Butterfly effect - Wikipedia
1 december 2017 fri.
https://en.m.wikipedia.org › wiki › Butter...
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The butterfly effect is a concept that states that "small causes can have larger effects". This concept was initially used in theories about weather prediction but later the term became a ...
Butterfly effect (disambiguation)
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Read in another languageButterfly effect
For other uses, see Butterfly effect (disambiguation).
A plot of Lorenz's strange attractorfor values ρ=28, σ = 10, β = 8/3. The butterfly effect or sensitive dependence on initial conditions is the property of a dynamical system that, starting from any of various arbitrarily close alternativeinitial conditions on the attractor, the iterated points will become arbitrarily spread out from each other.
The butterfly effect is a concept that states that "small *** can have larger effects".
This concept was initially used in theories about weather prediction but later the term became a popular metaphor in science writing.[1]
In chaos theory, the butterfly effect is the sensitive dependence on initial conditions in which a small change in one state of a deterministic nonlinear system can result in large differences in a later state. [2]
The term itself was coined by Edward Lorenz, and is derived from the metaphorical example of the details of a tornado (exact time of formation, exact path taken) being influenced by minor perturbations such as the flapping of the wings of a distant butterfly several weeks earlier. Lorenz discovered the effect when he observed that runs of his weather model with initial condition data that was rounded in a seemingly inconsequential manner would fail to reproduce the results of runs with the unrounded initial condition data. A very small change in initial conditions had created a significantly different outcome. [3]
The idea that small causes may have large effects in general and in weather specifically was used from Henri Poincaré to Norbert Wiener. Edward Lorenz's work placed the concept of instability of the earth'satmosphere onto a quantitative base and linked the concept of instability to the properties of large classes of dynamic systems which are undergoing nonlinear dynamics and deterministic chaos.[1]
The butterfly effect can also be demonstrated by very simple systems. For example, therandomness of the outcomes of throwingdice depends on this characteristic to amplify small differences in initial conditions—the precise direction, thrust, and orientation of the throw—into significantly different dice paths and outcomes, which makes it virtually impossible to throw a dice exactly the same way twice.
HistoryEdit
In The Vocation of Man (1800), Fichte says that "you could not remove a single grain of sand from its place without thereby ... changing something throughout all parts of the immeasurable whole".
Chaos theory and the sensitive dependence on initial conditions were described in the literature in a particular case of the three-body problem by Henri Poincaré in 1890.[4] He later proposed that such phenomena could be common, for example, in meteorology.[5]
In 1898,[4] Jacques Hadamard noted general divergence of trajectories in spaces of negative curvature. Pierre Duhem discussed the possible general significance of this in 1908.[4]
The idea that one butterfly could eventually have a far-reaching ripple effect on subsequent historic events made its earliest known appearance in "A Sound of Thunder", a 1952 short story by Ray Bradbury about time travel.[6]
In 1961, Lorenz was running a numerical computer model to redo a weather prediction from the middle of the previous run as a shortcut. He entered the initial condition 0.506 from the printout instead of entering the full precision 0.506127 value. The result was a completely different weather scenario.[7]
Lorenz wrote:
"At one point I decided to repeat some of the computations in order to examine what was happening in greater detail. I stopped the computer, typed in a line of numbers that it had printed out a while earlier, and set it running again. I went down the hall for a cup of coffee and returned after about an hour, during which time the computer had simulated about two months of weather. The numbers being printed were nothing like the old ones. I immediately suspected a weak vacuum tube or some other computer trouble, which was not uncommon, but before calling for service I decided to see just where the mistake had occurred, knowing that this could speed up the servicing process. Instead of a sudden break, I found that the new values at first repeated the old ones, but soon afterward differed by one and then several units in the last decimal place, and then began to differ in the next to the last place and then in the place before that. In fact, the differences more or less steadily doubled in size every four days or so, until all resemblance with the original output disappeared somewhere in the second month. This was enough to tell me what had happened: the numbers that I had typed in were not the exact original numbers, but were the rounded-off values that had appeared in the original printout. The initial round-off errors were the culprits; they were steadily amplifying until they dominated the solution." (E. N. Lorenz, The Essence of Chaos, U. Washington Press, Seattle (1993), page 134)[8]
In 1963 Lorenz published a theoretical study of this effect in a highly cited, seminal paper called Deterministic Nonperiodic Flow[9][10] (the calculations were performed on a Royal McBee LGP-30 computer).[11][12] Elsewhere he stated:
One meteorologist remarked that if the theory were correct, one flap of a sea gull's wings would be enough to alter the course of the weather forever. The controversy has not yet been settled, but the most recent evidence seems to favor the sea gulls.[12]
Following suggestions from colleagues, in later speeches and papers Lorenz used the more poetic butterfly. According to Lorenz, when he failed to provide a title for a talk he was to present at the 139th meeting of theAmerican Association for the Advancement of Science in 1972, Philip Merilees concoctedDoes the flap of a butterfly’s wings in Brazil set off a tornado in Texas? as a title.[13] 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.[14]
The phrase refers to the idea that a butterfly's wings might create tiny changes in theatmosphere that may ultimately alter the path of a tornado or delay, accelerate or even prevent the occurrence of a tornado in another location. The butterfly does not power or directly create the tornado, but the term is intended to imply that the flap of the butterfly's wings can cause the tornado: in the sense that the flap of the wings is a part of the initial conditions; one set of conditions leads to a tornado while the other set of conditions doesn't. The flapping wing represents a small change in the initial condition of the system, which cascades to large-scale alterations of events (compare:domino effect). Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different—but it's also equally possible that the set of conditions without the butterfly flapping its wings is the set that leads to a tornado.
The butterfly effect presents an obvious challenge to prediction, since initial conditions for a system such as the weather can never be known to complete accuracy. This problem motivated the development ofensemble forecasting, in which a number of forecasts are made from perturbed initial conditions.[15]
Some scientists have since argued that the weather system is not as sensitive to initial conditions as previously believed.[16] David Orrell argues that the major contributor to weather forecast error is model error, with sensitivity to initial conditions playing a relatively small role.[17][18] Stephen Wolframalso notes that the Lorenz equations areaaaaaa
1 december 2017 fri.
https://en.m.wikipedia.org › wiki › Butter...
from en.m.wikipedia.org
The butterfly effect is a concept that states that "small causes can have larger effects". This concept was initially used in theories about weather prediction but later the term became a ...
Butterfly effect (disambiguation)
Butte
$$$$$$$$$$$$$$$$$$$
Open main menu
Search
Show my notifications
EditWatch this page
Read in another languageButterfly effect
For other uses, see Butterfly effect (disambiguation).
A plot of Lorenz's strange attractorfor values ρ=28, σ = 10, β = 8/3. The butterfly effect or sensitive dependence on initial conditions is the property of a dynamical system that, starting from any of various arbitrarily close alternativeinitial conditions on the attractor, the iterated points will become arbitrarily spread out from each other.
The butterfly effect is a concept that states that "small *** can have larger effects".
This concept was initially used in theories about weather prediction but later the term became a popular metaphor in science writing.[1]
In chaos theory, the butterfly effect is the sensitive dependence on initial conditions in which a small change in one state of a deterministic nonlinear system can result in large differences in a later state. [2]
The term itself was coined by Edward Lorenz, and is derived from the metaphorical example of the details of a tornado (exact time of formation, exact path taken) being influenced by minor perturbations such as the flapping of the wings of a distant butterfly several weeks earlier. Lorenz discovered the effect when he observed that runs of his weather model with initial condition data that was rounded in a seemingly inconsequential manner would fail to reproduce the results of runs with the unrounded initial condition data. A very small change in initial conditions had created a significantly different outcome. [3]
The idea that small causes may have large effects in general and in weather specifically was used from Henri Poincaré to Norbert Wiener. Edward Lorenz's work placed the concept of instability of the earth'satmosphere onto a quantitative base and linked the concept of instability to the properties of large classes of dynamic systems which are undergoing nonlinear dynamics and deterministic chaos.[1]
The butterfly effect can also be demonstrated by very simple systems. For example, therandomness of the outcomes of throwingdice depends on this characteristic to amplify small differences in initial conditions—the precise direction, thrust, and orientation of the throw—into significantly different dice paths and outcomes, which makes it virtually impossible to throw a dice exactly the same way twice.
HistoryEdit
In The Vocation of Man (1800), Fichte says that "you could not remove a single grain of sand from its place without thereby ... changing something throughout all parts of the immeasurable whole".
Chaos theory and the sensitive dependence on initial conditions were described in the literature in a particular case of the three-body problem by Henri Poincaré in 1890.[4] He later proposed that such phenomena could be common, for example, in meteorology.[5]
In 1898,[4] Jacques Hadamard noted general divergence of trajectories in spaces of negative curvature. Pierre Duhem discussed the possible general significance of this in 1908.[4]
The idea that one butterfly could eventually have a far-reaching ripple effect on subsequent historic events made its earliest known appearance in "A Sound of Thunder", a 1952 short story by Ray Bradbury about time travel.[6]
In 1961, Lorenz was running a numerical computer model to redo a weather prediction from the middle of the previous run as a shortcut. He entered the initial condition 0.506 from the printout instead of entering the full precision 0.506127 value. The result was a completely different weather scenario.[7]
Lorenz wrote:
"At one point I decided to repeat some of the computations in order to examine what was happening in greater detail. I stopped the computer, typed in a line of numbers that it had printed out a while earlier, and set it running again. I went down the hall for a cup of coffee and returned after about an hour, during which time the computer had simulated about two months of weather. The numbers being printed were nothing like the old ones. I immediately suspected a weak vacuum tube or some other computer trouble, which was not uncommon, but before calling for service I decided to see just where the mistake had occurred, knowing that this could speed up the servicing process. Instead of a sudden break, I found that the new values at first repeated the old ones, but soon afterward differed by one and then several units in the last decimal place, and then began to differ in the next to the last place and then in the place before that. In fact, the differences more or less steadily doubled in size every four days or so, until all resemblance with the original output disappeared somewhere in the second month. This was enough to tell me what had happened: the numbers that I had typed in were not the exact original numbers, but were the rounded-off values that had appeared in the original printout. The initial round-off errors were the culprits; they were steadily amplifying until they dominated the solution." (E. N. Lorenz, The Essence of Chaos, U. Washington Press, Seattle (1993), page 134)[8]
In 1963 Lorenz published a theoretical study of this effect in a highly cited, seminal paper called Deterministic Nonperiodic Flow[9][10] (the calculations were performed on a Royal McBee LGP-30 computer).[11][12] Elsewhere he stated:
One meteorologist remarked that if the theory were correct, one flap of a sea gull's wings would be enough to alter the course of the weather forever. The controversy has not yet been settled, but the most recent evidence seems to favor the sea gulls.[12]
Following suggestions from colleagues, in later speeches and papers Lorenz used the more poetic butterfly. According to Lorenz, when he failed to provide a title for a talk he was to present at the 139th meeting of theAmerican Association for the Advancement of Science in 1972, Philip Merilees concoctedDoes the flap of a butterfly’s wings in Brazil set off a tornado in Texas? as a title.[13] 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.[14]
The phrase refers to the idea that a butterfly's wings might create tiny changes in theatmosphere that may ultimately alter the path of a tornado or delay, accelerate or even prevent the occurrence of a tornado in another location. The butterfly does not power or directly create the tornado, but the term is intended to imply that the flap of the butterfly's wings can cause the tornado: in the sense that the flap of the wings is a part of the initial conditions; one set of conditions leads to a tornado while the other set of conditions doesn't. The flapping wing represents a small change in the initial condition of the system, which cascades to large-scale alterations of events (compare:domino effect). Had the butterfly not flapped its wings, the trajectory of the system might have been vastly different—but it's also equally possible that the set of conditions without the butterfly flapping its wings is the set that leads to a tornado.
The butterfly effect presents an obvious challenge to prediction, since initial conditions for a system such as the weather can never be known to complete accuracy. This problem motivated the development ofensemble forecasting, in which a number of forecasts are made from perturbed initial conditions.[15]
Some scientists have since argued that the weather system is not as sensitive to initial conditions as previously believed.[16] David Orrell argues that the major contributor to weather forecast error is model error, with sensitivity to initial conditions playing a relatively small role.[17][18] Stephen Wolframalso notes that the Lorenz equations areaaaaaa
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