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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 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.

Contents 1 Illustration 2 Mathematical definition 3 Popular media 3.1 Movies 3.2 Music 3.3 Literature and print 3.4 Interactive media 3.5 Television 4 References 5 See also 6 External links

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.

A dynamical system with evolution map 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 f^t, then 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

The definition does not require that all points from a neighborhood separate from the base point x.

The 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, so that 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 in time, which would create a phenomenon known as time paradox.

In 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 movie A Sound of Thunder (2005) is directly based on the 1952 science fiction short story of the same name by Ray Bradbury. As in the movie, when a hunter sent back to the prehistoric era carelessly kills a single butterfly, he causes a chain reaction that unpredictably alters history in disastrous ways.

British alternative rock band Muse's third album Absolution contains a song entitled "Butterflies and Hurricanes" which lyrically discusses the butterfly effect and describes how individuals can make a huge difference to major world events.

Portuguese gothic metal band Moonspell's fourth full length album The Butterfly Effect is inspired by chaos theory.

The Butterfly Effect is an Australian hard rock band from Brisbane.

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.^[3]

The butterfly effect was invoked by fictional chaotician Ian Malcolm in both the novel Jurassic Park and subsequent film adaptation. He used it to explain the inherent instability of (among other things) an amusement park with dinosaurs as the attraction.

In Terry Pratchett's "Men at Arms," Zorgo the Retrophrenologist approves of hunting butterflies with crossbows, claiming that he "suppose(s) it stops them creating all these damn thuderstorms."

The webcomic Kevin and Kell refers to Bradbury in the March 10, 1998 strip, which has Coney eating a butterfly while the family is in the Stone Age. A caption reads "When they return to 1998, they'll discover that a writer named Ray Bradbury never existed".

In the videogame Second Sight, main character John Vattic is able to change the present by having flashbacks to six months earlier, where he does things differently, affecting the future; only he remembers the alternate futures. The ending reveals that the 'past' sections are actually the present and the 'present' sections are John having precognitive visions of the future.

In The Simpsons Halloween episode, "Time and Punishment", Homer repeatedly travels back to the time of dinosaurs with a time machine (à la Bradbury's story). Each time there, Homer's actions (involving intentional and unintentional violence) drastically alter the current universe. Some of the changes include: A world dictator, a universe where his family is rich and classy and it rains donuts, and a seemingly normal universe, with the exception of everyone having long reptilian tongues.

In the Family Guy episode "Meet the Quagmires", Peter, with the help of Death, repeatedly travels back to the 80's to live up his teenage years and cancel a date with Lois which leads to her marrying Quagmire and Peter marrying Molly Ringwald among other things, causing two drastic changes of the present (Chevy Chase is host of The Tonight Show, and Al Gore is president of the U.S) and finally when things seem normal again it turns out Roger from American Dad! is living with them.

In a 2004 television episode of comedy sitcom Scrubs called "My Butterfly", the episode is shown in two parts: The first in which a butterfly lands on a woman sitting in the hospital's waiting room, and the second where time is rewound and the butterfly instead lands on the man next to her. Both halves of the episode show the noticeably (albeit sensationally) different outcomes that stem directly from the original choice of landing locations of this butterfly^[4].

In a first-season episode of the stop-motion animation show "Robot Chicken" titled Operation: Rich in Spirit there is a sketch where a young boy tries to explain the butterfly effect to a young girl. When the young girl squishes the butterfly, it causes earthquakes in Japan. A Japanese woman retaliates, stepping on a butterfly, which causes a volcano to erupt behind the children. The boy retaliates as well, ripping a butterfly in half, which causes Godzilla to terrorize Japan.

In a second season episode of CSI titled "Chaos Theory", the entire CSI team investigates a disappearance of a young woman at a local university. Forensics leads them to possible suspects, and possible suspects all have probable motives, but nothing seems to pan out. This leads our team to discuss the "Chaos Theory:" When combined, many seemingly-innocuous events may have a deadly outcome, and closure is not always within reach. Gil Grissom, played by William Petersen, states, "A butterfly flaps its wings in Brazil, we get a hurricane off the coast of Florida."

• Causality
• Chain reaction
• Determinism
• Domino effect
• Dynamical systems
• Fractal
• Rube Goldberg machine

• Butterfly Effect (Mathematical Recreations)
• 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.
• The Butterfly Effect. New Line Cinema's feature film The Butterfly Effect IMDB
• Eric W. Weisstein, Butterfly Effect at MathWorld.