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Butterfly effect
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For other uses, see Butterfly effect (disambiguation).
Point attractors in 2D phase space.
In chaos theory, the butterfly effect is the sensitive dependence on initial conditions; where a
small change at one place in a nonlinear system can result in large differences to a later state. For
example, the presence or absence of a butterfly flapping its wings could lead to creation or
absence of a hurricane.
Although the butterfly effect may appear to be an esoteric and unusual behavior, it is exhibited
by very simple systems: for example, a ball placed at the crest of a hill might roll into any of
several valleys depending on slight differences in initial position.
The term "butterfly effect" itself is related to the meteorological work of Edward Lorenz, who
popularized the term.
The butterfly effect is a common trope in fiction when presenting scenarios involving time travel
and with "what if" cases where one storyline diverges at the moment of a seemingly minor event
resulting in two significantly different outcomes.
Contents
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1 Theory
2 Origin of the concept and the term
3Illustration
4 Mathematical definition
5 Examples
6In popular culture
7 See also
8References
9Further reading
10External links
Theory
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.
Origin of the concept and the term
The term "butterfly effect" itself is related to the work of Edward Lorenz, and it is based in chaos
theory and sensitive dependence on initial conditions, already 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, say in meteorology. In 1898,[1]Jacques Hadamard noted
general divergence of trajectories in spaces of negative curvature, and 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 seems first to have appeared in "A
Sound of Thunder", a 1952 short story by Ray Bradbury about time travel (see Literature and
print here) although Lorenz made the term popular. 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 .506 instead of entering the full .506127. The result was a completely
different weather scenario.[2] Lorenz published his findings in a 1963 paper[3] for the New York
Academy of Sciences noting[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." Later speeches
and papers by Lorenz used the more poetic butterfly. According to Lorenz, when Lorenz 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.[4]
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. 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. While the butterfly does not "cause" the tornado in the
sense of providing the energy for the tornado, it does "cause" it in the sense that the flap of its
wings is an essential part of the initial conditions resulting in a tornado, and without that flap that
particular tornado would not have existed.
Illustration
The butterfly effect in the Lorenz attractor
time 0 ≤ t ≤ 30 (larger)
z coordinate (larger)
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 only by 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.
Mathematical definition
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 ft, then ft 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) < δ such that
d(fτ(x),fτ(y)) > exp(aτ)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
The butterfly effect is most familiar in terms of weather; it can easily be demonstrated in
standard weather prediction models, for example.[5]
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.[6][7] Some authors have argued that extreme (exponential)
dependence on initial conditions is not expected in pure quantum treatments;[8][9] however, the
sensitive dependence on initial conditions demonstrated in classical motion is included in the
semiclassical treatments developed by Martin Gutzwiller[10] and Delos and co-workers.[11]
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.[12] Poulin et al. present 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.”[13] 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.[14][15] This quantum butterfly effect has been
demonstrated experimentally.[16] Quantum and semiclassical treatments of system sensitivity to
initial conditions are known as quantum chaos.[8][14]
In popular culture
Main article: Butterfly effect in popular culture
See also
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Avalanche effect
Behavioral Cusp
Cascading failure
Causality
Chain reaction
Determinism
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Domino effect
Dynamical systems
Fractal
Innovation butterfly
Kessler Syndrome
Law of unintended consequences
Positive feedback
Snowball effect
Traffic congestion
Tropical cyclogenesis
References
1. ^ ab c Some Historical Notes: History of Chaos Theory
2. ^ Mathis, Nancy (2007). Storm Warning: The Story of a Killer Tornado. Touchstone. p.
x. ISBN 0743280532.
3. ^ Lorenz, Edward N. (March 1963). "Deterministic Nonperiodic Flow". Journal of the
Atmospheric Sciences 20 (2): 130–141. Bibcode 1963JAtS...20..130L. doi:10.1175/15200469(1963)020<0130:DNF>2.0.CO;2. ISSN 1520-0469.
http://journals.ametsoc.org/doi/abs/10.1175/15200469%281963%29020%3C0130%3ADNF%3E2.0.CO%3B2. Retrieved 3 June 2010.
4. ^ "The Butterfly Effects: Variations on a Meme". AP42 …and everything.
http://blog.ap42.com/2011/08/03/the-butterfly-effect-variations-on-a-meme/. Retrieved 3
August 2011.
5. ^ http://www.realclimate.org/index.php/archives/2005/11/chaos-and-climate/
6. ^ Heller, E. J.; Tomsovic, S. (July 1993). "Postmodern Quantum Mechanics". Physics
Today.
7. ^ Gutzwiller, Martin C. (1990). Chaos in Classical and Quantum Mechanics. New York:
Springer-Verlag. ISBN 0387971734.
8. ^ abRudnick, Ze'ev (January 2008). "What is... Quantum Chaos" (PDF). Notices of the
American Mathematical Society. http://www.ams.org/notices/200801/tx080100032p.pdf.
9. ^ Berry, Michael (1989). "Quantum chaology, not quantum chaos". Physica Scripta40
(3): 335. Bibcode 1989PhyS...40..335B. doi:10.1088/0031-8949/40/3/013.
10. ^ 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.
11. ^ 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.
12. ^ 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.
13. ^ 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.
14. ^ a bPoulin, David. "A Rough Guide to Quantum Chaos" (PDF).
http://www.iqc.ca/publications/tutorials/chaos.pdf.
15. ^ Peres, A. (1995). Quantum Theory: Concepts and Methods. Dordrecht: Kluwer
Academic.
16. ^ 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
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Devaney, Robert L. (2003). Introduction to Chaotic Dynamical Systems. Westview Press.
ISBN 0813340853.
Hilborn, Robert C. (2004). "Sea gulls, butterflies, and grasshoppers: A brief history of the
butterfly effect in nonlinear dynamics". American Journal of Physics 72 (4): 425–427.
Bibcode 2004AmJPh..72..425H. doi:10.1119/1.1636492.
External links
Look up butterfly effect in Wiktionary, the free dictionary.
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The meaning of the butterfly: Why pop culture loves the 'butterfly effect,' and gets it
totally wrong, Peter Dizikes, Boston Globe, June 8, 2008
From butterfly wings to single e-mail (Cornell University)
New England Complex Systems Institute - Concepts: Butterfly Effect
The Chaos Hypertextbook. An introductory primer on chaos and fractals
ChaosBook.org. Advanced graduate textbook on chaos (no fractals)
Weisstein, Eric W., "Butterfly Effect" from MathWorld.
Read more: http://www.answers.com/topic/butterfly-effect-2#ixzz1U5txOlWr
The butterfly effect is a term used in Chaos Theory to describe how tiny variations can affect giant
systems, and complex systems, like weather patterns. The term butterfly effect was applied in Chaos
Theory to suggest that the wing movements of a butterfly might have significant repercussions on wind
strength and movements throughout the weather systems of the world, and theoretically, could cause
tornadoes halfway around the world.
What the butterfly effect seems to posit, is that the prediction of the behavior of any large system is
virtually impossible unless one could account for all tiny factors, which might have a minute effect on
the system. Thus large systems like weather remain impossible to predict because there are too many
unknown variables to count.
The term "butterfly effect" is attributed to Edward Norton Lorenz, a mathematician and meteorologist,
who was one of the first proponents of Chaos Theory. Though he had been working on the theory for
some ten years, with the principal question as to whether a seagulls’ wing movements changes the
weather, he changed to the more poetic butterfly in 1973.
A speech he delivered was titled, “Does the Flap of a Butterfly’s Wings in Brazil Set off a Tornado in
Texas.” Actually, fellow scientist, Philip Merilees created the title. Lorenz had failed to provide a title
for his speech.
The concept of small variations producing the butterfly effect actually predates science and finds its
home in science fiction. Writers like Ray Bradbury were particularly interested in the types of problems
that might occur if one traveled back in time, trailing anachronisms. Could small actions taken in the
past dramatically affect the future?
Fictional treatments of the butterfly effect as applies to time travel are numerous. Many cite the 2005
Butterfly Effect film as a good example of the possible negative changes that small behaviors in the past
could have on the future, if one could time travel. Actually, a better and more critically accepted
treatment of this concept is the 2000 film Frequency. In the film a father and son communicate over time
through radio waves and attempt to change the past for the good.
In human behavior, one can certainly see how small changes could render behavior, or another complex
system, extremely unpredictable. Small actions or experiences stored in the unconscious mind, could
certainly affect a person’s behavior in unexpected ways.
One looks at teen suicide for example, where no instance of previous depression has occurred. Loved
ones are often left wondering what the many small factors were that precipitated a suicide. Further,
people often agonize about the small details they did not see as possible factors for an unexpected
suicide.
However, there are plenty of ways that such a behavior would be unanswerable according to the
butterfly effect. Minute actions and experiences dating from childhood stored in the unconscious mind
are not accessible when a person has died, and they may be hard to access without hypnosis or therapy
when a person is living.
Whether used in science, fiction, or social sciences, the butterfly effect remains theory. However, it does
seem a reasonable explanation for the unpredictability of events. As it relates to human behavior, it does
suggest that even the smallest actions may have huge consequences for good or ill.