digplanet beta 1: Athena
Share digplanet:

Agriculture

Applied sciences

Arts

Belief

Business

Chronology

Culture

Education

Environment

Geography

Health

History

Humanities

Language

Law

Life

Mathematics

Nature

People

Politics

Science

Society

Technology

Point attractors in 2D phase space

In chaos theory, the butterfly effect is the sensitive dependency on initial conditions in which a small change at one place in a deterministic nonlinear system can result in large differences in a later state. The name of the effect, coined by Edward Lorenz, is derived from the theoretical example of the details of a hurricane (exact time of formation, exact path taken) being influenced by minor perturbations equating to the flapping of the wings of a distant butterfly several weeks earlier. Lorenz discovered the effect when he observed that runs of his weather model with initial condition data that was rounded in a seemingly inconsequential manner would fail to reproduce the results of runs with the unrounded initial condition data. A very small change in initial conditions had created a significantly different outcome.

Although the butterfly effect may appear to be an unlikely behavior, it is exhibited by very simple systems. 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. Also the randomness of throwing a dice depends on this system's characteristic to amplify small differences in initial conditions - the throw - into significantly different dice paths and outcome, which makes it virtually impossible to throw a dice exactly the same way twice.

The butterfly effect is a common trope in fiction, especially in scenarios involving time travel. Additionally, works of fiction that involve points at which the storyline diverges during a seemingly minor event, resulting in a significantly different outcome than would have occurred without the divergence, are an example of the butterfly effect.

History[edit]

Chaos theory and the sensitive dependence on initial conditions was described in the literature in a particular case of the three-body problem by Henri Poincaré in 1890.[1] He later proposed that such phenomena could be common, for example, in meteorology.[2]

In 1898,[1] Jacques Hadamard noted general divergence of trajectories in spaces of negative curvature. Pierre Duhem discussed the possible general significance of this in 1908.[1] The idea that one butterfly could eventually have a far-reaching ripple effect on subsequent historic events first appears in "A Sound of Thunder", a 1952 short story by Ray Bradbury about time travel (see Literature and print here).

In 1961, Lorenz was using a numerical computer model to rerun a weather prediction, when, as a shortcut on a number in the sequence, he entered the decimal 0.506 instead of entering the full 0.506127. The result was a completely different weather scenario.[3] In 1963 Lorenz published a theoretical study of this effect in a well-known paper called Deterministic Nonperiodic Flow.[4] (As noted in the paper, the calculations were performed on a Royal McBee LPD-30 computing machine.[5]) Elsewhere he said[citation needed] that "One meteorologist remarked that if the theory were correct, one flap of a seagull's wings could change the course of weather forever." Following suggestions 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. 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.[6]

The phrase refers to the idea that a butterfly's wings might create tiny changes in the atmosphere that may ultimately alter the path of a tornado or delay, accelerate or even prevent the occurrence of a tornado in another location. Note that the butterfly does not power or directly create the tornado. The Butterfly effect does not convey the notion - as is often misconstrued - that the the flap of the butterfly's wings causes the tornado. The flap of the wings is a part of the initial conditions; one set of conditions leads to a tornado while the other set of conditions doesn't. The flapping wing represents a small change in the initial condition of the system, which causes a chain of events leading 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 - it's 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.[7]

Some scientists have since argued that the weather system is not as sensitive to initial condition as previously believed.[8] David Orrell argues that the major contributor to weather forecast error is model error, with sensitivity to initial conditions playing a relatively small role.[9][10] Stephen Wolfram also notes that the 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.[11]

Illustration[edit]

The butterfly effect in the Lorenz attractor
time 0 ≤ t ≤ 30 (larger) z coordinate (larger)
TwoLorenzOrbits.jpg LorenzCoordinatesSmall.jpg
These figures show two segments of the three-dimensional evolution of two trajectories (one in blue, the other in yellow) for the same period of time in the Lorenz attractor starting at two initial points that differ by only 10−5 in the x-coordinate. Initially, the two trajectories seem coincident, as indicated by the small difference between the z coordinate of the blue and yellow trajectories, but for t > 23 the difference is as large as the value of the trajectory. The final position of the cones indicates that the two trajectories are no longer coincident at t = 30.
A Java animation of the Lorenz attractor shows the continuous evolution.

Theory and mathematical definition[edit]

Recurrence, the approximate return of a system towards its initial conditions, together with sensitive dependence on initial conditions, are the two main ingredients for chaotic motion. They have the practical consequence of making complex systems, such as the weather, difficult to predict past a certain time range (approximately a week in the case of weather) since it is impossible to measure the starting atmospheric conditions completely accurately.

A dynamical system displays sensitive dependence on initial conditions if points arbitrarily close together separate over time at an exponential rate. The definition is not topological, but essentially metrical.

If M is the state space for the map f^t, then f^t displays sensitive dependence to initial conditions if for any x in M and any δ > 0, there are y in M, with 0 < d(x, y) < \delta such that

d(f^\tau(x), f^\tau(y)) > \mathrm{e}^{a\tau} \, d(x,y).

The definition does not require that all points from a neighborhood separate from the base point x, but it requires one positive Lyapunov exponent.

Examples[edit]

The butterfly effect is most familiar in terms of weather; it can easily be demonstrated in standard weather prediction models, for example.[12]

The potential for sensitive dependence on initial conditions (the butterfly effect) has been studied in a number of cases in semiclassical and quantum physics including atoms in strong fields and the anisotropic Kepler problem.[13][14] Some authors have argued that extreme (exponential) dependence on initial conditions is not expected in pure quantum treatments;[15][16] however, the sensitive dependence on initial conditions demonstrated in classical motion is included in the semiclassical treatments developed by Martin Gutzwiller[17] and Delos and co-workers.[18]

Other authors suggest that the butterfly effect can be observed in quantum systems. Karkuszewski et al. consider the time evolution of quantum systems which have slightly different Hamiltonians. They investigate the level of sensitivity of quantum systems to small changes in their given Hamiltonians.[19] Poulin et al. presented a quantum algorithm to measure fidelity decay, which "measures the rate at which identical initial states diverge when subjected to slightly different dynamics". They consider fidelity decay to be "the closest quantum analog to the (purely classical) butterfly effect".[20] Whereas the classical butterfly effect considers the effect of a small change in the position and/or velocity of an object in a given Hamiltonian system, the quantum butterfly effect considers the effect of a small change in the Hamiltonian system with a given initial position and velocity.[21][22] This quantum butterfly effect has been demonstrated experimentally.[23] Quantum and semiclassical treatments of system sensitivity to initial conditions are known as quantum chaos.[15][21]

See also[edit]

References[edit]

  1. ^ a b c Some Historical Notes: History of Chaos Theory
  2. ^ Steves, Bonnie; Maciejewski, AJ (September 2001). The Restless Universe Applications of Gravitational N-Body Dynamics to Planetary Stellar and Galactic Systems. USA: CRC Press. ISBN 0750308222. Retrieved January 6, 2014. 
  3. ^ Mathis, Nancy (2007). Storm Warning: The Story of a Killer Tornado. Touchstone. p. x. ISBN 978-0-7432-8053-2. 
  4. ^ Lorenz, Edward N. (March 1963). "Deterministic Nonperiodic Flow". Journal of the Atmospheric Sciences 20 (2): 130–141. Bibcode:1963JAtS...20..130L. doi:10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2. ISSN 1520-0469. Retrieved 3 June 2010. 
  5. ^ "Part19". Cs.ualberta.ca. 1960-11-22. Retrieved 2014-06-08. 
  6. ^ "The Butterfly Effects: Variations on a Meme". AP42 ...and everything. Retrieved 3 August 2011. 
  7. ^ Woods, Austin (2005). Medium-range weather prediction: The European approach; The story of the European Centre for Medium-Range Weather Forecasts. New York: Springer. p. 118. ISBN 978-0387269283. 
  8. ^ Orrell, David; Smith, Leonard; Barkmeijer, Jan; Palmer, Tim (2001). "Model error in weather forecasting". Nonlinear Proc. Geoph. 9: 357–371. 
  9. ^ Orrell, David (2002). "Role of the metric in forecast error growth: How chaotic is the weather?". Tellus 54A: 350–362. doi:10.3402/tellusa.v54i4.12159. 
  10. ^ Orrell, David (2012). Truth or Beauty: Science and the Quest for Order. New Haven: Yale University Press. p. 208. ISBN 978-0300186611. 
  11. ^ Wolfram, Stephen (2002). A New Kind of Science. Wolfram Media. p. 998. ISBN 978-1579550080. 
  12. ^ "Chaos and Climate". RealClimate. Retrieved 2014-06-08. 
  13. ^ Heller, E. J.; Tomsovic, S. (July 1993). "Postmodern Quantum Mechanics". Physics Today. 
  14. ^ Gutzwiller, Martin C. (1990). Chaos in Classical and Quantum Mechanics. New York: Springer-Verlag. ISBN 0-387-97173-4. 
  15. ^ a b Rudnick, Ze'ev (January 2008). "What is...Quantum Chaos" (PDF). Notices of the American Mathematical Society. 
  16. ^ Berry, Michael (1989). "Quantum chaology, not quantum chaos". Physica Scripta 40 (3): 335. Bibcode:1989PhyS...40..335B. doi:10.1088/0031-8949/40/3/013. 
  17. ^ Gutzwiller, Martin C. (1971). "Periodic Orbits and Classical Quantization Conditions". Journal of Mathematical Physics 12 (3): 343. Bibcode:1971JMP....12..343G. doi:10.1063/1.1665596. 
  18. ^ Gao, J. & Delos, J. B. (1992). "Closed-orbit theory of oscillations in atomic photoabsorption cross sections in a strong electric field. II. Derivation of formulas". Phys. Rev. A 46 (3): 1455–1467. Bibcode:1992PhRvA..46.1455G. doi:10.1103/PhysRevA.46.1455. 
  19. ^ Karkuszewski, Zbyszek P.; Jarzynski, Christopher; Zurek, Wojciech H. (2002). "Quantum Chaotic Environments, the Butterfly Effect, and Decoherence". Physical Review Letters 89 (17): 170405. arXiv:quant-ph/0111002. Bibcode:2002PhRvL..89q0405K. doi:10.1103/PhysRevLett.89.170405. 
  20. ^ Poulin, David; Blume-Kohout, Robin; Laflamme, Raymond & Ollivier, Harold (2004). "Exponential Speedup with a Single Bit of Quantum Information: Measuring the Average Fidelity Decay". Physical Review Letters 92 (17): 177906. arXiv:quant-ph/0310038. Bibcode:2004PhRvL..92q7906P. doi:10.1103/PhysRevLett.92.177906. PMID 15169196. 
  21. ^ a b Poulin, David. "A Rough Guide to Quantum Chaos" (PDF). 
  22. ^ Peres, A. (1995). Quantum Theory: Concepts and Methods. Dordrecht: Kluwer Academic. 
  23. ^ Lee, Jae-Seung & Khitrin, A. K. (2004). "Quantum amplifier: Measurement with entangled spins". Journal of Chemical Physics 121 (9): 3949. Bibcode:2004JChPh.121.3949L. doi:10.1063/1.1788661. 

Further reading[edit]

External links[edit]


Original courtesy of Wikipedia: http://en.wikipedia.org/wiki/Butterfly_effect — Please support Wikipedia.
This page uses Creative Commons Licensed content from Wikipedia. A portion of the proceeds from advertising on Digplanet goes to supporting Wikipedia.
1000000 videos foundNext > 

The Butterfly Effect (2004) Trailer

The Butterfly Effect Trailer taken from The Butterfly Effect [DVD] This Video Clip Is In No Way Profitable, All Copyright Reserved To Their Owners. Ripped Us...

The Butterfly Effect by Andy Andrews

All of our choices matter. This speech by Andy Andrews is an amazing testiment to how we can positively or negatively effect those around us, even for genera...

Different endings of Butterfly effect 1 compilation

My photography page: http://www.bit.ly/aceshooting ----------------------------------------------------------------------- Different endings of Butterfly effect 1 compilation: 4 endings: 1-...

Chaos Theory / Butterfly Effect

Chaos theory, which speculates that the smallest of events can have the hugest of consequences. Although a butterfly flapping its wings has remained constant...

The Butterfly Effect

A young man struggling to access sublimated childhood memories finds a technique that allows him to travel back to the past. Occupying his childhood body, he is able to change history. But...

The Butterfly Effect ending (HD)

An amazing movie and amazing song. The name's song is Stop Crying Your Heart Out performed by Oasis.

Butterfly Effect, The

A young man struggling to access sublimated childhood memories finds a technique that allows him to travel back to the past. Occupying his childhood body, he...

The Butterfly Effect - Final Conversation of Kings (Full Album)

Band: The Butterfly Effect Album: Final Conversation Of Kings (2008) 01. Worlds on Fire 00:00 02. Room Without a View 07:04 03. Final Conversation 11:16 04. ...

The best scene of: Butterfly Effect

Thumper playing pool.

Prizmmy☆ / 「Butterfly Effect」 MV (2014.2.5 ON SALE)

http://mv.avex.jp/prizmmy/ Prizmmy☆の記念すべき10枚目のシングル「Butterfly Effect」!

1000000 videos foundNext > 

847 news items

TheWrap

Entertainment Weekly
Fri, 04 Jul 2014 10:14:41 -0700

Ashton Kutcher's 2004 movie The Butterfly Effect is still making ripples a decade later: The production company Benderspink filed a lawsuit Thursday claiming that it's still owed shares of the film's profits. The suit names FilmEngine, Highwire ...

Calgary Herald (blog)

Calgary Herald (blog)
Tue, 22 Jul 2014 09:17:23 -0700

What if we could make significant change with a single dollar – the Butterfly Effect of giving. Can your single dollar create waves that will be felt around the world? Or rather, that can generate obvious change on the issue that you want to address? I ...
 
MediaPost Communications
Thu, 17 Jul 2014 08:23:08 -0700

If you are not familiar with The Butterfly Effect, it is the “sensitive dependency on initial conditions in which a small change at one place in a deterministic nonlinear system can result in large differences in a later state.” At this point you may ...
 
The Daily Telegraph
Sun, 20 Jul 2014 05:22:30 -0700

TOMMY Berry has revealed for the first time how he lost his passion for racing after the death of his twin brother, Nathan Berry. Berry makes his comeback to the Sydney riding ranks at Canterbury on Wednesday. The 23-year-old had a season to remember ...
 
News 12 Long Island
Thu, 03 Jul 2014 17:42:38 -0700

Tijuana Fulford says she was inspired to start the Butterfly Effect Project after learning some girls in her community were unable to participate in Girl Scouts because their families could not afford dues. Fulford says her program is open to all girls ...
 
Air Alamo
Thu, 10 Jul 2014 20:30:38 -0700

He may only be one man, but Carmelo Anthony could have a heck of an impact on the NBA landscape once his decision is made. If he signs with the Chicago Bulls. Rumors kickstarted again recently when credible sources indicated the Chicago Bulls aren't ...
 
New Zealand Listener
Wed, 09 Jul 2014 16:37:30 -0700

The butterfly effect. What was to have been a 20th-birthday celebration exhibition for the Holloway Press is now a wake. By Guy Somerset In Books. Print Share. Tweet. 10th July, 2014. Holloway Press · Peter Simpson. Publishing is a delicate ecosystem ...

ArtLyst

ArtLyst
Thu, 17 Jul 2014 02:45:57 -0700

A work of art by the best known YBA artist Damien Hirst is to be offered for sale at Sotheby's in Doha, Qatar, on 13th October 2014. Carrying a price estimate of $1,000,000–1,500,000 'Tranquility' explores the subject of the struggle between life and ...
Loading

Oops, we seem to be having trouble contacting Twitter

Talk About Butterfly effect

You can talk about Butterfly effect with people all over the world in our discussions.

Support Wikipedia

A portion of the proceeds from advertising on Digplanet goes to supporting Wikipedia. Please add your support for Wikipedia!