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For other uses, see Butterfly effect (disambiguation).
Point attractors in 2D phase space.
.The butterfly effect is a metaphor that encapsulates the concept of sensitive dependence on initial conditions in chaos theory; namely that small differences in the initial condition of a dynamical system may produce large variations in the long term behavior of the system.^ The idea of "The Butterfly Effect" comes from the science of chaos theory.

 ^ The Butterfly Effect theory describes the phenomenon of how small variations in a dynamic system can subsequently cause larger and more complex variations over the long term.
  • Butterfly Effect in Virtualized Environments | Virtualization Journal 10 February 2010 12:50 UTC virtualization.sys-con.com [Source type: General]
  • Butterfly Effect in Virtualized Environments | JAVA Developer's Journal 10 February 2010 12:50 UTC java.sys-con.com [Source type: General]
  • Butterfly Effect in Virtualized Environments | SYS-CON MEDIA 10 February 2010 12:50 UTC www.sys-con.com [Source type: Academic]

 ^ This article will explain this concept called "Sensitive Dependence on Initial Condition", otherwise know as the "Butterfly Effect".
  • Butterfly Effect - Chaos - Sensitive Dependence - Effect on the Future - Free Will - Time Travel 10 February 2010 12:50 UTC www.perkel.com [Source type: Original source]

 .Although this 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.^ This is sometimes presented as esoteric behavior, but can be 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.
  • BlogStar — The Labor Day Morning Movie: The Butterfly Effect 10 February 2010 12:50 UTC www.blogstar.com [Source type: FILTERED WITH BAYES]

 ^ 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.
  • BlogStar — The Labor Day Morning Movie: The Butterfly Effect 10 February 2010 12:50 UTC www.blogstar.com [Source type: FILTERED WITH BAYES]

 ^ In a deterministic setting the system would choose one or the other of these attractors depending on the initial conditions and would subsequently remain trapped therein.
  • Butterfly effect - Scholarpedia 10 February 2010 12:50 UTC www.scholarpedia.org [Source type: Academic]

 .The butterfly effect is a common trope in fiction when presenting scenarios involving time travel and with "what if" scenarios where one storyline diverges at the moment of a seemingly minor event resulting in two significantly different outcomes.^ The result was a completely different weather scenario.
  • The "Butterfly Effect" and Societal Transformation :: Contact :: Care2 Groups 10 February 2010 12:50 UTC www.care2.com [Source type: General]

 ^ As a result, The Butterfly Effect has no effect.
  • The Butterfly Effect - Ask.com 10 February 2010 12:50 UTC www.ask.com [Source type: General]

 ^ One Response to “The butterfly effect” .
  • The butterfly effect | Free Newsletter 10 February 2010 12:50 UTC www.earlytorise.com [Source type: General]

Contents

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.^ 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.
  • BlogStar — The Labor Day Morning Movie: The Butterfly Effect 10 February 2010 12:50 UTC www.blogstar.com [Source type: FILTERED WITH BAYES]

 ^ The concept of sensitive dependence on initial condition is very powerful indeed.
  • Butterfly Effect - Chaos - Sensitive Dependence - Effect on the Future - Free Will - Time Travel 10 February 2010 12:50 UTC www.perkel.com [Source type: Original source]

 ^ 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.
  • BlogStar — The Labor Day Morning Movie: The Butterfly Effect 10 February 2010 12:50 UTC www.blogstar.com [Source type: FILTERED WITH BAYES]

 .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.^ Like the weather, human systems are complex and people are sensitive to conditions.

 ^ Error growth and the prediction of complex systems .
  • Butterfly effect - Scholarpedia 10 February 2010 12:50 UTC www.scholarpedia.org [Source type: Academic]

 ^ 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.
  • BlogStar — The Labor Day Morning Movie: The Butterfly Effect 10 February 2010 12:50 UTC www.blogstar.com [Source type: FILTERED WITH BAYES]

Origin of the concept and the term

.The term "butterfly effect" itself is related to the work of Edward Lorenz, and 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].^ The idea of "The Butterfly Effect" comes from the science of chaos theory.

 ^ Lorenz’s broad strokes on the evolution of chaos theory have now been named The Law Of Sensitive Dependence Upon Initial Conditions.
  • Blog - The Butterfly Effect - Andy Andrews - Inspirational Author & Speaker 10 February 2010 12:50 UTC www.andyandrews.com [Source type: General]

 ^ This article will explain this concept called "Sensitive Dependence on Initial Condition", otherwise know as the "Butterfly Effect".
  • Butterfly Effect - Chaos - Sensitive Dependence - Effect on the Future - Free Will - Time Travel 10 February 2010 12:50 UTC www.perkel.com [Source type: Original source]

 He even later proposed that such phenomena could be common, say in meteorology. In 1898[2] Jacques Hadamard noted general divergence of trajectories in spaces of negative curvature, and Pierre Duhem discussed the possible general significance of this in 1908[3]. .The idea that one butterfly could eventually have a far-reaching ripple effect on subsequent historic events seems first to have appeared in a 1952 short story by Ray Bradbury about time travel (see Literature and print here) although Lorenz made the term popular.^ The butterfly effect was introduced in Ray Bradbury's 1952 short story "A Sound of Thunder," when time travelers change the world beyond measure by accidentally killing a butterfly in prehistoric times.
  • Barack Obama's Butterfly Effect - Political Punch 10 February 2010 12:50 UTC blogs.abcnews.com [Source type: General]

 ^ Butterfly Effect is one of my all-time favorite movies.
  • We Love Butterfly Effect! 10 February 2010 12:50 UTC www.squidoo.com [Source type: FILTERED WITH BAYES]

 ^ It’s that Butterfly Effect is a horror movie first, and a time travel movie second.
  • : RevolutionSF - The Butterfly Effect : Review 10 February 2010 12:50 UTC www.revolutionsf.com [Source type: General]

 .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 computer would hold.^ 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 computer would hold.
  • The "Butterfly Effect" and Societal Transformation :: Contact :: Care2 Groups 10 February 2010 12:50 UTC www.care2.com [Source type: General]

 ^ The print out, however, rounded the results to three numbers past the decimal point and so he input .506 instead of entering the full .506127.

 ^ What Lorenz discovered - now known as the butterfly effect - is that with even an almost infinitely power computer and with weather sensors every 6 inches - that the weather still can't be predicted because any variation at all would eventually lead to a radically different result given enough time.
  • Butterfly Effect - Chaos - Sensitive Dependence - Effect on the Future - Free Will - Time Travel 10 February 2010 12:50 UTC www.perkel.com [Source type: Original source]

 .The result was a completely different weather scenario.^ The result was a completely different weather scenario.
  • The "Butterfly Effect" and Societal Transformation :: Contact :: Care2 Groups 10 February 2010 12:50 UTC www.care2.com [Source type: General]

 ^ When he made slight changes in the initial conditions that he entered, the resulting weather conditions were very different.
  • THE BUTTERFLY EFFECT 10 February 2010 12:50 UTC www.greatdreams.com [Source type: FILTERED WITH BAYES]

 ^ The tiny difference in starting values produced completely different results.

 [4] Lorenz published his findings in a 1963 paper for the New York Academy of Sciences noting 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, upon failing 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.^ On December 29, 1972 Lorenz presented a talk in the 139th meeting of the American Association for the Advancement of Science held in Washington, D.C. entitled .
  • Butterfly effect - Scholarpedia 10 February 2010 12:50 UTC www.scholarpedia.org [Source type: Academic]

 ^ Lorenz gave a lecture in 1972 titled: “ Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?

 ^ He posed the question: Can the flap of a butterfly's wings over Brazil spawn a tornado over Texas?
  • THE BUTTERFLY EFFECT 10 February 2010 12:50 UTC www.greatdreams.com [Source type: FILTERED WITH BAYES]

 .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.^ In 1969 Edward Lorenz, the famed meteorologist and Chaos Theory proponent, introduced the concept now known as the "Butterfly Effect" when he posed the famous question: "Does the flap of a butterfly's wings in Brazil set off a tornado in Texas?"
  • Butterfly Effect in Virtualized Environments | SYS-CON MEDIA 10 February 2010 12:50 UTC www.sys-con.com [Source type: Academic]

 ^ Peppermill, you titled your post "The Butterfly Effect" which refers to the concept that a small movement--like a butterfly's wing--can have an increasing effects as they spread throughout the environment.  A flap of a butterfly's wing can cause (or prevent) a hurricane.
  • Re: The Butterfly Effect! - Barnes & Noble Book Clubs 10 February 2010 12:50 UTC bookclubs.barnesandnoble.com [Source type: FILTERED WITH BAYES]

 ^ Lorenz gave a lecture in 1972 titled: “ Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?

 [5]
.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 a certain location.^ The phrase refers to the Chaos Theory work of Edward Lorenz, suggesting 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 a certain location.
  • What do American Idol & the Butterfly Effect Have to do With Your Destiny? | Sparkplug CEO 10 February 2010 12:50 UTC www.sparkplugging.com [Source type: General]

 ^ The phrase is derived from the understanding that each and every tiny change (a flap of a butterfly's wings...

 ^ The phrase 'butterfly effect' refers to the idea that a butterfly's wings might create tiny changes in the atmosphere that ultimately cause a tornado to appear.
  • We Love Butterfly Effect! 10 February 2010 12:50 UTC www.squidoo.com [Source type: FILTERED WITH BAYES]

 .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).^ Small changes can lead to huge consequences.
  • Butterfly Effect | dougbelshaw.com/blog 10 February 2010 12:50 UTC dougbelshaw.com [Source type: FILTERED WITH BAYES]

 ^ Small changes, if properly situated, can cascade and have large scale effects.
  • The "Butterfly Effect" and Societal Transformation :: Contact :: Care2 Groups 10 February 2010 12:50 UTC www.care2.com [Source type: General]

 ^ Small changes -> large effects .
  • Butterfly Effect | dougbelshaw.com/blog 10 February 2010 12:50 UTC dougbelshaw.com [Source type: FILTERED WITH BAYES]

 .Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different.^ The propensity of a system to be sensitive to initial conditions, and for the system becoming unpredictable over time, this idea gave rise to the notion of a butterfly flapping its wings in one area of the world causing a tornado in another remote area of the world...
  • Bodybuilding.com - Butterfly Effect: The Basics Of The Thyroid - Part 1. - Mark Subsinsky 10 February 2010 12:50 UTC www.bodybuilding.com [Source type: FILTERED WITH BAYES]

 ^ It goes something like this: "does the flap of a butterfly's wing in Brazil set off a tornado in Texas?"
  • Frank's Reel Movie Reviews - Movie Review - The Butterfly Effect 10 February 2010 12:50 UTC www.franksreelreviews.com [Source type: FILTERED WITH BAYES]

 ^ By the time of the 1972 meeting, he had examined and refined that idea for his talk, "Predictability: Does the Flap of a Butterfly's Wings in Brazil set off a Tornado in Texas?"
  • Butterfly Effect 10 February 2010 12:50 UTC www.rsc.org [Source type: Academic]

 .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.^ The butterfly flapping its wings may cause a hurricane in Japan.

 ^ In 1969 Edward Lorenz, the famed meteorologist and Chaos Theory proponent, introduced the concept now known as the "Butterfly Effect" when he posed the famous question: "Does the flap of a butterfly's wings in Brazil set off a tornado in Texas?"
  • Butterfly Effect in Virtualized Environments | SYS-CON MEDIA 10 February 2010 12:50 UTC www.sys-con.com [Source type: Academic]

 ^ It goes something like this: "does the flap of a butterfly's wing in Brazil set off a tornado in Texas?"
  • Frank's Reel Movie Reviews - Movie Review - The Butterfly Effect 10 February 2010 12:50 UTC www.franksreelreviews.com [Source type: FILTERED WITH BAYES]

Illustration

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 only by 10−5 in the x-coordinate.^ The red and blue colored lines of this figure correspond to the succession in time of variable emanating from two initial conditions differing from the reference trajectory (black line) by errors of amplitudes and , respectively.
  • Butterfly effect - Scholarpedia 10 February 2010 12:50 UTC www.scholarpedia.org [Source type: Academic]

 ^ In a deterministic setting the system would choose one or the other of these attractors depending on the initial conditions and would subsequently remain trapped therein.
  • Butterfly effect - Scholarpedia 10 February 2010 12:50 UTC www.scholarpedia.org [Source type: Academic]

 ^ Any one can ask any of our Corporate Vendors that we do business with and they will tell you that I am tough but at the same time I give credit due for an execellent job done and show our loyalty in return.
  • Customer Service and the Butterfly Effect 10 February 2010 12:50 UTC michaelhyatt.com [Source type: General]

 .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.^ Red and blue lines denote the trajectories emanating from two initial conditions differing from the reference trajectory by (red curve) and (blue curve).
  • Butterfly effect - Scholarpedia 10 February 2010 12:50 UTC www.scholarpedia.org [Source type: Academic]

 ^ The tiniest of changes of the initial inputs – from .506127 to .506 – resulted in two very different weather predictions.

 ^ I remember this funny conversation between two divers on how to tell the difference: .

 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.^ See the Lorenz Attractor in Action Java Applet THE MANDELBROT SET CROP FORMATION .
  • THE BUTTERFLY EFFECT 10 February 2010 12:50 UTC www.greatdreams.com [Source type: FILTERED WITH BAYES]

Mathematical definition

.A dynamical system with evolution map ft displays sensitive dependence on initial conditions if points arbitrarily close become separate with increasing t.^ The concept of sensitive dependence on initial condition is very powerful indeed.
  • Butterfly Effect - Chaos - Sensitive Dependence - Effect on the Future - Free Will - Time Travel 10 February 2010 12:50 UTC www.perkel.com [Source type: Original source]

 ^ Lorenz’s broad strokes on the evolution of chaos theory have now been named The Law Of Sensitive Dependence Upon Initial Conditions.
  • Blog - The Butterfly Effect - Andy Andrews - Inspirational Author & Speaker 10 February 2010 12:50 UTC www.andyandrews.com [Source type: General]

 ^ Ridiculed for a generation, the butterfly effect was eventually proved by scientists and given the status of a law: ”The law of sensitive dependence on initial conditions.” It basically means that everything you do—and don’t do—matters.
  • Customer Service and the Butterfly Effect 10 February 2010 12:50 UTC michaelhyatt.com [Source type: General]

 If M is the state space for the map ft, then ft displays sensitive dependence to initial conditions if there is a δ>0 such that for every point x∈M and any neighborhood N containing x there exist a point y from that neighborhood N and a time τ such that the distance
d(f^ au(x), f^ au(y)) > \delta \,.
The definition does not require that all points from a neighborhood separate from the base point x.

Examples in semiclassical and quantum physics

.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.^ This butterfly effect has been studied mathematically.
  • THE BUTTERFLY EFFECT 10 February 2010 12:50 UTC www.greatdreams.com [Source type: FILTERED WITH BAYES]

 ^ This article will explain this concept called "Sensitive Dependence on Initial Condition", otherwise know as the "Butterfly Effect".
  • Butterfly Effect - Chaos - Sensitive Dependence - Effect on the Future - Free Will - Time Travel 10 February 2010 12:50 UTC www.perkel.com [Source type: Original source]

 ^ The concept of sensitive dependence on initial condition is very powerful indeed.
  • Butterfly Effect - Chaos - Sensitive Dependence - Effect on the Future - Free Will - Time Travel 10 February 2010 12:50 UTC www.perkel.com [Source type: Original source]

 [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.^ The moment of conception is extremely sensitive to initial condition.
  • Butterfly Effect - Chaos - Sensitive Dependence - Effect on the Future - Free Will - Time Travel 10 February 2010 12:50 UTC www.perkel.com [Source type: Original source]

 ^ The concept of sensitive dependence on initial condition is very powerful indeed.
  • Butterfly Effect - Chaos - Sensitive Dependence - Effect on the Future - Free Will - Time Travel 10 February 2010 12:50 UTC www.perkel.com [Source type: Original source]

 ^ Lorenz’s broad strokes on the evolution of chaos theory have now been named The Law Of Sensitive Dependence Upon Initial Conditions.
  • Blog - The Butterfly Effect - Andy Andrews - Inspirational Author & Speaker 10 February 2010 12:50 UTC www.andyandrews.com [Source type: General]

 [11]
.Other authors suggest that the butterfly effect can be observed in quantum systems.^ "The butterfly effect is the idea that in a chaotic system, a very small change to the system applied at a certain point in time makes the future change in a very dramatic way.
  • THE BUTTERFLY EFFECT 10 February 2010 12:50 UTC www.greatdreams.com [Source type: FILTERED WITH BAYES]

 ^ The Butterfly Effect theory describes the phenomenon of how small variations in a dynamic system can subsequently cause larger and more complex variations over the long term.
  • Butterfly Effect in Virtualized Environments | JAVA Developer's Journal 10 February 2010 12:50 UTC java.sys-con.com [Source type: General]

 ^ Simply put, the butterfly effect is the notion that something as small as a flap of a butterfly’s wings can make a big impact – like causing a tornado on the other side of the world.

 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.^ "The butterfly effect is the idea that in a chaotic system, a very small change to the system applied at a certain point in time makes the future change in a very dramatic way.
  • THE BUTTERFLY EFFECT 10 February 2010 12:50 UTC www.greatdreams.com [Source type: FILTERED WITH BAYES]

 ^ We see that adults shall master to whatever level that they can in this given life; and ascending children shall be born that shall anchor a new vibrational level for all to operate within.
  • The "Butterfly Effect" and Societal Transformation :: Contact :: Care2 Groups 10 February 2010 12:50 UTC www.care2.com [Source type: General]

 ^ And that is that any small perturbation, any small tap given to a chaotic system is amplified quickly in time."
  • THE BUTTERFLY EFFECT 10 February 2010 12:50 UTC www.greatdreams.com [Source type: FILTERED WITH BAYES]

 [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.^ Just as the Brazilian butterfly may set off a tornado a world away, any change to the virtualized objects that are the components of the virtual infrastructure can have severe and unintended consequences to virtualized systems, networks, and data centers.
  • Butterfly Effect in Virtualized Environments | Virtualization Journal 10 February 2010 12:50 UTC virtualization.sys-con.com [Source type: General]
  • Butterfly Effect in Virtualized Environments | JAVA Developer's Journal 10 February 2010 12:50 UTC java.sys-con.com [Source type: General]
  • Butterfly Effect in Virtualized Environments | SYS-CON MEDIA 10 February 2010 12:50 UTC www.sys-con.com [Source type: Academic]

 ^ Small variations of the initial condition of a nonlinear dynamical system may produce large variations in the long term behavior of the system.
  • We Love Butterfly Effect! 10 February 2010 12:50 UTC www.squidoo.com [Source type: FILTERED WITH BAYES]

 ^ The phrase 'butterfly effect' refers to the idea that a butterfly's wings might create tiny changes in the atmosphere that ultimately cause a tornado to appear.
  • We Love Butterfly Effect! 10 February 2010 12:50 UTC www.squidoo.com [Source type: FILTERED WITH BAYES]

 [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.^ Lorenz’s broad strokes on the evolution of chaos theory have now been named The Law Of Sensitive Dependence Upon Initial Conditions.
  • Blog - The Butterfly Effect - Andy Andrews - Inspirational Author & Speaker 10 February 2010 12:50 UTC www.andyandrews.com [Source type: General]

 ^ Small variations of the initial condition of a nonlinear dynamical system may produce large variations in the long term behavior of the system.
  • We Love Butterfly Effect! 10 February 2010 12:50 UTC www.squidoo.com [Source type: FILTERED WITH BAYES]

 ^ There is a term in chaos theory describing the how infinitesmal variations of the initial condition of a dynamical system may produce large variations in the long term behavior of the system.
  • Barack Obama's Butterfly Effect - Political Punch 10 February 2010 12:50 UTC blogs.abcnews.com [Source type: General]

 [8][17]

Appearances in popular culture

Main article: Butterfly effect in popular culture

See also

References

  1. ^ Some Historical Notes: History of Chaos Theory
  2. ^ Some Historical Notes: History of Chaos Theory
  3. ^ Some Historical Notes: History of Chaos Theory
  4. ^ Mathis, Nancy: "Storm Warning: The Story of a Killer Tornado", page x. Touchstone, 2007. ISBN 0-7432-8053-2
  5. ^ "Butterfly Effects - Variations on a Meme". clearnightsky.com. http://clearnightsky.com/node/428. 
  6. ^ Postmodern Quantum Mechanics, EJ Heller, S Tomsovic, Physics Today, July 1993
  7. ^ Martin C. Gutzwiller, Chaos in Classical and Quantum Mechanics, (1990) Springer-Verlag, New York ISBN=0-387-97173-4.
  8. ^ a b What is... Quantum Chaos by Ze'ev Rudnick (January 2008, Notices of the American Mathematical Society)
  9. ^ Quantum chaology, not quantum chaos, Michael Berry, 1989, Phys. Scr., 40, 335-336 doi: 10.1088/0031-8949/40/3/013.
  10. ^ Martin C. Gutzwiller (1971). "Periodic Orbits and Classical Quantization Conditions". Journal of Mathematical Physics 12: 343. doi:10.1063/1.1665596
  11. ^ Closed-orbit theory of oscillations in atomic photoabsorption cross sections in a strong electric field. II. Derivation of formulas, J Gao and JB Delos, Phys. Rev. A 46, 1455 - 1467 (1992)
  12. ^ Quantum Chaotic Environments, the Butterfly Effect, and Decoherence, Zbyszek P. Karkuszewski, Christopher Jarzynski, and Wojciech H. Zurek, Physical Review Letters VOLUME 89, NUMBER 17 (2002).
  13. ^ Exponential speed-up with a single bit of quantum information: Testing the quantum butterfly effect, David Poulin, Robin Blume-Kohout, Raymond Laflamme, and Harold Ollivier http://arxiv.org/PS_cache/quant-ph/pdf/0310/0310038v1.pdf
  14. ^ A Rough Guide to Quantum Chaos, David Poulin, http://www.iqc.ca/publications/tutorials/chaos.pdf
  15. ^ A. Peres, Quantum Theory: Concepts and Methods ~Kluwer Academic, Dordrecht, 1995.
  16. ^ Quantum amplifier: Measurement with entangled spins, Jae-Seung Lee and A. K. Khitrin, JOURNAL OF CHEMICAL PHYSICS VOLUME 121, NUMBER 9 (2004)
  17. ^ A Rough Guide to Quantum Chaos, David Poulin, http://www.iqc.ca/publications/tutorials/chaos.pdf

Further reading

  • Robert L. Devaney (2003). .Introduction to Chaotic Dynamical Systems.^ In a chaotic dynamical system this happens transiently along the directions associated to its vanishing Lyapunov exponents, prior to the stage where the directions associated to the positive exponents are taking over.
    • Butterfly effect - Scholarpedia 10 February 2010 12:50 UTC www.scholarpedia.org [Source type: Academic]

     Westview Press. ISBN 0-8133-4085-3.     
  • Robert C. Hilborn (2004). "Sea gulls, butterflies, and grasshoppers: A brief history of the butterfly effect in nonlinear dynamics". American Journal of Physics 72: 425–427. doi:10.1119/1.1636492. 

External links

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