Butterfly effect



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. 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.

Theory
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 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. Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different. Of course the butterfly cannot literally cause a tornado. The kinetic energy in a tornado is enormously larger than the energy in the turbulence of a butterfly. The kinetic energy of a tornado is ultimately provided by the sun and the butterfly can only influence certain details of weather events in a chaotic manner.

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).

Origin of the concept and the term
The term "butterfly effect" itself is related to the work of Edward Lorenz, based in Chaos Theory and sensitive dependence on initial conditions, first described in the literature by Jacques Hadamard in 1890 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 Lorenz made popular the term. 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. 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.

Illustration

 * {|class="wikitable" width=535px

! colspan=3|The butterfly effect in the Lorenz attractor
 * colspan=2 align="center"| time 0 ≤ t ≤ 30 (larger)
 * align="center" | z coordinate (larger)
 * colspan=2 align="center"|[[Image:TwoLorenzOrbitsSmall.jpg]]
 * align="center"|[[Image:LorenzCoordinatesSmall.jpg]]
 * colspan=3| 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.
 * align="center" colspan=3| A Java animation of the Lorenz attractor shows the continuous evolution.
 * }
 * colspan=3| 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.
 * align="center" colspan=3| A Java animation of the Lorenz attractor shows the continuous evolution.
 * }
 * }

Mathematical definition
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
 * $$d(f^\tau(x), f^\tau(y)) > \delta \,.$$

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

Appearances in popular culture
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. 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.

Movies
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.

The 1998 British movie Sliding Doors runs two parallel stories of the same woman, Helen (Gwyneth Paltrow). In one universe, Helen manages to catch a London Underground train home on time, and in the other she misses it. This small event influenced her life dramatically.

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 surviving 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.

In the 1990 movie "Havana" with Robert Redford and Lena Olin, Redford even makes a direct reference to: "And a butterfly can flutter its wings over a flower in China and cause a hurricane in the Caribbean. I believe it. They can even calculate the odds. It just isn't likely and it takes so long." He's referring to the probability of the two of their characters ever getting together. Redford's character was a gambler in late 50's Cuba and Olin was spotted earlier in the movie looking at books on the Theory of Numbers and Probability in the apartment of Redford's character.

Literature and print
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.

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 thunderstorms."

Interactive media

 * 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.

Television
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 A Sound of Thunder). 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. This 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 United States) 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.

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 the team to discuss the "Chaos Theory": when combined, many seemingly innocuous events may have a deadly outcome, and closure is not always within reach.

The third season premiere of Heroes is entitled "The Butterfly Effect."

A Malcolm in the Middle episode shows Hal and Lois arguing about which one of them will take Malcolm and Reese to bowling and which one will stay at home with Dewey. After that, the episode will show two timelines: one where Lois takes them and another one where Hal takes them. An event from the timeline where Lois goes to the bowling is shown as a flashback in a later episode, implying that timeline to be the one in canon.