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| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Butterfly effect | 3/4 | https://en.wikipedia.org/wiki/Butterfly_effect | reference | science, encyclopedia | 2026-05-05T10:54:43.276736+00:00 | kb-cron |
which, unlike most chaotic maps, has a closed-form solution:
x
n
=
sin
2
(
2
n
θ
π
)
{\displaystyle x_{n}=\sin ^{2}(2^{n}\theta \pi )}
where the initial condition parameter
θ
{\displaystyle \theta }
is given by
θ
=
1
π
sin
−
1
(
x
0
1
/
2
)
{\displaystyle \theta ={\tfrac {1}{\pi }}\sin ^{-1}(x_{0}^{1/2})}
. For rational
θ
{\displaystyle \theta }
, after a finite number of iterations
x
n
{\displaystyle x_{n}}
maps into a periodic sequence. But almost all
θ
{\displaystyle \theta }
are irrational, and, for irrational
θ
{\displaystyle \theta }
,
x
n
{\displaystyle x_{n}}
never repeats itself – it is non-periodic. This solution equation clearly demonstrates the two key features of chaos – stretching and folding: the factor 2n shows the exponential growth of stretching, which results in sensitive dependence on initial conditions (the butterfly effect), while the squared sine function keeps
x
n
{\displaystyle x_{n}}
folded within the range [0, 1].
== In physical systems ==
=== In weather ===
==== 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.