Talk:Butterfly effect

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This article has been mentioned by a media organization: Jim Six (June 23, 2005). "The root of chaos is in my back yard". NJ.Com.

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The explanation is wrong. Sensitive dependence from initial condition does NOT refer to the fact that small changes in the starting condition will lead you to different attractors. But that with Chaotic systems the system is globally stable but locally unpredictable. In other words, a slight difference causes a huge difference while inside the attractor. The error grow exponentially, but it still remains inside the same attractor. And in fact the concept of butterfly effect came from Lorentz who discovered attractors where this was possible. Before him there were no "strange attractors", and attractors were either cycles or pointed attractors. And the error would not grow exponentially. — Preceding unsigned comment added by 87.10.200.244 (talk) 12:12, 17 August 2012 (UTC)[reply]

Any support for making this a separate article?100TWdoug (talk) 05:24, 25 March 2008 (UTC)[reply]

Agreed. There are enough movie or TV plotlines that center on the BFly Effect (or some version of it) that it might almost constitute a separate genre, similar to the "Zombie" movie category. Plenty of examples listed in the Wiki article now, but there are many others, including Twilight Zone episodes (e.g., "Back There") and Star Trek episodes like "City on the Edge of Forever." C d h (talk) 03:00, 30 April 2008 (UTC)[reply]

Absolutely. Separating (or even deleting) the popular media references would be a great improvement, because the text is completely unbalanced, with around a dozen paragraphs on popular media, mostly focusing on time travel, versus about three on the core content. This detracts from the article as a whole. This is a significant topic and deserves proper encyclopedic treatment. —Preceding unsigned comment added by 69.86.228.200 (talk) 04:16, 29 May 2008 (UTC)[reply]

Yup, I'm gonna split these. Whole section needs nuked from orbit, but there's less drama involved in splitting it. Chris Cunningham (not at work) - talk 15:21, 21 July 2008 (UTC)[reply]

is this true?....(Jessy2000 (talk) 20:26, 26 January 2011 (UTC))[reply]

The introduction implies the butterfly is exclusive to nonlinear dynamics, then the example is given of a ball placed on top of a hill. I'm thinking of the "hill" z=C-x^2-y^2, which gives a linear dynamical system. 22:55, 7 May 2008 (UTC)

I hate to interrupt your type up, but I'm very confused on this subject and you sound like you know what's going on.cold you help me

                                      your's thankfully
                                   (Jessy2000 (talk) 20:30, 26 January 2011 (UTC))[reply]

This article doesn't seem to really explain the nature of the Butterfly effect as it is understood in nonlinear dynamics. It looks as if the colloquial misconception is just being reproduced. The Butterfly effect refers to how perturbations which are small relative to the attractor grow exponentially before being damped out by the overall behavior of the system. It's this kind of response which makes it impossible to make long-term predictions based upon the local behavior of a system.

Basically, it looks to me like the page is simply putting forth the Jurassic Park explanation which is incorrect; a butterfly can beat it's wings in Peking all it likes and the weather in New York won't be any different. —Preceding unsigned comment added by Gmcastil (talk • contribs) 22:10, 25 August 2008 (UTC)[reply]

If a butterfly flapping it's wings could influence weather patterns then with all the planes flying (propellor and jet propulsion) we would constantly live in a hurricane unlike any before. Sounds like GLOBAL WARMING to me. —Preceding unsigned comment added by 24.199.250.34 (talk) 20:42, 16 December 2008 (UTC)[reply]

I agree, the actual bulk of this article needs citations if I am to be convinced that choas theory is really about small changes becoming uncontrollable. My understanding has been that it is small ERRORS that make things impossible to predict after an amount of time. 86.12.7.205 (talk) 16:30, 17 June 2009 (UTC)[reply]

I may be wrong and don't really know where to find an answer to this question. The butterfly effect is listed in the dictionary as "a chaotic effect created by something seemingly insignificant, the phenomenon whereby a small change in one part of a complex system can have a large effect somewhere else". Is this like a one or two degree temperature drop causing fish population to decrease? I suppose temp. drop is a significant effect, so what would be considered insig.? —Preceding unsigned comment added by 130.39.202.154 (talk) 16:46, 14 September 2009 (UTC)[reply]

An insignificant effect in your example might be a particular fish swimming in one direction for a moment rather than another. Conceivably through a long chain of events that could cause the fish population as a whole to increase or decrease one way or another. If such a chain (not a specific one, but in general) were likely to happen, then we would say the fish population would be chaotic and it would be impossible to predict how it would develop. sorsoup (talk) 13:12, 23 May 2010 (UTC)[reply]

No no no! The butterfly effect is not that a clap of butterfly wings _will always cause_ a storm. It's just that the very small thing (movement of wing) will cause some other things (air will move), which will cause a bunch of other things (more air will move, a flower leaf might move a little, causing a drop to fall off that would otherwise stick to it), and all these effects cascading _MIGHT_ just be the difference in a storm emerging or no storm emerging. Even though the butterfly wing is small and it's effect seems ignorable, in our reality the effect is almost infinitely complex and therefore unpredictable. But I think it's also important to notice that the wing moving is just part of the same reality in which a storm or no storm will emerge. The connection (or lack thereof) is more a matter of human interpretation or synchronicity.--Burt777 (talk) 12:00, 23 February 2010 (UTC)[reply]

Indeed. My understanding is that the flap of a butterflies wings could well (and often does) cause a hurricane, but it can and does also, with equal probability prevent a hurricane. So more butterflies, or more airplanes doesn't mean we should expect more hurricanes in future, just at different specific places and times. Possibly even more or less generally if climate, as distinct from weather, is chaotic, but we wouldn't be able to say whether to expect each flap to cause more hurricanes, less, or not change the frequency of them. sorsoup (talk) 13:12, 23 May 2010 (UTC)[reply]

It's a metaphor, you dodo brain. It's hyperbole and a few other choice things as well. It's certainly not meant to be taken as a concrete statement of fact. Just because chaos theory has latched onto it as a convenient example doesn't mean it is scientific fact. —Preceding unsigned comment added by 131.215.115.31 (talk • contribs) 15:47, October 7, 2010

It's an inappropriate illustration of the chaos idea. A hurricane has a huge vorticity which cannot be created by any number of butterflies because wing flapping does not create such vorticity. Atmospheric vorticity is created thermodynamically by the sun's rays - isn't that so Prof. Lorenz?JFB80 (talk) 18:18, 12 May 2011 (UTC)[reply]

Wrong; you're using "creation" in the wrong sense William M. Connolley (talk) 19:10, 12 May 2011 (UTC)[reply]

Have you heard of William Thomson's theorem that in a perfect fluid vorticity can neither be created or destroyed? (by butterflies also) In a real fluid it is generated thermodynamically. The butterfly could alter the flow pattern but vorticity remains constant. JFB80 (talk) 16:29, 13 May 2011 (UTC)[reply]

*plonk* Short Brigade Harvester Boris (talk) 16:44, 13 May 2011 (UTC)[reply]

Air isn't a perfect fluid, so that theorem is irrelevant William M. Connolley (talk) 18:20, 13 May 2011 (UTC)[reply]

It is not irrelevant because conservation of vorticity (or circulation) was sufficiently accurate to be used in an early computer method of numerical Weather Forecasting used for predicting hurricanes(by von Neumann etc. See Barotropic vorticity equation) Further Bjerknes showed how the conservation law must be modified in real fluids, the modification depending on thermodynamic factors (solar radiation)as I said. This is a difficult subject and it needs careful thought. JFB80 (talk) 19:06, 14 May 2011 (UTC)[reply]

The example in the intro of a ball at the crest of a hill does not exhibit sensitive dependence on initial conditions in the sense that the term is usually used in the dynamical systems literature. The example should be removed and replaced by an example of a chaotic system. —Preceding unsigned comment added by 99.188.91.58 (talk) 04:15, 3 March 2009 (UTC)[reply]

I agree 100%. I was baffled when I saw that example. A much better example would be the Plinko game from The Price Is Right: if you drop the Plinko chip from a specific and exact spot 5 times, it could end up in a different compartment every time (because the starting location varied by about the width of a hair--or less--each time).--208.63.102.196 (talk) 18:51, 2 June 2009 (UTC)[reply]

• On second thought, that's actually more of an example of chaos....But it still works.--208.63.102.196 (talk) 18:54, 2 June 2009 (UTC)[reply]

As it is, this article is meaningless to non-experts in chaos theory. Example, "The butterfly effect is a phrase that encapsulates the more technical notion of sensitive dependence on initial conditions in chaos theory."bob bobato (talk) 17:40, 10 August 2009 (UTC)[reply]

It is generally considered bad form to remove sourced material whose relevance is established through citations without justifying such removal on the talk page. "I don't like it" is not justification. "I don't think it is relevant" is not justification.140.32.16.101 (talk) 22:09, 11 August 2009 (UTC)[reply]

Wrong again. "I don't think it is relevant" is one of the best justifications for removal. In fact, in this case, it clearly isn't relevant. — Arthur Rubin (talk) 22:21, 11 August 2009 (UTC)[reply]

Further reasons.

1. WP:LEDE specifies it shouldn't be in the lede unless it's also in the article.
2. Our article "Quantum chaos" doesn't (and probably shouldn't) mention the butterfly effect. Unless I misread it, quantum chaos is a theory describing how quantum mechanics may produce macroscopic chaotic effects. That doesn't suggest relevance to the butterfly effect.
3. Even if a number of authors use the butterfly effect in their titles for works related to quantum chaos, that doesn't mean that the author believes there is a connection. One would have to read the articles to determine whether the authors just put in popular buzz-words.

— Arthur Rubin (talk) 22:28, 11 August 2009 (UTC)[reply]

Riddle me this: if Wikipedia articles cannot be used as sources to support content, how can the absence of support in a Wikipedia article be used to justify removal of content, especially when inclusion of that material is supported with citations of sources that meet Wikipedia standards for reliability?

I’ve addressed the issues raised above by adding a section on the butterfly effect in semiclassical and quantum physics, and by including quotes demonstrating that the citations are actually discussing the butterfly effect rather than merely including buzz words.140.32.16.101 (talk) 14:55, 12 August 2009 (UTC)[reply]

Quite. Regardless, it should NOT be in the lead, no matter how well-referenced it is (and I haven't checked yet). — Arthur Rubin (talk) 22:17, 12 August 2009 (UTC)[reply]

New section tagged as giving quantum physics an WP:UNDUE weight over classical physics, and the comment in the lead about quantum chaos still seems inappropriate, but now plausible. — Arthur Rubin (talk) 08:02, 14 August 2009 (UTC)[reply]

Neutrality requires that the article should fairly represent all significant viewpoints that have been published by a reliable source, and should do so in proportion to the prominence of each.(From WP:UNDUE)

The section on semiclassical and quantum physics is a small proportion of the overall article, and better supported with reliable sources than the article as a whole. It is also clear that there is very little debate about the validity of the sources being used to support the relationship between the butterfly effect and semiclassical/quantum physics.

Wikipedia is not bound by limits of space that limit published encyclopedia articles. Rather than waste time on edit wars over "proper weight", the Project is more rapidly advanced by collaborative editing with a spirit of generosity and openness to seeing as much material presented as possible. If you feel one POV is being represented by too many citations, the solution is to do your research to see if there are other peer reviewed sources representing your preferred POV that have not been included. Seek to bring balance to the article by adding material, not deleting it.(From WP:Scientific standards)

The new section does not disparage classical physics or indicate that the butterfly effect is not applicable or important in classical physics. If editors believe classical physics is deserving of more attention than it is currently receiving, they are welcome to improve the article by increasing its coverage, as long as they support the added coverage with reliable sources. Should there be more complete coverage of the butterfly effect in classical physics? Absolutely.140.32.16.101 (talk) 15:43, 14 August 2009 (UTC)[reply]

The Butterfly Effect as described doesn't occur in the real world because air is a real gas possessing, among other things, viscosity. Viscosity causes disturbances to decay and disappear with time. The vorticity generated by the flapping of a butterfly's wings is damped out by viscosity and cannot affect the weather anywhere else on the planet. The disturbance caused by a jet plane's passage is damped out and the air mass through which it just flew returns to its original quiescent state. --Virgil H. Soule (talk) 16:01, 26 September 2009 (UTC)[reply]

You're thinking about it classically. Lorenz's simulation had changes of less 0.0001 degree of temperature causing the pattern of weather over the entire system to change. The effects of a butterfly flapping are damped out, but in the process it changes the local system some, which cascade over into the larger system until it changes the whole system. Even classically, you can't damp out something without changing yourself.--Prosfilaes (talk) 23:42, 9 July 2010 (UTC)[reply]

Exactly! Even though momentum will carry the movement forward, it is so infinitively smaller than the other natural forces which works all the time. The very reason to why the our planet keeps itself in a order so complex as this is because there are natural systems which maintains order. Also a reason to suspect that this is no mere rock we are sitting on.... --Nabo0o (talk) 15:22, 9 March 2010 (UTC)[reply]

The article states that using a butterfly as an example of the effect first appeared in Bradbury's short story. However, I've found an earlier use of it from 1903. The Universe a Vast Electric Organism by George Woodward Warder quotes Prof. Garrett P. Serviss in the New York American, May 16, 1903: "Lord Kelvin has been credited with the statement that the fluttering of a butterfly's wing sets up vibrations that shake the universe." Unfortunately I have not been able to find the source of Serviss' paraphrased quote. Lyle zapato (talk) 15:54, 13 December 2009 (UTC)[reply]

Rene Daumal's short novel Mount Analogue, published the same year as Bradbury's story (1952), described a catastrophe initiated by the destruction of a wasp's nest. I am pleased to hear of the Lord Kelvin attribution, which I suspect refers further back, possibly to Nataraja's dance. —Preceding unsigned comment added by 131.215.115.31 (talk • contribs) 15:47, October 7, 2010

The opening paragraph needs work. According to http://www.editcentral.com/gwt1/EditCentral.html, this paragraph has a flesh-kincaid grade level is 16.5. Shouldn't this be written at a more understandable level? I was trying to reference this article as a definition of the butterfly effect, but I think it's written at too high a grade level and should be simplified. —Preceding unsigned comment added by 24.45.84.38 (talk) 15:54, 15 February 2010 (UTC)[reply]

Memory is a bit dim after 38 years, but I'm fairly sure the term was alluded to in Thom's "Structural Stability & Morphogenesis", as the shape of a representation one of his seven canonical types of unstable system, a butterfly catastrophe, where the attractor is in figure-8 pattern (not quite the Lorenz attractor illustrated), which coincidentally describes a "butterfly effect". If someone is in a position to check the reference, perhaps it is worth a mention. That's Stabilité structurelle et morphogenèse, René Thom, Interéditions, Paris, 1972 ; 1977 (ISBN 2-7296-0081-7); more in Catastrophe_theory.

Memethuzla (talk) 09:30, 3 April 2011 (UTC)[reply]

I've clumsily tried to clean up that paragraph. I'm trying to convey the sense that one set of initial conditions leads to one set of events, and another set of initial conditions (which differs only by the flap of a butterfly's wings) leads to another very different set of events; and that it's not a butterfly causing the tornado. This is an important point because there are many see also links linke domino effect / cascade / etc which could lead people to make the popular mistake that the flapping wing air is "amplified" through the system into a tornado. — Preceding unsigned comment added by 90.208.122.8 (talk) 14:45, 8 January 2012 (UTC)[reply]

In relation to this: "A ball placed at the crest of a hill may roll into any of several valleys fully depending on slight differences in initial position." .... This explanation only works if the initial conditions of the ball is the entire world. But there are also the valleys and the laws of nature, so the formulation "fully depending" is false. The butterfly does not cause the storm; the butterfly plus everything around it (movement + state) plus the laws of nature does. The butterfly does not cause the storm - the effect only applies to changes in initial conditions everything else held constant. I think some of the reason why this effect is getting a lot of attention is a misconception of "fully depending"; we need to make the distinction between what the effect is and what it is not more clear. Narssarssuaq (talk) 02:08, 22 July 2012 (UTC) At worst, the entire fascination with the unseeming "butterfly effect" relies on rampant reductionism which nobody has spotted, which leads to a notion that the laws of nature are breached. I mean, dice or the shuffling of poker cards are much more commonplace examples of exactly the same phenomenon. Narssarssuaq (talk) 02:16, 22 July 2012 (UTC)[reply]

I find that the butterfly effect is distinct from randomness; but by using dice or cards, you are implying that it relies on the existence of a non deterministic system.Embrittled (talk) 02:55, 22 July 2012 (UTC)[reply]

In order to discuss randomness, we need to understand what randomness truly is. And nobody does understand that. Randomness is, in fact, one of the toughest phenomena of all to model; cf. random seed. In reply to your question: Nothing in the butterfly effect, dice or cards are displays of perfect randomness. Narssarssuaq (talk) 19:10, 23 July 2012 (UTC)[reply]

I object to this formulation. To use the concepts actuality and potentiality, the chaos inherent to the butterfly effect arises from the vast potentiality in the whole system observed, i.e. the immense amount of possible intermediary states between start state and end state. Hence, from the perspective of potentiality, the systems are not very simple, in fact they are quite the opposite. The essence of this argument could be important to add. Narssarssuaq (talk) 09:42, 14 September 2012 (UTC)[reply]

Hello all, I am not a usual participant in editing Wikipedia, but I thought I'd help citing the claim that Poincaré did in fact foresee the relation which Lorenz proved in the 60s.

“A very small unknown cause determines a considerable effect which we cannot understand. We therefore say that the effect is due to chance. If we knew exactly the laws of nature and the situation of the universe at the initial moment, we could predict exactly the situation of that same universe at a succeeding moment. but even if it were the case that the natural laws had no longer any secret for us, we could still only know the initial situation approximately. If that enabled us to predict the succeeding situation with the same approximation, that is all we require, and we should say that the phenomenon had been predicted, that it is governed by laws. But it is not always so; it may happen that small differences in the initial conditions produce very great ones in the final phenomena. A small error in the former will produce an enormous error in the latter. Prediction becomes impossible, and we have the fortuitous phenomenon.”

"Why do meteorologists have such a hard time in foreseeing the weather with a reasonable degree of precision? Why do showers and storms seem to occur at random, so that many people find it absolutely natural to pray for rain or good weather while they would praying for an eclipse utterly ridiculous? We see that great perturbations generally occur in regions where the atmosphere is unstable. Meteorologists are well aware of the instability of the equilibrium and that somewhere there will be a hurricane, but where? They cannot tell, because a tenth of a degree more or less at any point will determine a hurricane here instead of there, and there will be devastations in areas that would have been spared. If one had known this tenth of a degree one could have foreseen the event, but observations were neither sufficiently frequent nor sufficiently precise, and for this reason everything seems to be due to the intervention of hazard. "

Poincaré Science et méthode 1903

I really hope I helped!

It would add to the article if it explained when the expression "Butterfly effect" was first used. Bradbury created the metaphor, Lorenz used it... but who first referred to critical dependence as the 'butterfly effect'? — Preceding unsigned comment added by 60.229.33.229 (talk) 05:16, 4 November 2013 (UTC)[reply]

S added:

The physicist Stephen Wolfram notes that, while sensitivity to initial conditions is certainly a feature of some mathematical equations, perturbations of the sort produced by the flapping of a butterfly's wings will tend to dissipate due to the effect of viscosity, which was not included in Lorenz's equations.<ref>{{Cite book |last=Wolfram|first=Stephen |title=A New Kind of Science |page=998 |location= |publisher=Wolfram Media |year=2002 |isbn=978-1579550080 }}</ref>

I don't believe this. Not necessarily that he didn't say it, but that its true. Viscosity simply doesn't remove these perturbations. Oh course, if Wolfram has a careful mathematical proof then I'll revise my opinion, but I bet its just hand-waving William M. Connolley (talk) 21:36, 2 January 2014 (UTC)[reply]

But you dont give a proof, you just make an assertion. Are you not hand-waving too? What was stated was a point which needs proper consideration. Can you quote any work done to establish your statement? JFB80 (talk) 22:46, 2 January 2014 (UTC)[reply]

Wolfram is raising a basic point here which needs to be acknowledged in this page. As it currently reads, the page seems to imply that sensitivity to initial conditions in the Lorenz system is evidence for similar sensitivity in the atmosphere. But as Wolfram points out, the Lorenz equations are highly simplified, and in particular do not adequately account for viscous effects. One can debate the strength of these damping effects, but they have to be mentioned, and clearly it is not safe to conclude from Lorenz's results with a toy model that the atmospheric system is highly sensitive to initial condition (I agree with JFB80 that this is where the hand-waving has traditionally come in). I have tried again to reword this sentence, but Wolfram's objection is reasonable and should be noted. Sucnidea (talk) 23:26, 2 January 2014 (UTC)[reply]

Wolfram is making a basic error, which shouldn't be in the article. I'm not clear why exactly you think this point is so fundamental. Viscous effects aren't in the Lorenz equations because they are irrelevant. Their addition in the real atmosphere makes no difference. I've run real GCMs, with viscous effects included, and you get exactly what you expect from Lorentz: even the tiniest perturbations amplify. As they obviously must William M. Connolley (talk) 08:30, 3 January 2014 (UTC)[reply]

The basic error in this page is that sensitivity in the Lorenz equations appears to imply similar sensitivity in the atmospheric system. Wolfram's point stands. The grounds are that (a) viscosity is not properly accounted for, and (b) viscosity will have a damping influence. So you can't assume sensitivity to initial conditions in the atmosphere unless these viscosity terms are fully taken into account, and are shown to be negligible. I have reworded the sentence to correct a typo but have omitted the phrase 'without grounds' since he does explain his reasoning in his book (cited). Sucnidea (talk) 13:56, 3 January 2014 (UTC)[reply]

You're not reading what I'm writing. And I don't think you understand what you're talking about. Simply repeating that viscosity-is-damping goes nowhere William M. Connolley (talk) 18:16, 3 January 2014 (UTC)[reply]

I support Sucnidea. You dont answer the point about the approximate nature of the Lorenz equations. Are they not based on a crude truncation of a Fourier series? If you do that sort of thing of course you will have sensitivity to initial conditions. Concerning viscosity, if you neglect it then you have a perfect fluid in which circulation is conserved by Kelvin's theorem and there is no way in which it can be generated by any number of butterflies. Certainly not for a tornado in Brazil. JFB80 (talk) 21:30, 3 January 2014 (UTC)[reply]

The threshold question would seem to be "can we find a reliable source that says 'Wolfram is making a basic error'?" If so, it's easy - the text can say "The physicist Stephen Wolfram says X ... [insert reliable source] says Wolfram is making a basic error ..." If not, it gets much more difficult. Instead of inserting text based on original research, we're talking about excluding text based on original research.EastTN (talk) 19:51, 9 January 2014 (UTC)[reply]

My understanding from this discussion is that we should keep the Wolfram remark, so I have put it back in. Sucnidea (talk) 16:36, 4 March 2014 (UTC)[reply]

Your understanding is wrong William M. Connolley (talk) 22:16, 4 March 2014 (UTC)[reply]

According to this discussion you cannot neglect viscosity (see comment from JFB80), and if you have a reliable source which contradicts Wolfram then you can add that (see comment from EastTN). Sucnidea (talk) 20:49, 7 March 2014 (UTC)[reply]

Grumble. Wolfram's expert opinions (as an expert in computational mathematics, rather than as a physicist) should be in the article unless implausible and contradicted by reliable sources. I say this even though he's often wrong. — Arthur Rubin (talk) 16:18, 18 March 2014 (UTC)[reply]

Yeah, great idea. Lets fill the article with stuff we know is crap. That'll really help our readers, eh? Follow All Rules and remember: this is a bureaucracy William M. Connolley (talk) 17:46, 18 March 2014 (UTC)[reply]

You're often wrong, also.... Even if what he says is wrong, it's still an expert opinion, and could be mentioned as such. — Arthur Rubin (talk) 18:49, 18 March 2014 (UTC)[reply]

I second that. JFB80 (talk) 14:22, 19 March 2014 (UTC)[reply]

The question of non-locality is being overlooked here. Hypothetically, the butterfly flapping its wings is not only effecting local conditions, but also non-local conditions that are then amplified through feedback in a chain of cause-and-effect events. Interconnectedness is perhaps chaos theory's most important aspect. - Benjamin Franklin 75.74.157.29 (talk) 11:51, 17 May 2014 (UTC)[reply]

"For example, a ball placed at the crest of a hill may roll into any surrounding valley depending on, among other things, slight differences in its initial position."...this is a very poor example. JohnInLongmont — Preceding unsigned comment added by 71.229.248.190 (talk) 16:10, 15 May 2014 (UTC)[reply]

I disagree, I think it's a fine example, especially for the layman. - Benjamin Franklin 75.74.157.29 (talk) 11:44, 17 May 2014 (UTC)[reply]

The name of the effect is also derived from the Lorentz attractor graph being shaped like a butterfly's wings (see diagram).<ref)Gleick, James Chaos Theory</ref) - Benjamin Franklin 75.74.157.29 (talk) 11:43, 17 May 2014 (UTC)[reply]

Regarding non-locality, hypothetically, "the butterfly flapping its wings" is not only effecting local conditions, but also non-local conditions that are then amplified through feedback in a chain of cause-and-effect events (causality). Interconnectedness is perhaps chaos theory's most important aspect. - Benjamin Franklin 75.74.157.29 (talk) 12:53, 17 May 2014 (UTC)[reply]