In
chaos theory, the
butterfly effect is the
sensitive dependency on initial conditions in which a small change at one place in a deterministic
nonlinear system can result in large differences in a later state. The name of the effect, coined by
Edward Lorenz, is derived from the theoretical example of the details of a hurricane (exact time of formation, exact path taken) being influenced by minor perturbations equating to 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.
Although the butterfly effect may appear to be an unlikely behavior, it is exhibited by very simple systems. For example, a ball placed at the crest of a hill may roll into any surrounding valley depending on, among other things, slight differences in its initial position. Also the randomness of throwing a
dice depends on this system's characteristic to amplify small differences in initial conditions - the throw - into significantly different dice paths and outcome, which makes it virtually impossible to throw a dice exactly the same way twice.
The butterfly effect is a common
trope in fiction, especially in scenarios involving
time travel. Additionally, works of fiction that involve points at which the storyline diverges during a seemingly minor event, resulting in a significantly different outcome than would have occurred without the divergence, are an example of the butterfly effect.
Chaos theory and the sensitive dependence on initial conditions was described in the literature in a particular case of the
three-body problem by
Henri Poincaré in 1890.
[1] He later proposed that such phenomena could be common, for example, in meteorology.
[2]
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 0.506 instead of entering the full 0.506127. The result was a completely different weather scenario.
[3] In 1963 Lorenz published a theoretical study of this effect in a well-known paper called
Deterministic Nonperiodic Flow.
[4] (As noted in the paper, the calculations were performed on a
Royal McBee LPD-30 computing machine.
[5]) Elsewhere he said
[citation needed] that "One meteorologist remarked that if the theory were correct, one flap of a
seagull's wings could change the course of weather forever." 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 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. 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.
[6]
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 another location. Note that the butterfly does not power or directly create the tornado. The Butterfly effect does not convey the notion - as is often misconstrued - that the flap of the butterfly's wings
causes the tornado. 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 causes a chain of events leading 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 - it's 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 of
ensemble forecasting, in which a number of forecasts are made from perturbed initial conditions.
[7]
Some scientists have since argued that the weather system is not as sensitive to initial condition as previously believed.
[8] David Orrell argues that the major contributor to weather forecast error is model error, with sensitivity to initial conditions playing a relatively small role.
[9][10] Stephen Wolfram also notes that the Lorenz equations are highly simplified and do not contain terms that represent viscous effects; he believes that these terms would tend to damp out small perturbations.
[11]
Illustration[edit]
| 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 by only 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. |
Theory and mathematical definition[edit]
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) since it is impossible to measure the starting atmospheric conditions completely accurately.
A
dynamical system displays sensitive dependence on initial conditions if points arbitrarily close together separate over time at an exponential rate. The definition is not topological, but essentially metrical.
If
M is the
state space for the map
, then
displays sensitive dependence to initial conditions if for any x in
M and any δ > 0, there are y in
M, with
such that
The definition does not require that all points from a neighborhood separate from the base point
x, but it requires one positive
Lyapunov exponent.
Examples[edit]
The butterfly effect is most familiar in terms of
weather; it can easily be demonstrated in standard weather prediction models, for example.
[12]
The potential for sensitive dependence on initial conditions (the butterfly effect) has been studied in a number of cases in semiclassical and
quantum physics including atoms in strong fields and the anisotropic
Kepler problem.
[13][14] Some authors have argued that extreme (exponential) dependence on initial conditions is not expected in pure quantum treatments;
[15][16] however, the sensitive dependence on initial conditions demonstrated in classical motion is included in the semiclassical treatments developed by
Martin Gutzwiller[17] and Delos and co-workers.
[18]
Other authors suggest that the butterfly effect can be observed in quantum systems. Karkuszewski et al. consider the time evolution of quantum systems which have slightly different Hamiltonians. They investigate the level of sensitivity of quantum systems to small changes in their given Hamiltonians.
[19] Poulin et al. presented a quantum algorithm to measure fidelity decay, which "measures the rate at which identical initial states diverge when subjected to slightly different dynamics". They consider fidelity decay to be "the closest quantum analog to the (purely classical) butterfly effect".
[20] Whereas the classical butterfly effect considers the effect of a small change in the position and/or velocity of an object in a given
Hamiltonian system, the quantum butterfly effect considers the effect of a small change in the Hamiltonian system with a given initial position and velocity.
[21][22] This quantum butterfly effect has been demonstrated experimentally.
[23] Quantum and semiclassical treatments of system sensitivity to initial conditions are known as
quantum chaos.
[15][21]
