Editing Predictability (section)

==In mathematics==
In [[stochastic analysis]] a [[stochastic process|random process]] is a [[predictable process]] if it is possible to know the next state from the present time.

The branch of mathematics known as [[Chaos theory|Chaos Theory]] focuses on the behavior of systems that are highly sensitive to initial conditions. It suggests that a small change in an initial condition can completely alter the progression of a system. This phenomenon is known as the [[butterfly effect]], which claims that a butterfly flapping its wings in Brazil can cause a tornado in Texas. The nature of chaos theory suggests that the predictability of any system is limited because it is impossible to know all of the minutiae of a system at the present time. In principle, the deterministic systems that chaos theory attempts to analyze can be predicted, but uncertainty in a forecast increases exponentially with elapsed time.<ref>''Sync: The Emerging Science of Spontaneous Order'', Steven Strogatz, Hyperion, New York, 2003, pages 189-190.</ref>

As documented in,<ref>{{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 |volume=2 |issue=3 |pages=1250–1259 |doi=10.3390/encyclopedia2030084 |issn=2673-8392|doi-access=free }}</ref> three major kinds of butterfly effects within Lorenz studies include: the sensitive dependence on initial conditions,<ref name=":0">{{Cite journal |last=Lorenz |first=Edward N. |date=1963-03-01 |title=Deterministic Nonperiodic Flow |journal=Journal of the Atmospheric Sciences |language=EN |volume=20 |issue=2 |pages=130–141 |doi=10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2 |bibcode=1963JAtS...20..130L |issn=0022-4928|doi-access=free }}</ref><ref name=":1">{{Cite book |last=Lorenz |first=Edward |title=The Essence of Chaos |publisher=University of Washington Press |year=1993 |location=Seattle, WA, USA |pages=227p}}</ref> the ability of a tiny perturbation to create an organized circulation at large distances,<ref>{{Cite web |last=Lorenz |first=Edward |date=2022-08-17 |title=Predictability: Does the flap of a butterfly's wings in Brazil set off a tornado in Texas? |url=https://eapsweb.mit.edu/sites/default/files/Butterfly_1972.pdf |website=MIT}}</ref> and the hypothetical role of small-scale processes in contributing to finite predictability.<ref>{{Cite journal |last=Lorenz |first=Edward N. |date=1969-01-01 |title=The predictability of a flow which possesses many scales of motion |journal=Tellus |volume=21 |issue=3 |pages=289–307 |doi=10.3402/tellusa.v21i3.10086 |bibcode=1969Tell...21..289L |issn=0040-2826|doi-access=free }}</ref><ref>{{Cite journal |last1=Palmer |first1=T N |last2=Döring |first2=A |last3=Seregin |first3=G |date=2014-08-19 |title=The real butterfly effect |url=http://dx.doi.org/10.1088/0951-7715/27/9/r123 |journal=Nonlinearity |volume=27 |issue=9 |pages=R123–R141 |doi=10.1088/0951-7715/27/9/r123 |bibcode=2014Nonli..27R.123P |s2cid=122339502 |issn=0951-7715|url-access=subscription }}</ref><ref>{{Cite journal |last1=Shen |first1=Bo-Wen |last2=Pielke |first2=Roger A. |last3=Zeng |first3=Xubin |date=2022-05-07 |title=One Saddle Point and Two Types of Sensitivities within the Lorenz 1963 and 1969 Models |journal=Atmosphere |language=en |volume=13 |issue=5 |pages=753 |doi=10.3390/atmos13050753 |bibcode=2022Atmos..13..753S |issn=2073-4433|doi-access=free }}</ref> The three kinds of butterfly effects are not exactly the same.