Butterfly effect

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For other uses, see Butterfly effect (disambiguation).

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 (but not 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).

1 History
2 Illustration
3 Mathematical definition
4 Popular media
5 References
6 See also
7 External links

[edit] History

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, 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 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.

[edit] Illustration

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.

[edit] 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.

[edit] Popular media

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.

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]

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] 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 change the future (i.e. Homer's present; for example, Homer squishing a mosquito results in Ned Flanders ruling the world, and Homer wiping out the dinosaurs with a sneeze results in his extraordinary wealth, well-behaved kids, his sisters-in-law dead, and donuts to rain from the sky).[4]

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 claims "When they return to 1998, they'll discover that a writer named Ray Bradbury never existed".

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. 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. [5]

The butterfly effect was also invoked by fictional chaotician 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.^{[citation needed]}

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 person 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.[6]

[edit] References

^ http://www.wolframscience.com/reference/notes/971c

Robert L. Devaney (2003). Introduction to Chaotic Dynamical Systems. Westview Press. ISBN 0-8133-4085-3.
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.