Editing Chaos theory (section)

===Sensitivity to initial conditions===
{{Main|Butterfly effect}}
[[File:SensInitCond.gif|thumb|Lorenz equations used to generate plots for the y variable. The initial conditions for ''x'' and ''z'' were kept the same but those for ''y'' were changed between '''1.001''', '''1.0001''' and '''1.00001'''. The values for <math>\rho</math>, <math>\sigma</math> and <math>\beta</math> were '''45.91''', '''16''' and '''4 ''' respectively. As can be seen from the graph, even the slightest difference in initial values causes significant changes after about 12 seconds of evolution in the three cases. This is an example of sensitive dependence on initial conditions.]]

'''Sensitivity to initial conditions''' means that each point in a chaotic system is arbitrarily closely approximated by other points that have significantly different future paths or trajectories. Thus, an arbitrarily small change or perturbation of the current trajectory may lead to significantly different future behavior.<ref name=":1" />

Sensitivity to initial conditions is popularly known as the "[[butterfly effect]]", so-called because of the title of a paper given by [[Edward Lorenz]] in 1972 to the [[American Association for the Advancement of Science]] in Washington, D.C., entitled ''Predictability: Does the Flap of a Butterfly's Wings in Brazil set off a Tornado in Texas?''.<ref>{{Cite web|url=http://news.mit.edu/2008/obit-lorenz-0416|title=Edward Lorenz, father of chaos theory and butterfly effect, dies at 90|website=MIT News|date=16 April 2008 |access-date=2019-11-24}}</ref> The flapping wing represents a small change in the initial condition of the system, which causes a chain of events that prevents the predictability of large-scale phenomena. Had the butterfly not flapped its wings, the trajectory of the overall system could have been vastly different.

As suggested in Lorenz's book entitled ''The Essence of Chaos'', published in 1993,<ref name="Lorenz"/>{{rp|8}} "sensitive dependence can serve as an acceptable definition of chaos". In the same book, Lorenz defined the 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."<ref name="Lorenz"/>{{rp|23}} The above definition is consistent with the sensitive dependence of solutions on initial conditions (SDIC). An idealized skiing model was developed to illustrate the sensitivity of time-varying paths to initial positions.<ref name="Lorenz"/>{{rp|189–204}} A predictability horizon can be determined before the onset of SDIC (i.e., prior to significant separations of initial nearby trajectories).<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>

A consequence of sensitivity to initial conditions is that if we start with a limited amount of information about the system (as is usually the case in practice), then beyond a certain time, the system would no longer be predictable. This is most prevalent in the case of weather, which is generally predictable only about a week ahead.<ref name="RGW">{{cite book |author=Watts, Robert G. |title=Global Warming and the Future of the Earth |url=https://archive.org/details/globalwarmingfut00watt_399 |url-access=limited |publisher=Morgan & Claypool |year=2007 |page=[https://archive.org/details/globalwarmingfut00watt_399/page/n22 17] }}</ref> This does not mean that one cannot assert anything about events far in the future—only that some restrictions on the system are present. For example, we know that the temperature of the surface of the earth will not naturally reach {{convert|100|C|F}} or fall below {{convert|-130|C|F}} on earth (during the current [[geologic era]]), but we cannot predict exactly which day will have the hottest temperature of the year.

In more mathematical terms, the [[Lyapunov exponent]] measures the sensitivity to initial conditions, in the form of rate of exponential divergence from the perturbed initial conditions.<ref>{{Cite web|url=http://mathworld.wolfram.com/LyapunovCharacteristicExponent.html|title=Lyapunov Characteristic Exponent|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-11-24}}</ref> More specifically, given two starting [[trajectory|trajectories]] in the [[phase space]] that are infinitesimally close, with initial separation <math>\delta \mathbf{Z}_0</math>, the two trajectories end up diverging at a rate given by

:<math> | \delta\mathbf{Z}(t) | \approx e^{\lambda t} | \delta \mathbf{Z}_0 |,</math>

where <math>t</math> is the time and <math>\lambda</math> is the Lyapunov exponent. The rate of separation depends on the orientation of the initial separation vector, so a whole spectrum of Lyapunov exponents can exist. The number of Lyapunov exponents is equal to the number of dimensions of the phase space, though it is common to just refer to the largest one. For example, the maximal Lyapunov exponent (MLE) is most often used, because it determines the overall predictability of the system. A positive MLE, coupled with the solution's boundedness, is usually taken as an indication that the system is chaotic.<ref name=":2" />

In addition to the above property, other properties related to sensitivity of initial conditions also exist. These include, for example, [[Measure (mathematics)|measure-theoretical]] [[Mixing (mathematics)|mixing]] (as discussed in [[ergodicity|ergodic]] theory) and properties of a [[Kolmogorov automorphism|K-system]].<ref name="WerndlCharlotte" />