Butterfly effect

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The butterfly effect ( English butterfly effect is) a phenomenon of the nonlinear dynamics . It occurs in non-linear dynamic , deterministic systems and manifests itself in the fact that it cannot be foreseen how small changes in the initial conditions of the system will affect the development of the system in the long term.

The eponymous illustration of this effect using the example of the weather comes from Edward N. Lorenz “Can the flapping of a butterfly's wings in Brazil trigger a tornado in Texas?” The analogy is reminiscent of the snowball effect, in which small effects result in a chain reaction and even disaster amplify. However, the butterfly effect is about the unpredictability of the long-term effects.

Origin of the name

The catchy term butterfly effect comes from the American meteorologist Edward N. Lorenz , who gave a lecture to the American Association for the Advancement of Science in 1972 with the title Predictability: Does the Flap of a Butterfly's Wings in Brazil set off a Tornado in Texas? held. In its original form, however, he used the flapping of the wings of a seagull instead of a butterfly .

Scientific background

Sensitivity of the Lorenz attractor to changes in the initial conditions. A dense point cloud quickly spreads over the entire attractor.
Experimental demonstration of the butterfly effect with several images of the same double pendulum. In each recording, the initial deflection of the double pendulum is almost the same, but after a while the dynamic behavior begins to differ significantly.

Lorenz carried out preliminary work on the theory with a paper from 1963 in which he carried out a calculation for weather forecasting with the computer. In connection with long-term weather forecasts, he used a simplified convection model to investigate the behavior of liquids or gases when they are heated: here initially rolls form (hot gas rises on one side, loses heat and then sinks again on the other side) further heat supply become unstable.

He characterized this behavior using the three connected differential equations . He projected the numerical result into phase space and received that strange attractor that later became known as the Lorenz attractor : an infinitely long trajectory in three-dimensional space that does not intersect itself and has the shape of two butterfly wings.

Lorenz came across the chaotic behavior of his model by chance. In order to save computing time, he had the numerical solution of the above. Equations are based on intermediate results of calculations that have already been carried out, but only take three decimal places into account, although the computer calculated with an accuracy of six decimal places. The result was increasing deviations over time between the old and new calculations, which led Lorenz to his statements about the sensitivity to the initial conditions. From almost the same starting point, the weather curves diverged until they finally showed nothing in common.

In his first calculation he specified a starting value for an iteration to six decimal places (0.506127), in the second calculation to three (0.506), and although these values ​​only differed by about 1 / 10,000, they differed in the further course Calculation greatly decreases over time from the first.

The butterfly effect occurs in systems that exhibit deterministic chaotic behavior . These systems have the property that arbitrarily small differences in the initial conditions ( clinamen ) lead to major differences in the system over time; they are therefore sensitive to the initial values. This phenomenon can be quantified using the so-called Lyapunov exponent .

Examples

meteorology

Since the initial conditions can only ever be determined experimentally with finite accuracy, a consequence of this effect for such systems is that it is impossible to predict their behavior over a longer period of time. For example, the weather for a day can be forecast relatively accurately, while a forecast for a month is hardly possible. Even if the entire surface of the earth were covered with sensors , these were only slightly apart from each other, reached up to the highest altitudes of the earth's atmosphere and provided exact data, even an unlimited powerful computer would not be able to make precise long-term forecasts of weather developments. Since the computer model does not record the spaces between the sensors, there are slight divergences between the model and reality, which then reinforce each other positively and lead to large differences.

For example, the data from 1000 weather stations can be used to make reasonably reliable forecasts over a period of four days. Corresponding forecasts over eleven days would already require 100 million measuring stations evenly distributed over the earth. The plan becomes absurd when the forecast is to extend over a month; because then 10 20 weather stations would be required, that is, one on every five square millimeters of the earth's surface ( lit .: Heiden).

However, the Lorenz model is actually much more chaotic than the actual course of the weather. The equations are much more unstable than the basic physical equations. The mathematician Vladimir Igorewitsch Arnold gives two weeks as a principal upper limit for the weather forecast.

Tent illustration

Butterfly effect with the tent picture

A minimal example of the butterfly effect is the tent image .

In the diagram, the difference between the values ​​of two such images with slightly different starting parameters (here: 0.506 and 0.506127) is plotted against the number of iterations (shown in the diagram as "time"). Both images have the same control parameter , which was selected so that the tent image shows chaotic behavior (recognizable in the corresponding bifurcation diagram ). The maximum possible deviation is ± 1. The two images are therefore completely different after just a few iterations.

Planetary orbits

If more than two celestial bodies are gravitationally bound to one another, minimal changes in the initial situation can lead to large, unpredictable changes in orbits and positions over time. This behavior is the subject of the three-body problem .

Artistic processing

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Fiction

Movie

TV Shows

Video games

literature

  • Edward N. Lorenz: The Essence of Chaos. University of Washington Press, Seattle WA 1993, ISBN 0-295-97270-X .
  • Uwe an der Heiden : Chaos and order, chance and necessity. In: Günter Küppers (Ed.): Chaos and Order. Forms of self-organization in nature and society (= Reclams Universal-Bibliothek. 9434). Reclam, Stuttgart 1996, ISBN 3-15-009434-8 , p. 111.

Individual evidence

  1. ^ Edward N. Lorenz, Predictability: Does the flap of a butterfly's wings in Brazil set off a tornado in Texas? , Title of paper presented in 1972 at the American Association for the Advancement of Science Annual Meeting ; according to Science 320, 2008, p. 431
  2. ^ First published in Edward Lorenz: The Essence of Chaos. Seattle 1993, Appendix 1, pp. 181-184.
  3. ^ Edward N. Lorenz: Deterministic Nonperiodic Flow . In: Journal of the Atmospheric Sciences . 20, No. 2, March 1963, pp. 130-141. doi : 10.1175 / 1520-0469 (1963) 020 <0130: DNF> 2.0.CO; 2 . Retrieved June 3, 2010.
  4. ↑ Dominik Weber: Life is Strange: The Butterfly Time Travel in the Test - Games, PlayStation 3, PlayStation 4, Xbox 360, Xbox One. In: GamingNerd.net. January 28, 2016, accessed on December 23, 2019 (German).