you just a tiny bit over the edge, the result will be enormously different. I don't think it would be hard to express this in a mathematical model. $\endgroup$ – WillO Feb 23 '17 at 19:01
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    $\begingroup$ @WillO I believe that sensitive dependence in the meaning used for chaos is different from the existence of thresholds for behavior. In chaotic systems exponential divergence is the norm—found in the bulk of the phases space rather than just at a few surfaces. $\endgroup$ – dmckee --- ex-moderator kitten Feb 24 '17 at 22:58
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    $\begingroup$ -1 Not clear what you are asking, since you have not provided any statement of what the Butterfly Effect is. David Hamman's answer to "Is the butterfly effect real?" shows that the popular interpretation is not what scientists mean by it. To most people the popular statement is obviously false, not true. $\endgroup$ – sammy gerbil Apr 7 '18 at 12:47
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Large change in the final outcomes because of the small changes in the initial condition is a hallmark of chaos. However, that is not the sufficient condition for a motion to be called chaotic. An equally important aspect is that the motion should be bounded in the phase space. When such divergence of trajectories happens in a bounded system, it is indeed surprising. As an example, consider the following function:

$$x(t) = Ae^{t}$$

What is the behavior of this function for $A = 0$, for $A$ that is slightly positive and for $A$ that is slightly negative? It is easy to see that one solution remains constant in time whereas the other two go to $\pm\infty$. As you said in the comments, this is not that surprising. However, now try to preserve the same property of divergence so that trajectories don't fly off to infinity and always remain within a bounded region. This is an essence of the butterfly effect or chaos. The trajectories do diverge "locally" so that the small change in the initial conditions does give rise to very different futures (and hence we can't predict weather exactly) but still the motion remains bounded and hence repeats itself approximately (which is why seasons still exist!).

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I think two things are (or were until people knew this stuff) reasonably surprising. I'll use weather as the example for both (since I work somewhere that does weather prediction).

Firstly, it is not intuitively obvious just how badly things blow up. I don't think it's easy to understand, unless you already know, that weather prediction two days out might be completely tractable with a computer you can more-or-less buy off the shelf, while weather prediction a month or a year out might require more computing resources than will ever exist in the universe. Indeed, I think that the existence of problems that have that kind of blow-up is one of the great discoveries of the 20th century: I don't think that, if you asked someone in 1900 whether such problems existed at all, they would have said yes: but it turns out that not only do they exist as fairly abstract computational problems, they are really common in quite mundane physics.

Secondly, and perhaps even more surprisingly is how simple these systems can be. Of course weather prediction is hard: there are a huge number of variables in the system so it's not surprising it's hard. But it is surprising that the most absurdly simplified model of weather it is possible to imagine -- a system with three variables evolving over time -- is also effectively unpredictable. If that is not surprising to you then you have lost your sense of wonder, because it is a deeply astonishing fact.

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