======================================================================
= Butterfly effect =
======================================================================
Introduction
======================================================================
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.
The term is closely associated with the work of the mathematician and
meteorologist Edward Norton Lorenz. He noted that the butterfly effect
is derived from the example of the details of a tornado (the exact
time of formation, the exact path taken) being influenced by minor
perturbations such as a distant butterfly flapping its wings several
weeks earlier. Lorenz originally used a seagull causing a storm but
was persuaded to make it more poetic with the use of a butterfly and
tornado by 1972. He discovered the effect when he observed runs of his
weather model with initial condition data that were rounded in a
seemingly inconsequential manner. He noted that the weather model
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.
The idea that small causes may have large effects in weather was
earlier acknowledged by the French mathematician and physicist Henri
Poincaré. The American mathematician and philosopher Norbert Wiener
also contributed to this theory. Lorenz's work placed the concept of
'instability' of the Earth's atmosphere 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.
The concept of the butterfly effect has since been used outside the
context of weather science as a broad term for any situation where a
small change is supposed to be the cause of larger consequences.
History
======================================================================
In 'The Vocation of Man' (1800), Johann Gottlieb Fichte says "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 numerous forms of literature. This is evidenced by the
case of the three-body problem by Poincaré in 1890. He later proposed
that such phenomena could be common, for example, in meteorology.
In 1898, Jacques Hadamard noted general divergence of trajectories in
spaces of negative curvature. Pierre Duhem discussed the possible
general significance of this in 1908.
In 1950, Alan Turing noted: "The displacement of a single electron by
a billionth of a centimetre at one moment might make the difference
between a man being killed by an avalanche a year later, or escaping."
The idea that the death of one butterfly could eventually have a
far-reaching ripple effect on subsequent historical events made its
earliest known appearance in "A Sound of Thunder", a 1952 short story
by Ray Bradbury. "A Sound of Thunder" features time travel.
More precisely, though, almost the exact idea and the exact phrasing
—of a tiny insect's wing affecting the entire atmosphere's winds— was
published in a children's book which became extremely successful and
well-known globally in 1962, the year before Lorenz published:
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.
Lorenz wrote:
In 1963, Lorenz published a theoretical study of this effect in a
highly cited, seminal paper called 'Deterministic Nonperiodic Flow'
(the calculations were performed on a Royal McBee LGP-30 computer).
Elsewhere he stated:
Following proposals 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.
The phrase refers to the effect of a butterfly's wings creating 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. 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 of an
interconnected complex web; one set of conditions leads to a tornado,
while the other set of conditions doesn't. The flapping wing creates 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 of
ensemble forecasting, in which a number of forecasts are made from
perturbed initial conditions.
Some scientists have since argued that the weather system is not as
sensitive to initial conditions as previously believed. David Orrell
argues that the major contributor to weather forecast error is model
error, with sensitivity to initial conditions playing a relatively
small role. 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. Recent studies using generalized Lorenz models that
included additional dissipative terms and nonlinearity suggested that
a larger heating parameter is required for the onset of chaos.
While the "butterfly effect" is often explained as being synonymous
with sensitive dependence on initial conditions of the kind described
by Lorenz in his 1963 paper (and previously observed by Poincaré), the
butterfly metaphor was originally applied to work he published in 1969
which took the idea a step further. Lorenz proposed a mathematical
model for how tiny motions in the atmosphere scale up to affect larger
systems. He found that the systems in that model could only be
predicted up to a specific point in the future, and beyond that,
reducing the error in the initial conditions would not increase the
predictability (as long as the error is not zero). This demonstrated
that a deterministic system could be "observationally
indistinguishable" from a non-deterministic one in terms of
predictability. Recent re-examinations of this paper suggest that it
offered a significant challenge to the idea that our universe is
deterministic, comparable to the challenges offered by quantum
physics.
In the book entitled 'The Essence of Chaos' published in 1993, Lorenz
defined butterfly effect as: "The phenomenon that a small alteration
in the state of a dynamical system will cause subsequent states to
differ greatly from the states that would have followed without the
alteration." This feature is the same as sensitive dependence of
solutions on initial conditions (SDIC) in . In the same book, Lorenz
applied the activity of skiing and developed an idealized skiing model
for revealing the sensitivity of time-varying paths to initial
positions. A predictability horizon is determined before the onset of
SDIC.
Illustrations
======================================================================
: colspan=3|The butterfly effect in the Lorenz attractor
time 0 ≤ 't' ≤ 30 (larger) 'z' coordinate (larger)
|300px |300px
|colspan=3 | These figures show two segments of the three-dimensional
evolution of two trajectories (one in blue, and 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.
An animation of the Lorenz attractor shows the continuous evolution.
Theory and mathematical definition
======================================================================
Recurrence, the approximate return of a system toward 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. Lorenz defined sensitive dependence as follows:
'The property characterizing an orbit (i.e., a solution) if most other
orbits that pass close to it at some point do not remain close to it
as time advances.'
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 distance such that and such that
:
for some positive parameter 'a'. The definition does not require that
all points from a neighborhood separate from the base point 'x', but
it requires one positive Lyapunov exponent. In addition to a positive
Lyapunov exponent, boundedness is another major feature within chaotic
systems.
The simplest mathematical framework exhibiting sensitive dependence on
initial conditions is provided by a particular parametrization of the
logistic map:
:
which, unlike most chaotic maps, has a closed-form solution:
:
where the initial condition parameter is given by . For rational ,
after a finite number of iterations maps into a periodic sequence.
But almost all are irrational, and, for irrational , never repeats
itself - it is non-periodic. This solution equation clearly
demonstrates the two key features of chaos - stretching and folding:
the factor 2'n' shows the exponential growth of stretching, which
results in sensitive dependence on initial conditions (the butterfly
effect), while the squared sine function keeps folded within the
range [0, 1].
Overview
==========
The butterfly effect is most familiar in terms of weather; it can
easily be demonstrated in standard weather prediction models, for
example. The climate scientists James Annan and William Connolley
explain that chaos is important in the development of weather
prediction methods; models are sensitive to initial conditions. They
add the caveat: "Of course the existence of an unknown butterfly
flapping its wings has no direct bearing on weather forecasts, since
it will take far too long for such a small perturbation to grow to a
significant size, and we have many more immediate uncertainties to
worry about. So the direct impact of this phenomenon on weather
prediction is often somewhat wrong."
Differentiating types of butterfly effects
============================================
The concept of the butterfly effect encompasses several phenomena. The
two kinds of butterfly effects, including the sensitive dependence on
initial conditions, and the ability of a tiny perturbation to create
an organized circulation at large distances, are not exactly the same.
In Palmer et al., a new type of butterfly effect is introduced,
highlighting the potential impact of small-scale processes on finite
predictability within the Lorenz 1969 model. Additionally, the
identification of ill-conditioned aspects of the Lorenz 1969 model
points to a practical form of finite predictability. These two
distinct mechanisms suggesting finite predictability in the Lorenz
1969 model are collectively referred to as the third kind of butterfly
effect. The authors in have considered Palmer et al.'s suggestions
and have aimed to present their perspective without raising specific
contentions.
The third kind of butterfly effect with finite predictability, as
discussed in, was primarily proposed based on a convergent geometric
series, known as Lorenz's and Lilly's formulas. Ongoing discussions
are addressing the validity of these two formulas for estimating
predictability limits in.
A comparison of the two kinds of butterfly effects and the third kind
of butterfly effect has been documented. In recent studies, it was
reported that both meteorological and non-meteorological linear models
have shown that instability plays a role in producing a butterfly
effect, which is characterized by brief but significant exponential
growth resulting from a small disturbance.
Recent debates on butterfly effects
=====================================
The first kind of butterfly effect (BE1), known as SDIC (Sensitive
Dependence on Initial Conditions), is widely recognized and
demonstrated through idealized chaotic models. However, opinions
differ regarding the second kind of butterfly effect, specifically the
impact of a butterfly flapping its wings on tornado formation, as
indicated in two 2024 articles. In more recent discussions published
by 'Physics Today', it is acknowledged that the second kind of
butterfly effect (BE2) has never been rigorously verified using a
realistic weather model. While the studies suggest that BE2 is
unlikely in the real atmosphere, its invalidity in this context does
not negate the applicability of BE1 in other areas, such as pandemics
or historical events.
For the third kind of butterfly effect, the limited predictability
within the Lorenz 1969 model is explained by scale interactions in one
article and by system ill-conditioning in another more recent study.
Finite predictability in chaotic systems
==========================================
According to Lighthill (1986), the presence of SDIC (commonly known as
the butterfly effect) implies that chaotic systems have a finite
predictability limit. In a literature review, it was found that
Lorenz's perspective on the predictability limit can be condensed into
the following statement:
* (A). The Lorenz 1963 model qualitatively revealed the essence of a
finite predictability within a chaotic system such as the atmosphere.
However, it did not determine a precise limit for the predictability
of the atmosphere.
* (B). In the 1960s, the two-week predictability limit was originally
estimated based on a doubling time of five days in real-world models.
Since then, this finding has been documented in Charney et al. (1966)
and has become a consensus.
Recently, a short video has been created to present Lorenz's
perspective on predictability limit.
A recent study refers to the two-week predictability limit, initially
calculated in the 1960s with the Mintz-Arakawa model's five-day
doubling time, as the "Predictability Limit Hypothesis." Inspired by
Moore's Law, this term acknowledges the collaborative contributions of
Lorenz, Mintz, and Arakawa under Charney's leadership. The hypothesis
supports the investigation into extended-range predictions using both
partial differential equation (PDE)-based physics methods and
Artificial Intelligence (AI) techniques.
Revised perspectives on chaotic and non-chaotic systems
=========================================================
By revealing coexisting chaotic and non-chaotic attractors within
Lorenz models, Shen and his colleagues proposed a revised view that
"weather possesses chaos and order", in contrast to the conventional
view of "weather is chaotic". As a result, sensitive dependence on
initial conditions (SDIC) does not always appear. Namely, SDIC appears
when two orbits (i.e., solutions) become the chaotic attractor; it
does not appear when two orbits move toward the same point attractor.
The above animation for double pendulum motion provides an analogy.
For large angles of swing the motion of the pendulum is often chaotic.
By comparison, for small angles of swing, motions are non-chaotic.
Multistability is defined when a system (e.g., the double pendulum
system) contains more than one bounded attractor that depends only on
initial conditions. The multistability was illustrated using kayaking
in Figure on the right side (i.e., Figure 1 of ) where the appearance
of strong currents and a stagnant area suggests instability and local
stability, respectively. As a result, when two kayaks move along
strong currents, their paths display SDIC. On the other hand, when two
kayaks move into a stagnant area, they become trapped, showing no
typical SDIC (although a chaotic transient may occur). Such features
of SDIC or no SDIC suggest two types of solutions and illustrate the
nature of multistability.
By taking into consideration time-varying multistability that is
associated with the modulation of large-scale processes (e.g.,
seasonal forcing) and aggregated feedback of small-scale processes
(e.g., convection), the above revised view is refined as follows:
"The atmosphere possesses chaos and order; it includes, as examples,
emerging organized systems (such as tornadoes) and time varying
forcing from recurrent seasons."
In quantum mechanics
======================
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. Some authors have argued that
extreme (exponential) dependence on initial conditions is not expected
in pure quantum treatments; however, the sensitive dependence on
initial conditions demonstrated in classical motion is included in the
semiclassical treatments developed by Martin Gutzwiller and John B.
Delos and co-workers. The random matrix theory and simulations with
quantum computers prove that some versions of the butterfly effect in
quantum mechanics do not exist.
Other authors suggest that the butterfly effect can be observed in
quantum systems. Zbyszek P. 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. David 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". 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. This quantum butterfly effect
has been demonstrated experimentally. Quantum and semiclassical
treatments of system sensitivity to initial conditions are known as
quantum chaos.
In popular culture
======================================================================
The butterfly effect has appeared across mediums such as literature
(for instance, 'A Sound of Thunder'), films and television (such as
'The Simpsons'), video games (such as 'Life Is Strange'), webcomics
(such as 'Homestuck'), AI-driven expansive language models, and more.
See also
======================================================================
* Avalanche effect
* Behavioral cusp
* Cascading failure
* Catastrophe theory
* Causality
* Chain reaction
* Clapotis
* Determinism
* Domino effect
* Dynamical system
* Fractal
* Great Stirrup Controversy
* Innovation butterfly
* Kessler syndrome
* Norton's dome
* Numerical analysis
* Point of divergence
* Positive feedback
* Potentiality and actuality
* Representativeness heuristic
* Ripple effect
* Snowball effect
* Traffic congestion
* Tropical cyclogenesis
* Unintended consequences
Further reading
======================================================================
* James Gleick, 'Chaos: Making a New Science', New York: Viking, 1987.
368 pp.
*
*
* Bradbury, Ray. "A Sound of Thunder." Collier's. 28 June 1952
External links
======================================================================
* [https://vimeo.com/287523707/ Weather and Chaos: The Work of Edward
N. Lorenz]. A short documentary that explains the "butterfly effect"
in context of Lorenz's work.
* [https://hypertextbook.com/chaos/ The Chaos Hypertextbook]. An
introductory primer on chaos and fractals
*
* [https://necsi.edu/butterfly-effect New England Complex Systems
Institute - Concepts: Butterfly Effect]
* [https://chaosbook.org/ ChaosBook.org]. Advanced graduate textbook
on chaos (no fractals)
*
License
=========
All content on Gopherpedia comes from Wikipedia, and is licensed under CC-BY-SA
License URL: http://creativecommons.org/licenses/by-sa/3.0/
Original Article: http://en.wikipedia.org/wiki/Butterfly_effect