Editing Butterfly effect (section)

===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."<ref>{{cite web |title=Chaos and Climate |publisher=RealClimate |date=4 November 2005 |url=https://www.realclimate.org/index.php/archives/2005/11/chaos-and-climate/ |access-date=2014-06-08 |url-status=live |archive-url=https://web.archive.org/web/20140702105624/http://www.realclimate.org/index.php/archives/2005/11/chaos-and-climate/ |archive-date=2014-07-02}}</ref>

==== 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,<ref name=":0" /> and the ability of a tiny perturbation to create an organized circulation at large distances,<ref name=":1" /> are not exactly the same.<ref>{{cite journal |last=Shen |first=Bo-Wen |date=2014-05-01 |title=Nonlinear Feedback in a Five-Dimensional Lorenz Model |url=https://journals.ametsoc.org/view/journals/atsc/71/5/jas-d-13-0223.1.xml |journal=Journal of the Atmospheric Sciences |language=EN |volume=71 |issue=5 |pages=1701–1723 |doi=10.1175/JAS-D-13-0223.1 |bibcode=2014JAtS...71.1701S |s2cid=123683839 |issn=0022-4928}}</ref> In Palmer et al.,<ref name=":3" /> 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.<ref name=":7" /> These two distinct mechanisms suggesting finite predictability in the Lorenz 1969 model are collectively referred to as the third kind of butterfly effect.<ref name=":8" /> The authors in <ref name=":8" /> 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,<ref name=":3" /> 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.<ref>{{Cite journal |last1=Shen |first1=Bo-Wen |last2=Pielke Sr. |first2=Roger |last3=Zeng |first3=Xubin |date=2024-07-24 |title=Revisiting Lorenz's and Lilly's Empirical Formulas for Predictability Estimates |url=https://egusphere.copernicus.org/preprints/2024/egusphere-2024-2228/ |journal=EGUsphere |language=English |pages=1–0 |doi=10.13140/RG.2.2.32941.15849}}</ref>

A comparison of the two kinds of butterfly effects<ref name=":1" /><ref name=":0" /> and the third kind of butterfly effect<ref name=":2" /><ref name=":3" /><ref name=":4" /> has been documented.<ref name=":8">{{cite journal |last1=Shen |first1=Bo-Wen |last2=Pielke |first2=Roger A. |last3=Zeng |first3=Xubin |last4=Cui |first4=Jialin |last5=Faghih-Naini |first5=Sara |last6=Paxson |first6=Wei |last7=Atlas |first7=Robert |date=2022-07-04 |title=Three Kinds of Butterfly Effects within Lorenz Models |journal=[[Encyclopedia (journal)|Encyclopedia]] |language=en |volume=2 |issue=3 |pages=1250–1259 |issn=2673-8392 |doi=10.3390/encyclopedia2030084 |doi-access=free}}</ref> In recent studies,<ref name=":7" /><ref>{{Cite journal |last1=Saiki |first1=Yoshitaka |last2=Yorke |first2=James A. |date=2023-05-02 |title=Can the Flap of a Butterfly's Wings Shift a Tornado into Texas&mdash;Without Chaos? |journal=Atmosphere |language=en |volume=14 |issue=5 |pages=821 |doi=10.3390/atmos14050821 |bibcode=2023Atmos..14..821S |issn=2073-4433 |doi-access=free }}</ref> 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.<ref name=":9">{{Cite journal |last1=Pielke Sr. |first1=Roger |last2=Shen |first2=Bo-Wen |last3=Zeng |first3=Xubin |date=2024-05-01 |title=The Butterfly Effect: Can a butterfly in Brazil cause a tornado in Texas? |url=https://doi.org/10.1080/00431672.2024.2329521 |journal=Weatherwise |volume=77 |issue=3 |pages=14–18|doi=10.1080/00431672.2024.2329521 }}</ref><ref>{{Cite journal |last=Palmer |first=Tim |date=2024-05-01 |title=The real butterfly effect and maggoty apples |journal=Physics Today |volume=77 |issue=5 |pages=30–35 |doi=10.1063/pt.eike.hsbz |issn=0031-9228|doi-access=free |bibcode=2024PhT....77e..30P }}</ref> In more recent discussions published by ''Physics Today'',<ref name=":10">{{Cite journal |last1=Pielke |first1=Roger A. |last2=Shen |first2=Bo-Wen |last3=Zeng |first3=Xubin |date=2024-09-01 |title=Butterfly effects |journal=Physics Today |language=en |volume=77 |issue=9 |pages=10 |doi=10.1063/pt.ifge.djjy |issn=0031-9228|doi-access=free |bibcode=2024PhT....77Q..10P }}</ref><ref>{{Cite journal |last=Palmer |first=Tim |date=2024-09-01 |title=Butterfly effects |url=https://pubs.aip.org/physicstoday/article/77/9/10/3309181/Butterfly-effects |journal=Physics Today |language=en |volume=77 |issue=9 |pages=10 |doi=10.1063/pt.oktn.zdwa |bibcode=2024PhT....77R..10P |issn=0031-9228}}</ref> 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,<ref name=":9" /><ref name=":10" /> its invalidity in this context does not negate the applicability of BE1 in other areas, such as pandemics or historical events.<ref>{{Cite journal |last1=Shen |first1=Bo-Wen |last2=Pielke |first2=Roger |last3=Xubin Zeng |date=2024-09-02 |title=Summary of Two Kinds of Butterfly Effects |url=https://rgdoi.net/10.13140/RG.2.2.32401.24163 |journal=Technical Report |language=en |doi=10.13140/RG.2.2.32401.24163}}</ref>

For the third kind of butterfly effect, the limited predictability within the Lorenz 1969 model is explained by scale interactions in one article<ref name=":3" /> and by system ill-conditioning in another more recent study.<ref name=":7" />

==== Finite predictability in chaotic systems ====
According to Lighthill (1986),<ref>{{Cite journal |date=1986-09-08 |title=The recently recognized failure of predictability in Newtonian dynamics |url=http://dx.doi.org/10.1098/rspa.1986.0082 |journal=Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences |volume=407 |issue=1832 |pages=35–50 |doi=10.1098/rspa.1986.0082 |bibcode=1986RSPSA.407...35L |s2cid=86552243 |issn=0080-4630 |last1=Lighthill |first1=James }}</ref> the presence of SDIC (commonly known as the butterfly effect) implies that chaotic systems have a finite predictability limit. In a literature review,<ref>{{Cite journal |last1=Shen |first1=Bo-Wen |last2=Pielke |first2=Roger A. |last3=Zeng |first3=Xubin |last4=Zeng |first4=Xiping |date=2023-07-22 |title=Lorenz's View on the Predictability Limit of the Atmosphere |journal=Encyclopedia |language=en |volume=3 |issue=3 |pages=887–899 |doi=10.3390/encyclopedia3030063 |issn=2673-8392 |doi-access=free }}{{Creative Commons text attribution notice|cc=by4|from this source=yes}}</ref> 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)<ref>{{Cite book |date=1966-01-01 |title=The Feasibility of a Global Observation and Analysis Experiment |url=http://dx.doi.org/10.17226/21272 |doi=10.17226/21272|isbn=978-0-309-35922-1 }}</ref><ref>{{Cite journal |last=GARP |date=1969-03-01 |title=A Guide to GARP |journal=Bull. Amer. Meteor. Soc. |volume=50 |issue=3 |pages=136–141|doi=10.1175/1520-0477-50.3.136 |bibcode=1969BAMS...50..136. |doi-access=free }}</ref> and has become a consensus.

Recently, a short video has been created to present Lorenz's perspective on predictability limit.<ref>{{Cite web |last1=Shen |first1=Bo-Wen |last2=Pielke, Sr. |first2=Roger |last3=Zeng |first3=Xubin |last4=Zeng |first4=Xiping |date=2023-09-13 |title=Lorenz's View on the Predictability Limit. |url=https://encyclopedia.pub/video/video_detail/916 |access-date=2023-09-13 |website=Encyclopedia pub}}</ref>

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."<ref>{{Cite journal |last1=Shen |first1=Bo-Wen |last2=Pielke |first2=Roger A. |last3=Zeng |first3=Xubin |last4=Zeng |first4=Xiping |date=2024-07-16 |title=Exploring the Origin of the Two-Week Predictability Limit: A Revisit of Lorenz's Predictability Studies in the 1960s |journal=Atmosphere |language=en |volume=15 |issue=7 |pages=837 |doi=10.3390/atmos15070837 |doi-access=free |bibcode=2024Atmos..15..837S |issn=2073-4433}}</ref> 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".<ref>{{cite journal |last1=Shen |first1=Bo-Wen |last2=Pielke |first2=Roger A. |last3=Zeng |first3=Xubin |last4=Baik |first4=Jong-Jin |last5=Faghih-Naini |first5=Sara |last6=Cui |first6=Jialin |last7=Atlas |first7=Robert |date=2021-01-01 |title=Is Weather Chaotic?: Coexistence of Chaos and Order within a Generalized Lorenz Model |journal=Bulletin of the American Meteorological Society |language=EN |volume=102 |issue=1 |pages=E148–E158 |doi=10.1175/BAMS-D-19-0165.1 |bibcode=2021BAMS..102E.148S |s2cid=208369617 |issn=0003-0007 |doi-access=free}}</ref><ref>{{cite book |last1=Shen |first1=Bo-Wen |last2=Pielke |first2=R. A. Sr. |last3=Zeng |first3=X. |last4=Baik |first4=J.-J. |last5=Faghih-Naini |first5=S. |last6=Cui |first6=J. |last7=Atlas |first7=R. |last8=Reyes |first8=T. A. L. |title=13th Chaotic Modeling and Simulation International Conference |chapter=Is Weather Chaotic? Coexisting Chaotic and Non-chaotic Attractors within Lorenz Models |date=2021 |editor-last=Skiadas |editor-first=Christos H. |editor2-last=Dimotikalis |editor2-first=Yiannis |chapter-url=https://link.springer.com/chapter/10.1007/978-3-030-70795-8_57 |series=Springer Proceedings in Complexity |language=en |location=Cham |publisher=Springer International Publishing |pages=805–825 |doi=10.1007/978-3-030-70795-8_57 |isbn=978-3-030-70795-8 |s2cid=245197840}}</ref><ref>{{cite journal |last=Anthes |first=Richard A. |date=2022-08-14 |title=Predictability and Predictions |journal=Atmosphere |language=en |volume=13 |issue=8 |pages=1292 |doi=10.3390/atmos13081292 |bibcode=2022Atmos..13.1292A |issn=2073-4433 |doi-access=free}}</ref> 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.<ref>{{citation |last1=Richter |first1=P. H. |title=Chaos in Classical Mechanics: The Double Pendulum |last2=Scholz |first2=H.-J. |date=1984 |work=Stochastic Phenomena and Chaotic Behaviour in Complex Systems |series=Springer Series in Synergetics |volume=21 |pages=86–97 |place=Berlin, Heidelberg |publisher=Springer Berlin Heidelberg |isbn=978-3-642-69593-3 |doi=10.1007/978-3-642-69591-9_9 |url=https://link.springer.com/chapter/10.1007/978-3-642-69591-9_9 |access-date=2022-07-11}}</ref><ref>{{cite journal |last=Shinbrot |first=Troy, Celso A Grebogi, Jack Wisdom, James A Yorke |date=1992 |title=Chaos in a double pendulum |url=https://doi.org/10.1119/1.16860 |journal=American Journal of Physics |volume=60 |issue=6 |pages=491–499 |doi=10.1119/1.16860 |bibcode=1992AmJPh..60..491S}}</ref> 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 <ref name=":6">{{cite journal |last1=Shen |first1=Bo-Wen |last2=Pielke Sr. |first2=Roger Pielke |last3=Zeng |first3=Xubin |last4=Cui |first4=Jialin |last5=Faghih-Naini |first5=Sara |last6=Paxson |first6=Wei |last7=Kesarkar |first7=Amit |last8=Zeng |first8=Xiping |last9=Atlas |first9=Robert |date=2022-11-12 |title=The Dual Nature of Chaos and Order in the Atmosphere |journal=Atmosphere |language=en |volume=13 |issue=11 |pages=1892 |doi=10.3390/atmos13111892 |bibcode=2022Atmos..13.1892S |issn=2073-4433 |doi-access=free}} [[File:CC-BY icon.svg|50px]] Text was copied from this source, which is available under a [https://creativecommons.org/licenses/by/4.0/ Creative Commons Attribution 4.0 International License].</ref>) 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."<ref name=":6"/><ref>{{cite web |last=Shen |first=Bo-Wen |date=21 Feb 2023 |title=Exploring Chaos Theory for Monstability and Multistability |website=[[YouTube]] |url=https://www.youtube.com/watch?v=GXtpkf3QsPI}}</ref>