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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. 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.

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).

Although a butterfly flapping its wings has remained constant in the expression of this concept, the location of the butterfly, the consequences, and the location of the consequences have varied widely.

The butterfly effect in the Lorenz attractor time 0 ≤ t ≤ 30 TwoLorenzOrbits.jpg z coordinate LorenzCoordinatesBig.png 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. ## 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 term 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 a time paradox. The butterfly effect was also the name of the second episode of the third season of the NBC television show Heroes (TV_series). The character of Angela Petrelli says "You don't screw with time. It's called the butterfly effect. You step on a butterfly today, and three years from now a million people are wiped out."

## See also

## References

## Further reading

## External links

- The meaning of the butterfly: Why pop culture loves the 'butterfly effect,' and gets it totally wrong, Peter Dizikes, Boston Globe, June 8, 2008
- 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.
- Glass Bead Game using the Butterfly Effect
- The Lorenz Butterfly

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Last updated on Saturday October 11, 2008 at 14:51:37 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Saturday October 11, 2008 at 14:51:37 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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