In chaos theory, the butterfly effect is the sensitive dependence on initial conditions in which a small change in one state of a deterministic nonlinear system can result in large differences in a later state.
The term is closely associated with the work of mathematician and meteorologist Edward Norton Lorenz. He noted that the butterfly effect is derived from the metaphorical example of the details of a tornado (the exact time of formation, the exact path taken) being influenced by minor perturbations such as a distant butterfly flapping its wings several weeks earlier. Lorenz originally used a seagull causing a storm but was persuaded to make it more poetic with the use of a butterfly and tornado by 1972.[1][2] He discovered the effect when he observed runs of his weather model with initial condition data that were rounded in a seemingly inconsequential manner. He noted that the weather model 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.[3]
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The idea that small causes may have large effects in weather was earlier acknowledged by French mathematician and engineer Henri Poincar. American mathematician and philosopher Norbert Wiener also contributed to this theory. Lorenz's work placed the concept of instability of the Earth's atmosphere onto a quantitative base and linked the concept of instability to the properties of large classes of dynamic systems which are undergoing nonlinear dynamics and deterministic chaos.[4]
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. "A Sound of Thunder" features time travel.[8]
In 1963, Lorenz published a theoretical study of this effect in a highly cited, seminal paper called Deterministic Nonperiodic Flow[3][11] (the calculations were performed on a Royal McBee LGP-30 computer).[12][13] Elsewhere he stated:
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.[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.[14]
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.[15]
Some scientists have since argued that the weather system is not as sensitive to initial conditions as previously believed.[16] David Orrell argues that the major contributor to weather forecast error is model error, with sensitivity to initial conditions playing a relatively small role.[17][18] 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.[19] Recent studies using generalized Lorenz models that included additional dissipative terms and nonlinearity suggested that a larger heating parameter is required for the onset of chaos.[20]
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 Poincar), the butterfly metaphor was originally applied[1] to work he published in 1969[21] 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.[22][23]
In the book entitled The Essence of Chaos published in 1993,[24] 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 .[3] 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.[25]
The butterfly effect is most familiar in terms of weather; it can easily be demonstrated in standard weather prediction models, for example. The climate scientists James Annan and William Connolley explain that chaos is important in the development of weather prediction methods; models are sensitive to initial conditions. They add the caveat: "Of course the existence of an unknown butterfly flapping its wings has no direct bearing on weather forecasts, since it will take far too long for such a small perturbation to grow to a significant size, and we have many more immediate uncertainties to worry about. So the direct impact of this phenomenon on weather prediction is often somewhat wrong."[27] The two kinds of butterfly effects, including the sensitive dependence on initial conditions,[3] and the ability of a tiny perturbation to create an organized circulation at large distances,[1] are not exactly the same.[28] A comparison of the two kinds of butterfly effects[1][3] and the third kind of butterfly effect[21][22][23] has been documented.[29] In recent studies,[25][30] it was reported that both meteorological and non-meteorological linear models have shown that instability plays a role in producing a butterfly effect, which is characterized by brief but significant exponential growth resulting from a small disturbance.
According to Lighthill (1986),[31] the presence of SDIC (commonly known as the butterfly effect) implies that chaotic systems have a finite predictability limit. In a literature review,[32] it was found that Lorenz's perspective on the predictability limit can be condensed into the following statement:
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.[43][44] Some authors have argued that extreme (exponential) dependence on initial conditions is not expected in pure quantum treatments;[45][46] however, the sensitive dependence on initial conditions demonstrated in classical motion is included in the semiclassical treatments developed by Martin Gutzwiller[47] and John B. Delos and co-workers.[48] The random matrix theory and simulations with quantum computers prove that some versions of the butterfly effect in quantum mechanics do not exist.[49]
Other authors suggest that the butterfly effect can be observed in quantum systems. Zbyszek P. 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.[50] David 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".[51] 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.[52][53] This quantum butterfly effect has been demonstrated experimentally.[54] Quantum and semiclassical treatments of system sensitivity to initial conditions are known as quantum chaos.[45][52]
The Butterfly Effect is a 2004 American science fiction thriller film written and directed by Eric Bress and J. Mackye Gruber. It stars Ashton Kutcher, Amy Smart, Eric Stoltz, William Lee Scott, Elden Henson, Logan Lerman, Ethan Suplee, and Melora Walters. The title refers to the butterfly effect.
Roger Ebert wrote that he "enjoyed The Butterfly Effect, up to a point" and that the "plot provides a showcase for acting talent, since the actors have to play characters who go through wild swings." However, Ebert said that the scientific notion of the butterfly effect is used inconsistently: Evan's changes should have wider reverberations.[8] Sean Axmaker of the Seattle Post-Intelligencer called it a "metaphysical mess", criticizing the film's mechanics for being "fuzzy at best and just plain sloppy the rest of the time".[9] Mike Clark of USA Today also gave the film a negative review, stating, "Normally, such a premise comes off as either intriguing or silly, but the morbid subplots (there's prison sex, too) prevent Effect from becoming the unintentional howler it might otherwise be."[10] Additionally, Ty Burr of The Boston Globe went as far as saying, "whatever train-wreck pleasures you might locate here are spoiled by the vile acts the characters commit."[11] 2351a5e196
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