Butterfly effect
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Categories: Articles with trivia sections from February 2008 | Stability theory | Chaos theory | Physical phenomena
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Image:Sensitive-dependency.svg
Point attractors in 2D phase space.
For other uses, see Butterfly effect (disambiguation).
The butterfly effect is a phrase which encapsulates the more technical notion of sensitive dependence on initial conditions in chaos theory. Small variations of the initial condition of a nonlinear dynamical system may produce large variations in the long term behavior of the system. So 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. The phrase refers to the idea that a butterfly's wings might create tiny changes in the atmosphere that ultimately cause a tornado to appear (or prevent a tornado from appearing). The flapping wing represents a small change in the initial condition of the system, which causes a chain of events leading to large-scale phenomena. Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different. 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). Sensitive dependence on initial conditions was first described in the literature by Jacques Hadamard in 1890[1] and popularized by Pierre Duhem's 1906 book. The idea that one butterfly could have a far-reaching ripple effect on subsequent events seems first to have appeared in a 1952 short story by Ray Bradbury about time travel (see Popular Media below), although the term "butterfly effect" itself is related to the work of Edward Lorenz. 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 result was a completely different weather scenario.[2] Lorenz published his findings in a 1963 paper for the New York Academy of Sciences noted 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.
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Mathematical definitionA dynamical system with evolution map Failed to parse (Missing texvc executable; please see math/README to configure.): f^t displays sensitive dependence on initial conditions if points arbitrarily close become separate with increasing t. If M is the state space for the map Failed to parse (Missing texvc executable; please see math/README to configure.): f^t , then Failed to parse (Missing texvc executable; please see math/README to configure.): f^t 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
Popular mediaThe concept of the butterfly effect is sometimes used in popular media dealing with the idea of time travel, usually inaccurately. Most time travel depictions simply fail to address butterfly effects. According to the actual theory, if history could be "changed" at all (so that one is not invoking something like the Novikov self-consistency principle which would ensure a fixed self-consistent timeline), the mere presence of the time travelers in the past would be enough to change short-term events (such as the weather) and would also have an unpredictable impact on the distant future. Therefore, no one who travels into the past could ever return to the same version of reality he or she had come from and could have therefore not been able to travel back in time in the first place, which would create a phenomenon known as time paradox. MoviesIn arguably the earliest illustration of the butterfly effect in a story on film, an angel in It's a Wonderful Life (1946) shows George Bailey how rewriting history so that George was never born would detrimentally affect the lives of everyone in his hometown. In a subtle butterfly effect, snow is falling in one version of reality but not the other.[1][2] The French film Le Battement d'ailes du papillon (2000), translated as Happenstance in the English release, makes direct references to the butterfly effect in title, dialogue, and theme. In many cases, minor and seemingly inconsequential actions in the past are extrapolated over time and can have radical effects on the present time of the main characters. In the movie The Butterfly Effect (2004), Evan Treborn (Ashton Kutcher), when reading from his adolescent journals, is able to essentially "redo" parts of his past. As he continues to do this, he realizes that even though his intentions are good, the actions he takes always have unintended consequences. Despite its title, however, this movie does not seriously explore the implications of the butterfly effect; only the lives of the principal characters seem to change from one scenario to another. The greater world around them is mostly unaffected. Furthermore, the changes made in the past of the principal character are far from minor and in that sense the title of the film is a misnomer. An element of the butterfly effect in general terms is that differences in start conditions for different scenario outcomes are virtually undetectable, and consequences are not related to cause in a directly apparent way. On the other hand, in the movie "Run Lola Run" (Lola rennt in German-1998), the butterfly effect is represented more clearly. There, minor and almost sub-conscious actions in everyday life can be seen to have gross and wide spread effects upon the future. For example, the fact that Lola bumps into someone instead of passing by may lead to a painful death after suffering paralysis. As such, seemingly inconsequential actions can be seen to have drastic long-term results. The second episode in the "Back to the Future" trilogy also vividly illustrates the cascading and broad effects of what seemed a minor change in the course of events: because the loathsome Biff Tannen accidentally gets his hands on a record book from 2015, he is able to grow rich and corrupt "Marty Mcfly"'s home town. When McFly (Michael J. Fox) returns to 1985, he finds it utterly degraded from what had used to be. In the 2000 movie Frequency, a son, John Sullivan (James Caviezel), is presented with an opportunity to prevent the death of his father, Frank Sullivan (Dennis Quaid), through a miracle of nature in which they were both able to communicate across time 30 years using the same am radio, transmitting the signal via a freak occurrence of the Northern Lights. This one action, however, had several undesirable consequences, including the murder of his mother by a vicious killer known as the Nightingale who was supposed to have never been caught. In the original timeline, when the killer is lying unconscious in the hospital, he dies from a reaction of two medicines that were administered intravenously into his system. This was due to an oversight on his medical chart, in which the attending male nurse overlooked the fact that it stated that the patient had received a certain medication which could not be mixed with the other. In the alternate timeline, Frank visits his wife, a nurse, at the hospital immediately after surving the fire in which he was supposed to die. She alters her routine slightly to see him, and of all things she then sees the wrong medication being administered to the killer. She prevents this from happening, and so the killer survives to murder not only her, but several more people; all nurses. Also, this film illustrates a theoretical side effect of the butterfly effect, where John is able to remember the original future time, as well as other alternate futures that were created each time his father changed something in the past. Literature and print
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