Editing Butterfly effect (section)

==History==
{{See also|Chaos theory#A popular but inaccurate analogy for chaos}}
In ''[[The Vocation of Man]]'' (1800), [[Johann Gottlieb Fichte]] says "you could not remove a single grain of sand from its place without thereby ... changing something throughout all parts of the immeasurable whole".

[[Chaos theory]] and the sensitive dependence on initial conditions were described in numerous forms of literature. This is evidenced by the case of the [[three-body problem]] by Poincaré in 1890.<ref name="wolframscience.com">[https://www.wolframscience.com/reference/notes/971c Some Historical Notes: History of Chaos Theory] {{webarchive |url=https://web.archive.org/web/20060719234031/http://www.wolframscience.com/reference/notes/971c |date=2006-07-19}}</ref> He later proposed that such phenomena could be common, for example, in meteorology.<ref>{{cite book |last1=Steves |first1=Bonnie |last2=Maciejewski |first2=AJ |date=September 2001 |title=The Restless Universe Applications of Gravitational N-Body Dynamics to Planetary Stellar and Galactic Systems |url=https://books.google.com/books?id=-wa120qRW5wC |location=USA |publisher=CRC Press |isbn=0750308222 |access-date=January 6, 2014}}</ref>

In 1898, [[Jacques Hadamard]] noted general divergence of trajectories in spaces of negative curvature. [[Pierre Duhem]] discussed the possible general significance of this in 1908.<ref name="wolframscience.com"/>

In 1950, [[Alan Turing]] noted: "The displacement of a single electron by a billionth of a centimetre at one moment might make the difference between a man being killed by an avalanche a year later, or escaping."<ref name="turing1950">[https://academic.oup.com/mind/article/LIX/236/433/986238 Computing Machinery and Intelligence]</ref>

The idea that the death of one butterfly could eventually have a far-reaching [[ripple effect]] on subsequent historical events made its earliest known appearance in "[[A Sound of Thunder]]", a 1952 short story by [[Ray Bradbury]] in which a time traveller alters the future by inadvertently treading on a butterfly in the past.<ref>{{cite web |title=The Physics of Ray Bradbury's "A Sound of Thunder" |last=Flam |first=Faye |work=[[The Philadelphia Inquirer]] |date=2012-06-15 |url=https://www.inquirer.com/philly/blogs/evolution/Time-and-The-Physics-of-Ray-Bradbury--.html |access-date=2015-09-02 |url-status=live |archive-url=https://web.archive.org/web/20150924130717/http://www.philly.com/philly/blogs/evolution/Time-and-The-Physics-of-Ray-Bradbury--.html |archive-date=2015-09-24}}</ref>

More precisely, though, almost the exact idea and the exact phrasing —of a tiny insect's wing affecting the entire atmosphere's winds— was published in a children's book which became extremely successful and well-known globally in 1962, the year before Lorenz published:
{{blockquote|
"...whatever we do affects everything and everyone else, if even in the tiniest way. Why, when a housefly flaps his wings, a breeze goes round the world."

-- The Princess of Pure Reason |author=Norton Juster |source=''[[The Phantom Tollbooth]]''
}}

In 1961, Lorenz was running a numerical computer model to redo a weather prediction from the middle of the previous run as a shortcut. He entered the initial condition 0.506 from the printout instead of entering the full precision 0.506127 value. The result was a completely different weather scenario.<ref>{{cite book |last=Gleick |first=James |title=Chaos: Making a New Science |publisher=Viking |year=1987 |isbn=0-8133-4085-3 |page=16}}</ref>

Lorenz wrote:

{{blockquote|
At one point I decided to repeat some of the computations in order to examine what was happening in greater detail. I stopped the computer, typed in a line of numbers that it had printed out a while earlier, and set it running again. I went down the hall for a cup of coffee and returned after about an hour, during which time the computer had simulated about two months of weather. The numbers being printed were nothing like the old ones. I immediately suspected a weak [[vacuum tube]] or some other computer trouble, which was not uncommon, but before calling for service I decided to see just where the mistake had occurred, knowing that this could speed up the servicing process. Instead of a sudden break, I found that the new values at first repeated the old ones, but soon afterward differed by one and then several units in the last [decimal] place, and then began to differ in the next to the last place and then in the place before that. In fact, the differences more or less steadily doubled in size every four days or so, until all resemblance with the original output disappeared somewhere in the second month. This was enough to tell me what had happened: the numbers that I had typed in were not the exact original numbers, but were the rounded-off values that had appeared in the original printout. The initial round-off errors were the culprits; they were steadily amplifying until they dominated the solution. |author=E. N. Lorenz |source=''The Essence of Chaos'', University of Washington Press, Seattle (1993), page 134<ref>{{cite journal |title=Chaos at fifty |journal=Physics Today |volume=66 |issue=5 |pages=27–33 |doi=10.1063/PT.3.1977 |year=2013 |last1=Motter |first1=Adilson E. |last2=Campbell |first2=David K. |bibcode=2013PhT....66e..27M |arxiv=1306.5777 |s2cid=54005470}}</ref>
}}

In 1963, Lorenz published a theoretical study of this effect in a highly cited, seminal paper called ''Deterministic Nonperiodic Flow''<ref name=":0"/><ref>[https://scholar.google.com/scholar_lookup?title=Deterministic+non-periodic+flow&author=E.+N.+Lorenz&publication_year=1963 Google Scholar citation record]</ref> (the calculations were performed on a [[Royal McBee]] [[LGP-30]] computer).<ref>{{cite web |title=Part19 |publisher=Cs.ualberta.ca |date=1960-11-22 |url=http://www.cs.ualberta.ca/~smillie/ComputerAndMe/Part19.html |access-date=2014-06-08 |url-status=dead |archive-url=https://web.archive.org/web/20090717061640/http://www.cs.ualberta.ca/~smillie/ComputerAndMe/Part19.html |archive-date=2009-07-17}}</ref><ref name="Lorenz1963"/> Elsewhere he stated:
{{Blockquote|text=One meteorologist remarked that if the theory were correct, one flap of a [[gull|sea gull's]] wings would be enough to alter the course of the weather forever. The controversy has not yet been settled, but the most recent evidence seems to favor the sea gulls.<ref name="Lorenz1963">{{cite journal |last=Lorenz |first=Edward N. |date=1963 |title=The Predictability of Hydrodynamic Flow |journal=Transactions of the New York Academy of Sciences |volume=25 |issue=4 |pages=409–432 |url=https://eapsweb.mit.edu/sites/default/files/Predictability_hydrodynamic_flow_1963.pdf |access-date=1 September 2014 |url-status=live |archive-url=https://web.archive.org/web/20141010161512/http://eaps4.mit.edu/research/Lorenz/Predictability_hydrodynamic_flow_1963.pdf |archive-date=10 October 2014 |doi=10.1111/j.2164-0947.1963.tb01464.x}}</ref>}}

[[File:Polydamas swallowtail (Battus polydamas polydamas).JPG|thumb|A ''[[Battus polydamas]]'' butterfly in Brazil]]
Following proposals from colleagues, in later speeches and papers, Lorenz used the more poetic [[butterfly]]. According to Lorenz, when he failed to provide a title for a talk he was to present at the 139th meeting of the [[American Association for the Advancement of Science]] in 1972, Philip Merilees concocted ''Does the flap of a butterfly's wings in Brazil set off a tornado in Texas?'' as a title.<ref name=":1"/> Although a butterfly flapping its wings has remained constant in the expression of this concept, the location of the butterfly, the consequences, and the location of the consequences have varied widely.<ref>{{cite web |title=The Butterfly Effects: Variations on a Meme |url=http://blog.ap42.com/2011/08/03/the-butterfly-effect-variations-on-a-meme/ |access-date=3 August 2011 |work=AP42 ...and everything |url-status=dead |archive-url=https://web.archive.org/web/20111111132249/http://blog.ap42.com/2011/08/03/the-butterfly-effect-variations-on-a-meme/ |archive-date=11 November 2011}}</ref>

The phrase refers to the effect of a butterfly's wings creating tiny changes in the [[atmosphere of Earth|atmosphere]] that may ultimately alter the path of a [[tornado]] or delay, accelerate, or even prevent the occurrence of a tornado in another location. The butterfly does not power or directly create the tornado, but the term is intended to imply that the flap of the butterfly's wings can ''cause'' the tornado: in the sense that the flap of the wings is a part of the initial conditions of an interconnected complex web; one set of conditions leads to a tornado, while the other set of conditions doesn't. The flapping wing creates a small change in the initial condition of the system, which cascades to large-scale alterations of events (compare: [[domino effect]]). Had the butterfly not flapped its wings, the [[trajectory]] of the system might have been vastly different—but it's also equally possible that the set of conditions without the butterfly flapping its wings is the set that leads to a tornado.

The butterfly effect presents an obvious challenge to prediction, since initial conditions for a system such as the weather can never be known to complete accuracy. This problem motivated the development of [[ensemble forecasting]], in which a number of forecasts are made from perturbed initial conditions.<ref>{{cite book |last=Woods |first=Austin |title=Medium-range weather prediction: The European approach; The story of the European Centre for Medium-Range Weather Forecasts |url=https://archive.org/details/mediumrangeweath00wood |url-access=limited |page=[https://archive.org/details/mediumrangeweath00wood/page/n131 118] |location=New York |publisher=Springer |year=2005 |isbn=978-0387269283}}</ref>

Some scientists have since argued that the weather system is not as sensitive to initial conditions as previously believed.<ref>{{cite journal |last1=Orrell |first1=David |last2=Smith |first2=Leonard |last3=Barkmeijer |first3=Jan |last4=Palmer |first4=Tim |title=Model error in weather forecasting |journal=Nonlinear Processes in Geophysics |year=2001 |volume=9 |issue=6 |pages=357–371 |bibcode=2001NPGeo...8..357O |doi=10.5194/npg-8-357-2001 |doi-access=free}}</ref> [[David Orrell]] argues that the major contributor to weather forecast error is model error, with sensitivity to initial conditions playing a relatively small role.<ref>{{cite journal |last=Orrell |first=David |title=Role of the metric in forecast error growth: How chaotic is the weather? |journal=Tellus |year=2002 |volume=54A |issue=4 |pages=350–362 |bibcode=2002TellA..54..350O |doi=10.3402/tellusa.v54i4.12159 |doi-access=free}}</ref><ref>{{cite book |last=Orrell |first=David |title=Truth or Beauty: Science and the Quest for Order |page=208 |location=New Haven |publisher=Yale University Press |year=2012 |isbn=978-0300186611}}</ref> [[Stephen Wolfram]] also notes that the [[Lorenz system|Lorenz equations]] are highly simplified and do not contain terms that represent viscous effects; he believes that these terms would tend to damp out small perturbations.<ref>{{cite book |last=Wolfram |first=Stephen |title=A New Kind of Science |page=[https://archive.org/details/newkindofscience00wolf/page/998 998] |publisher=Wolfram Media |year=2002 |isbn=978-1579550080 |url-access=registration |url=https://archive.org/details/newkindofscience00wolf}}</ref> Recent studies using generalized [[Lorenz system|Lorenz models]] that included additional dissipative terms and nonlinearity suggested that a larger heating parameter is required for the onset of chaos.<ref>{{cite journal |last=Shen |first=Bo-Wen |date=2019 |title=Aggregated Negative Feedback in a Generalized Lorenz Model |journal=International Journal of Bifurcation and Chaos |volume=29 |issue=3 |pages=1950037–1950091 |bibcode=2019IJBC...2950037S |s2cid=132494234 |doi=10.1142/S0218127419500378 |doi-access=free}}</ref>

While the "butterfly effect" is often explained as being synonymous with sensitive dependence on initial conditions of the kind described by Lorenz in his 1963 paper (and previously observed by Poincaré), the butterfly metaphor was originally applied<ref name=":1"/> to work he published in 1969<ref name=":2">{{cite journal |last=Lorenz |first=Edward N. |date=June 1969 |title=The predictability of a flow which possesses many scales of motion |journal=Tellus |volume=XXI |issue=3 |pages=289–297 |bibcode=1969Tell...21..289L |doi=10.1111/j.2153-3490.1969.tb00444.x}}</ref> which took the idea a step further. Lorenz proposed a mathematical model for how tiny motions in the atmosphere scale up to affect larger systems. He found that the systems in that model could only be predicted up to a specific point in the future, and beyond that, reducing the error in the initial conditions would not increase the predictability (as long as the error is not zero). This demonstrated that a deterministic system could be "observationally indistinguishable" from a non-deterministic one in terms of predictability. Recent re-examinations of this paper suggest that it offered a significant challenge to the idea that our universe is deterministic, comparable to the challenges offered by quantum physics.<ref name=":3">{{cite web |title=The Butterfly Effect – What Does It Really Signify? |last=Tim |first=Palmer |website=Oxford U. Dept. of Mathematics Youtube Channel |date=19 May 2017 |url=https://www.youtube.com/watch?v=vkQEqXAz44I |access-date=13 February 2019 |url-status=live |archive-url=https://ghostarchive.org/varchive/youtube/20211031/vkQEqXAz44I |archive-date=2021-10-31}}{{cbignore}}</ref><ref name=":4">{{cite web |title=Edward N. Lorenz and the End of the Cartesian Universe |last=Emanuel |first=Kerry |website=MIT Department of Earth, Atmospheric, and Planetary Sciences Youtube channel |date=26 March 2018 |url=https://www.youtube.com/watch?v=FvWeK_PfDE4 |access-date=13 February 2019 |url-status=live |archive-url=https://ghostarchive.org/varchive/youtube/20211031/FvWeK_PfDE4 |archive-date=2021-10-31}}{{cbignore}}</ref>

In the book entitled ''The Essence of Chaos'' published in 1993,<ref name=":5"/> Lorenz defined butterfly effect as: "The phenomenon that a small alteration in the state of a dynamical system will cause subsequent states to differ greatly from the states that would have followed without the alteration." This feature is the same as sensitive dependence of solutions on initial conditions (SDIC) in .<ref name=":0"/> In the same book, Lorenz applied the activity of skiing and developed an idealized skiing model for revealing the sensitivity of time-varying paths to initial positions. A predictability horizon is determined before the onset of SDIC.<ref name=":7">{{cite journal |last1=Shen |first1=Bo-Wen |last2=Pielke |first2=Roger A. |last3=Zeng |first3=Xubin |date=2022-05-07 |title=One Saddle Point and Two Types of Sensitivities within the Lorenz 1963 and 1969 Models |journal=Atmosphere |volume=13 |issue=5 |pages=753 |bibcode=2022Atmos..13..753S |issn=2073-4433 |doi=10.3390/atmos13050753 |doi-access=free}}</ref>