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The butterfly effect is a phrase that encapsulates the more technical notion of sensitive dependence on initial conditions in chaos theory. Small variations of the initial condition of a dynamical system may produce large variations in the long term behavior of the system. 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. Image File history File links Sensitive-dependency. ...
Image File history File links Sensitive-dependency. ...
In dynamical systems, an attractor is a set to which the system evolves after a long enough time. ...
Phase space of a dynamical system with focal stability. ...
The butterfly effect may refer to: The butterfly effect a phrase describing the operation of some chaotic dynamical systems. ...
A plot of the trajectory Lorenz system for values r = 28, σ = 10, b = 8/3 In mathematics and physics, chaos theory describes the behavior of certain nonlinear dynamical systems that under certain conditions exhibit a phenomenon known as chaos. ...
In mathematics, boundary conditions are imposed on the solutions of ordinary differential equations and partial differential equations, to fit the solutions to the actual problem. ...
A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. ...
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, for that matter, 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. Families Superfamily Hesperioidea: Hesperiidae Superfamily Papilionoidea: Papilionidae Pieridae Nymphalidae Lycaenidae Riodinidae A butterfly is an insect of the Order Lepidoptera, and belongs to one of the superfamilies Hesperioidea (the skippers) or Papilionoidea (all other butterflies). ...
Saturns atmosphere is made up of hydorgen, helium and methane ...
Union City, Oklahoma tornado (1973) A tornado is a violently rotating column of air which is in contact with both a cumulonimbus (or, in rare cases, cumulus) cloud base and the surface of the earth. ...
Recurrence, the approximate return of a system towards its initial conditions, together with the 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. In mathematics, a measure-preserving transformation T on a probability space is said to be ergodic if the only measurable sets invariant under T have measure 0 or 1. ...
Complex systems have a number of properties, some of which are listed below. ...
Weather is an all-encompassing term used to describe all of the many and varied phenomena that can occur in the atmosphere of a planet. ...
History Sensitive dependence on initial conditions was first described in the literature by Hadamard and popularized by Duhem's 1906 book. The term butterfly effect is related to the work of Lorenz, who 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 AAAS in 1972, Philip Merilees concocted Does the flap of a butterfly’s wings in Brazil set off a tornado in Texas? as a title. This page is a candidate for speedy deletion. ...
Pierre Maurice Marie Duhem (10 June 1861 – 14 September 1916) French physicist and philosopher of science. ...
Edward Norton Lorenz (born May 23, 1917), a research meteorologist at MIT, observed that minute variations in the initial values of variables in his primitive computer weather model (c. ...
1963 (MCMLXIII) was a common year starting on Tuesday (the link is to a full 1963 calendar). ...
New York Academy of Sciences is a society of some 20,000 scientists of all disciplines from 150 countries. ...
Families Superfamily Hesperioidea: Hesperiidae Superfamily Papilionoidea: Papilionidae Pieridae Nymphalidae Lycaenidae Riodinidae A butterfly is an insect of the Order Lepidoptera, and belongs to one of the superfamilies Hesperioidea (the skippers) or Papilionoidea (all other butterflies). ...
AAAS is an acronym. ...
1972 (MCMLXXII) was a leap year starting on Saturday (the link is to a full 1972 calendar). ...
Illustration | The butterfly effect in the Lorenz attractor | | time 0 ≤ t ≤ 30
(larger) | z coordinate
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 | | 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. | Image File history File links Download high resolution version (1038x766, 211 KB) Summary Created by XaosBits using Mathematica and POV-Ray. ...
Image File history File links Download high resolution version (746x760, 35 KB) Summary Created with Mathematica and Illustrator Top: the z coordinate for the Lorenz model. ...
Image File history File links TwoLorenzOrbitsSmall. ...
Image File history File links LorenzCoordinatesSmall. ...
A plot of the trajectory Lorenz system for values r=28, σ = 10, b = 8/3 The Lorenz attractor, introduced by Edward Lorenz in 1963, is a non-linear three-dimensional deterministic dynamical system derived from the simplified equations of convection rolls arising in the dynamical equations of the atmosphere. ...
Mathematical definition A dynamical system with evolution map ft displays sensitive dependence on initial conditions if points arbitrarily close become separate with increasing t. 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 A dynamical system is a concept in mathematics where a fixed rule describes the time dependence of a point in a geometrical space. ...
 The definition does not require that all points from a neighborhood separate from the base point x.
Popular media The concept of the Butterfly effect is sometimes used in popular media dealing with the idea of time travel, usually inaccurately. In the 1952 short story by Ray Bradbury, "A Sound of Thunder", the killing of a butterfly during the time of dinosaurs causes the future to change in subtle but meaningful ways: e.g., the spelling of English, and the outcome of a political election. According to the actual theory, however, 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. Time travel is a concept that has long fascinated humanity—whether it is Merlin experiencing time backwards, or religious traditions like Mohammeds trip to Jerusalem and ascent to heaven, returning before a glass knocked over had spilt its contents. ...
Ray Bradbury in 1945. ...
A Sound of Thunder is a science fiction short story by Ray Bradbury, first published in Colliers magazine in 1952. ...
In a Simpsons episode about Homer going back to the time of dinosaurs with a time machine (a la Bradbury's story), Homer commits intentional and unintentional violence in the past, violence which drastically changes the future (i.e., Homer's present).
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, 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. 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. The Butterfly Effect is a 2004 American sci-fi/drama movie starring Ashton Kutcher, Amy Smart, Eric Stoltz, and others, distributed by New Line Cinema. ...
Ashton Kutcher in The Butterfly Effect, 2004 Christopher Ashton Kutcher (born February 7, 1978) is an American actor, model, and television producer. ...
Another movie which explores the butterfly effect (though not advertised as such) is Sliding Doors. The movie observes two parallel life paths of a woman named Helen, played by Gwyneth Paltrow. These two paths diverge when Helen attempts to catch a commuter train. In one life path she catches the train, and in another she is delayed for just a few seconds and barely misses the train. This results in two dramatically different sets of events. Sliding Doors is a 1998 film written and directed by former actor Peter Howitt. ...
Gwyneth Paltrow and Jude Law at a press conference for at the 2000 Berlinale, photo by Michael Weiner Gwyneth Kate Paltrow (born September 28, 1972) is an Academy Award-winning American actress. ...
The Butterfly effect was also invoked by fictional mathematician Ian Malcolm in both the novel and film versions of Jurassic Park. He used it to explain the inherent instability of (among other things) an amusement park with dinosaurs as the attraction - although this interpretation can also be taken to mean that zoo animals will always escape and kill their captors. Leonhard Euler is considered by many people to be one of the greatest mathematicians of all time A mathematician is a person whose primary area of study and research is mathematics. ...
This article is about the 1993 hit movie. ...
Giraffes in Sydneys Taronga Zoo Free monkeys islands at the São Paulo Zoo Panda enclosure at Chiang Mai Zoo Visitors feeding and petting tamed marmots at the Parc Animalier des Pyrenées Sea lions at the Melbourne Zoo For other uses of the term Zoo, see Zoo...
See also A plot of the trajectory Lorenz system for values r = 28, σ = 10, b = 8/3 In mathematics and physics, chaos theory describes the behavior of certain nonlinear dynamical systems that under certain conditions exhibit a phenomenon known as chaos. ...
In engineering and mathematics, a dynamical system is a deterministic process in which a functions value changes over time according to a rule that is defined in terms of the functions current value. ...
External links This is about the university. ...
monochrom is an international art-technology-philosophy group (founded in 1993). ...
References - Robert L. Devaney (2003). Introduction to Chaotic Dynamical Systems. Westview Press. ISBN 0813340853.
- 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.
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