Xaos nazariyasi sahifasining manbasini koʻrish

[[Fayl:Lorenz_attractor_yb.svg|right|thumb|201x201px|[[Lorenz tizimi|Lorenz attraktorining]] qiymatlar boʻyicha grafigi. {{Nobr|''r'' {{=}} 28}}, {{Nobr|σ {{=}} 10}}, {{Nobr|''b'' {{=}} {{sfrac|8|3}}}}]]
'''Xaos nazariyasi<ref name=":5">{{Veb manbasi |url=https://uz.eferrit.com/chaos-nazariyasi/ |sarlavha=Chaos nazariyasi |sana=eferrit.com |qaralgan sana=16.12.2024 |til=uz}}</ref>''' (yoki '''xaologiya'''<ref>chaology. (2024, June 22). Wiktionary. Retrieved 07:12, November 17, 2024 from https://en.wiktionary.org/w/index.php?title=chaology&oldid=80484855</ref>) – bu fanlararo ilmiy tadqiqot sohasi va [[Matematika|matematikaga]] oid boʻlim. Ushbu nazariya dinamik tizimlarning boshlangʻich sharoitlarga nisbatan yuqori sezgirlikka ega boʻlgan asosiy qonuniyatlari va deterministik tamoyillariga eʼtibor qaratadi{{sfn|Alimova. D.A|2018|p=131}}. Ilgari bu tizimlar butunlay tasodifiy tartibsizlik va tartibsizlik holatlariga ega deb hisoblangan<ref>{{Veb manbasi |url=https://www.britannica.com/science/chaos-theory |sarlavha=chaos theory {{!}} Definition & Facts |ish=Encyclopedia Britannica |til=en |qaralgan sana=2024-12-15}}</ref>. Xaos nazariyasiga koʻra, murakkab va xaotik tizimlarning koʻrinadigan tasodifiyligi ichida tub qonuniyatlar, oʻzaro bogʻliqlik, doimiy teskari aloqa zanjirlari, takrorlanish, oʻz-oʻziga oʻxshashlik, fraktallar va oʻz-oʻzini tashkil etish jarayonlari mavjud<ref name=":1">{{Veb manbasi |url=https://fractalfoundation.org/resources/what-is-chaos-theory/ |sarlavha=What is Chaos Theory? – Fractal Foundation |til=en-US |qaralgan sana=2024-12-15}}</ref>. Xaos nazariyasining asosiy tamoyillaridan biri boʻlgan [[kapalak effekti]]<ref name=":14">{{Veb manbasi |url=https://uz.eturbonews.com/tartibsizlik-davrida-turizm/ |sarlavha=Xaos davrida turizm |qaralgan sana=19.12.2024}}</ref> deterministik, chiziqli boʻlmagan tizimning bir holatidagi kichik oʻzgarishlar keyingi holatlarda katta farqlarga olib kelishi mumkinligini tavsiflaydi. Bu nazariyaga koʻra, boshlangʻich sharoitlarga nisbatan yuqori sezuvchanlik mavjud<ref>{{Veb manbasi |url=http://mathworld.wolfram.com/Chaos.html |sarlavha=Chaos |muallif=Weisstein |ism=Eric W. |ish=mathworld.wolfram.com |til=en |qaralgan sana=2024-12-15}}</ref>. Metaforik yoʻl bilan ifodalaganda [[Braziliya|Braziliyadagi]] bir kapalak qanotlarini qoqishi [[Texas|Texasda]] [[tornado]] keltirib chiqarishi mumkin deb hisoblangan<ref name="Lorenz">{{Kitob manbasi |last=Lorenz |first=Edward |title=The Essence of Chaos |publisher=University of Washington Press |year=1993 |isbn=978-0-295-97514-6 |url=https://books.google.com/books?id=j5Ub6sMCoOsC}}</ref><ref>{{Veb manbasi |url=https://geoffboeing.com/2015/03/chaos-theory-logistic-map/ |sarlavha=Chaos Theory and the Logistic Map |muallif=Boeing |ism=Geoff |sana=26 March 2015 |til=en |qaralgan sana=2020-05-17}}</ref><sup>:181-184</sup><ref name=":7">{{Cite journal|last=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}} [[Fayl:CC-BY_icon.svg|50x50px]] Text was copied from this source, which is available under a [[ccorg:licenses/by/4.0/|Creative Commons Attribution 4.0 International License]].</ref>.
Boshlangʻich sharoitlardagi kichik farqlar, masalan, oʻlchov xatoliklari yoki sonli hisoblashlardagi yaxlitlash xatolari tufayli, bunday dinamik tizimlar juda farqli natijalarga olib kelishi mumkin. Bu esa, umuman olganda, ularning uzoq muddatli xulq-atvorini bashorat qilishni imkonsiz qiladi<ref>{{Kitob manbasi |last=Kellert |first=Stephen H. |title=In the Wake of Chaos: Unpredictable Order in Dynamical Systems |url=https://archive.org/details/inwakeofchaosunp0000kell |publisher=University of Chicago Press |year=1993 |isbn=978-0-226-42976-2 |page=[https://archive.org/details/inwakeofchaosunp0000kell/page/32 32]}}</ref>. Bu holat, ushbu tizimlar deterministik boʻlsa ham sodir boʻlishi mumkin, yaʼni ularning kelajakdagi xatti-harakati noyob evolyutsiyaga amal qiladi va faqat boshlangʻich sharoitlar tomonidan toʻliq aniqlanadi, tasodifiy elementlar ishtirok etmaydi<ref name=":2">{{Citation |last=Bishop |first=Robert |title=Chaos |date=2017 |url=https://plato.stanford.edu/archives/spr2017/entries/chaos/ |work=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |access-date=2019-11-24 |edition=Spring 2017 |publisher=Metaphysics Research Lab, Stanford University}}</ref>. Boshqacha aytganda, ushbu tizimlarning deterministik tabiati ularni taxmin qilinadigan qilmaydi<ref>{{Harvard citation no brackets|Kellert|1993|p=62}}</ref><ref name="WerndlCharlotte">{{Cite journal|last=Werndl, Charlotte|title=What are the New Implications of Chaos for Unpredictability?|url=https://archive.org/details/sim_british-journal-for-the-philosophy-of-science_2009-03_60_1/page/n200|journal=The British Journal for the Philosophy of Science|volume=60|issue=1|pages=195–220|year=2009|doi=10.1093/bjps/axn053|arxiv=1310.1576}}</ref>. Bunday xatti-harakat determinizm xaosi yoki shunchaki xaos deb ataladi<ref>{{Veb manbasi |url=http://mpe.dimacs.rutgers.edu/2013/03/17/chaos-in-an-atmosphere-hanging-on-a-wall/ |sarlavha=Chaos in an Atmosphere Hanging on a Wall |muallif=Danforth |ism=Christopher M. |sana=April 2013 |ish=Mathematics of Planet Earth 2013 |qaralgan sana=2024-12-15}}</ref>.[[Fayl:Double-compound-pendulum.gif|thumb|Ikki sterjenli mayatnikning xaotik harakatini koʻrsatuvchi animatsiya. Oraliq energiya holatidagi oʻzgarishlarni koʻrsatadi.]]Xaotik xatti-harakatlar koʻplab tabiiy tizimlarda, jumladan suyuqlik oqimi, [[Aritmiya|yurak aritmiyasini buzilishi]], ob-havo va iqlimda mavjud<ref name="Lorenz1961">{{Cite journal|last=Lorenz, Edward N.|title=Deterministic non-periodic flow|journal=Journal of the Atmospheric Sciences|volume=20|pages=130–141|year=1963|doi=10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2|issue=2|bibcode=1963JAtS...20..130L}}</ref><ref name=":22">{{Citation |last=Bishop |first=Robert |title=Chaos |date=2017 |url=https://plato.stanford.edu/archives/spr2017/entries/chaos/ |work=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |access-date=2024-12-15 |edition=Spring 2017 |publisher=Metaphysics Research Lab, Stanford University}}</ref><ref>{{Kitob manbasi |last=Ivancevic |first=Vladimir G. |title=Complex nonlinearity: chaos, phase transitions, topology change, and path integrals |url=https://archive.org/details/complexnonlinear0000ivan |year=2008 |publisher=Springer |isbn=978-3-540-79356-4}}</ref>. Shuningdek, yoʻl harakati kabi sunʼiy tarkibiy qismlarga ega baʼzi tizimlarda oʻz-oʻzidan sodir boʻladi<ref name=":12">{{Veb manbasi |url=https://fractalfoundation.org/resources/what-is-chaos-theory/ |sarlavha=What is Chaos Theory? – Fractal Foundation |til=en-US |qaralgan sana=2024-12-15}}</ref>. Ushbu xatti-harakatni xaotik [[Matematik model|matematik modelni]] tahlil qilish yoki takrorlanish grafiklari va Puankare xaritalari kabi analitik usullar orqali oʻrganish mumkin. Xaos nazariyasi turli fanlarda, jumladan, [[meteorologiya]]<ref name=":22" />, [[antropologiya]]<ref name=":0">{{Kitob manbasi |title=On the order of chaos. Social anthropology and the science of chaos |last=Mosko M.S., Damon F.H. (Eds.) |publisher=Berghahn Books |year=2005 |location=Oxford}}</ref>, [[sotsiologiya]], atrof-muhit fanlari, [[informatika]], [[muhandislik]], [[Iqtisodiyot (fan)|iqtisodiyot]], [[Ekologiya|ekologiyasi]] va [[pandemiya]] [[Crisis management|krizislarini boshqarish]] sohalarda qoʻllanilishi mumkin<ref name="CT-REF-20">{{Veb manbasi |url=https://www.researchgate.net/publication/340775886 |sarlavha=Covid-19 Pandemic and Chaos Theory: Applications based on a Bibliometric Analysis |muallif=Piotrowski |ism=Chris |ish=researchgate.net |qaralgan sana=2024-12-15}}</ref><ref name="CT-REF-21">{{Kitob manbasi |last=Weinberger |first=David |title=Everyday Chaos – Technology, Complexity and How We're Thriving in a New World of Possibility |publisher=Harvard Business Review Press |year=2019 |isbn=9781633693968 |url=https://books.google.com/books?id=R7V2DwAAQBAJ}}</ref>. Nazariya murakkab dinamik tizimlar, xaos nazariyasining chekkasi va oʻz-oʻzini yigʻish jarayonlari kabi oʻrganish sohalari uchun asos boʻldi.

== Kirish ==
Xaos nazariyasi determinizm tizimlariga taalluqli boʻlib, ularning xatti-harakatlarini oldindan aytib berish mumkinligi bilan izohlanadi. Xaotik tizimlar maʼlum vaqtgacha prognoz qilinadi va keyin tasodifiy boʻlib qoladi. Xaotik tizimning xatti-harakatini samarali prognoz qilish mumkin boʻlgan vaqt hajmi uchta omilga bogʻliq: prognozda qanchalik noaniqlikka yoʻl qoʻyilishi mumkinligi, uning joriy holatini qanchalik aniq oʻlchash mumkinligi va tizimning dinamikasiga bogʻliq boʻlgan vaqt shkalasi Lyapunov vaqti deb ataladi. Lyapunov ning baʼzi misollari: tartibsiz elektr zanjirlari, taxminan 1 millisekund; ob-havo tizimlari, bir necha kun (isbotlanmagan); ichki quyosh tizimi, 4-5 million yil<ref>{{Cite journal|last=Wisdom|first1=Jack|last2=Sussman|first2=Gerald Jay|date=1992-07-03|title=Chaotic Evolution of the Solar System|journal=Science|language=en|volume=257|issue=5066|pages=56–62|doi=10.1126/science.257.5066.56|issn=1095-9203|pmid=17800710|bibcode=1992Sci...257...56S}}</ref>. Tartibsiz tizimlarda prognozdagi noaniqlik oʻtgan vaqt bilan eksponensial tarzda ortib boradi. Demak, matematik jihatdan prognoz vaqtini ikki baravar oshirish prognozdagi proporsional noaniqlikni kvadratga nisbatan koʻpaytiradi. Bu shuni anglatadiki, amalda Lyapunov vaqtining ikki yoki uch baravaridan ortiq vaqt oraligʻida mazmunli prognoz qilish mumkin emas. Mantiqiy bashorat qilish imkoni boʻlmaganda, tizim tasodifiy tarzda paydo boʻladi<ref>''Sync: The Emerging Science of Spontaneous Order'', Steven Strogatz, Hyperion, New York, 2003, pages 189–190.</ref>.

== Xaotik dinamika ==
[[Fayl:Chaos_Sensitive_Dependence.svg|thumb|x → 4 x (1 – x) va y → (x + y) [[Modulo|mod]] 1 bilan aniqlangan [[Xarita (matematika)|xarita]] boshlangʻich x holatlarga sezgirlikni koʻrsatadi. Bu yerda x va y qiymatlarining ikkita qatori vaqt oʻtishi bilan kichik boshlangʻich farqdan sezilarli darajada uzoqlashadi]]
Odatda, „xaos“ soʻzi „ tartibsizlik holati“ degan maʼnoni anglatadi<ref name=":53">{{Veb manbasi |url=https://uzbek.english-dictionary.help/english-to-uzbek-meaning-chaos |sarlavha=chaos - tartibsizlik |sana= |qaralgan sana=17.12.2024 |til=uz}}</ref><ref>Definition of {{linktext|chaos}} at [[Vikilugʻat|Wiktionary]];</ref><ref>{{Veb manbasi |url=https://www.dictionary.com/browse/chaos |sarlavha=Definition of chaos {{!}} Dictionary.com |ish=www.dictionary.com |til=en |qaralgan sana=2019-11-24}}</ref>. Ammo xaos nazariyasida bu atamaga aniqroq taʼrif berilgan. Garchi xaosning umumiy qabul qilingan matematik taʼrifi mavjud boʻlmasa-da, dastlab Robert L. Devaney tomonidan ishlab chiqilgan umumiy taʼrifga koʻra, dinamik tizimni xaotik deb tasniflash uchun u quyidagi xususiyatlarga ega boʻlishi kerak<ref>{{Kitob manbasi |title=A First Course in Dynamics: With a Panorama of Recent Developments |last=Hasselblatt |first=Boris |year=2003 |publisher=Cambridge University Press |isbn=978-0-521-58750-1}}</ref>:

# [[Kapalak effekti|Boshlangʻich sharoitlarga sezgir boʻlishi kerak]],
# Topologik jihatdan tranzitiv boʻlishi kerak,
# [[Zich to‘plam|Zich]] [[Davriy nuqta|davriy orbitlarga]] ega boʻlishi kerak.

Baʼzi hollarda yuqoridagi oxirgi ikkita xususiyat aslida boshlangʻich sharoitlarga sezgirlikni nazarda tutishi koʻrsatilgan<ref>{{Kitob manbasi |last=Elaydi, Saber N. |title=Discrete Chaos |publisher=Chapman & Hall/CRC |year=1999 |isbn=978-1-58488-002-8 |page=137}}</ref><ref>{{Kitob manbasi |last=Basener, William F. |title=Topology and its applications |publisher=Wiley |year=2006 |isbn=978-0-471-68755-9 |page=42}}</ref>. Diskret vaqt holida bu [[Metrik fazo|metrik fazolardagi]] barcha [[Uzluksiz funksiya|uzluksiz]] xaritalar uchun toʻgʻri keladi<ref>{{Cite journal|last=Banks|last2=Brooks|last3=Cairns|last4=Davis|last5=Stacey|title=On Devaney's definition of chaos|url=https://archive.org/details/sim_american-mathematical-monthly_1992-04_99_4/page/n29|journal=The American Mathematical Monthly|volume=99|issue=4|date=1992|pages=332–334|doi=10.1080/00029890.1992.11995856}}</ref>. Bunday hollarda, bu koʻpincha eng muhim amaliy xususiyat boʻlsa-da, taʼrifda „dastlabki sharoitlarga sezgirlik“ni taʼkidlash shart emas.

Agar eʼtibor faqat intervallarga qaratilsa, ikkinchi xususiyat qolgan ikkitasini ham oʻz ichiga oladi<ref>{{Cite journal|last=Vellekoop, Michel|last2=Berglund, Raoul|title=On Intervals, Transitivity = Chaos|journal=The American Mathematical Monthly|volume=101|issue=4|pages=353–5|date=April 1994|jstor=2975629|doi=10.2307/2975629}}</ref>. Xaosning muqobil va odatda zaifroq taʼrifi esa yuqoridagi roʻyxatda faqat dastlabki ikkita xususiyatga asoslanadi<ref>{{Kitob manbasi |last=Medio, Alfredo |title=Nonlinear Dynamics: A Primer |url=https://archive.org/details/nonlineardynamic00medi |publisher=Cambridge University Press |year=2001 |isbn=978-0-521-55874-7 |page=[https://archive.org/details/nonlineardynamic00medi/page/n175 165]}}</ref>.

== Boshlangʻich sharoitlarga sezgirlik ==
{{Main|Kapalak effekti}}
Boshlangʻich shartlarga sezgirlik degani, xaotik tizimdagi har bir nuqta boshqa nuqtalar tomonidan tasodifiy ravishda juda yaqinlashib, kelajakdagi yoʻnalishlari yoki harakat yoʻli sezilarli darajada farq qilishidir. Shunday qilib, hozirgi yoʻnalishdagi juda kichik oʻzgarish yoki kelajakdagi xatti-harakatlarini sezilarli darajada farq qilishiga olib kelishi mumkin<ref name=":13">{{Veb manbasi |url=https://fractalfoundation.org/resources/what-is-chaos-theory/ |sarlavha=What is Chaos Theory? – Fractal Foundation |til=en-US |qaralgan sana=2019-11-24}}</ref>.

Boshlangʻich shartlarga sezgirlikning natijasi shundaki, agar tizim haqidagi maʼlumotlar cheklangan boʻlsa (bu amalda odatda shunday boʻladi), maʼlum bir vaqt oʻtgach tizimni prognoz qilish imkonsiz boʻladi. Bu hodisa ob-havo bilan aloqador boʻlib, ob-havo odatda bir hafta oldindan prognoz qilinishi mumkin<ref name="RGW">{{Kitob manbasi |last=Watts, Robert G. |title=Global Warming and the Future of the Earth |url=https://archive.org/details/globalwarmingfut00watt_399 |publisher=Morgan & Claypool |year=2007 |page=[https://archive.org/details/globalwarmingfut00watt_399/page/n22 17]}}</ref>. Bu, uzoq kelajakdagi voqealar haqida hech narsa aytib boʻlmaydi degan maʼnoni anglatmaydi – faqat tizimga baʼzi cheklovlar mavjudligini bildiradi. Masalan, biz yer yuzasidagi harorat hech qachon 100 °C (212 °F) ga koʻtarilmasligi yoki −130 °C (−202 °F) ga tushmasligini bilamiz (hozirgi [[Geoxronologik shkala|geologik davrda]]), ammo yilning eng issiq kuni qaysi kun boʻlishini aniq prognoz qilishimiz mumkin emas.

=== Davriy boʻlmaganlik ===
[[Fayl:LogisticTopMixing1-6.gif|thumb|Animatsiya doiraviy boshlangʻich shartlarning birinchi va oltinchi iteratsiyasini koʻrsatadi. Koʻrinib turibdiki, iteratsiyalar davomida aralashish sodir boʻladi. Oltinchi iteratsiya nuqtalarning fazalar fazosida deyarli toʻliq sochilganligini koʻrsatadi. Agar biz iteratsiyalarni davom ettirganimizda edi, aralashtirish bir xil va qaytmas boʻlar edi.]]
Xaotik tizim oʻzgaruvchining qiymatlari ketma-ketligiga ega boʻlishi mumkin, ular aynan takrorlanadi va ushbu ketma-ketlikning istalgan nuqtasidan boshlab davriy harakatni beradi. Biroq, bunday davriy ketma-ketliklar tortish emas, balki itarishdir, yaʼni agar evolyutsion oʻzgaruvchi ketma-ketlikdan tashqarida boʻlsa, qanchalik yaqin boʻlmasin, u ketma-ketlikka kirmaydi va aslida undan uzoqlashadi. Shunday qilib, deyarli barcha boshlangʻich shartlar uchun oʻzgaruvchi davriy boʻlmagan xatti-harakatlar bilan xaotik ravishda rivojlanadi.

=== Topologik aralashuv ===
[[Fayl:Chaos_Topological_Mixing.png|thumb|x → 4 x (1 – x) va y → (x + y) mod 1 bilan aniqlangan xaritada ham topologik aralashuv kuzatiladi. Bu yerda koʻk soha dinamika orqali avval binafsha rangga, keyin pushti va qizil sohalarga va oxir-oqibat fazo boʻylab tarqalgan vertikal chiziqlar bulutiga aylanadi.]]
Topologik aralashuv (yoki topologik uzluksizlikning zaif sharti) shuni anglatadiki, tizim vaqt oʻtishi bilan oʻz faza maydonining har qanday berilgan sohasi yoki ochiq toʻplami oxir-oqibat boshqa bir berilgan soha bilan kesishadi. Bu matematik „aralashuv“ tushunchasi oddiy intuitiv tushunchaga mos keladi va rangli boʻyoqlar yoki suyuqliklarning aralashishi xaotik tizimga misol boʻladi.

=== Davriy orbitalar zichligi ===
Xaotik sistemaning zich davriy orbitalarga ega boʻlishi fazoning har bir nuqtasiga ixtiyoriy ravishda davriy orbitalar yaqinlashganini anglatadi<ref name="Devaney">{{Harvard citation no brackets|Devaney|2003}}</ref>. x → 4 x (1 – x) orqali aniqlangan bir oʻlchovli logistik xarita davriy orbitalar zichligiga ega boʻlgan eng sodda tizimlardan biridir. Masalan: <math>\tfrac{5-\sqrt{5}}{8}</math>→<math>\tfrac{5+\sqrt{5}}{8}</math>→<math>\tfrac{5-\sqrt{5}}{8}</math> (taxminan 0.3454915 → 0.9045085 → 0.3454915) 2 davrli (noturgʻun) orbita boʻlib, shunga oʻxshash orbitalar 4, 8, 16 davrlar va boshqalar uchun mavjud<ref>{{Harvard citation no brackets|Alligood|Sauer|Yorke|1997}}</ref>.

Sharkovskiyning teoremasi Li va Yorkening<ref>{{Cite journal|last=Li|first1=T.Y.|last2=Yorke|first2=J.A.|title=Period Three Implies Chaos|journal=[[American Mathematical Monthly]]|volume=82|pages=985–92|year=1975|url=http://pb.math.univ.gda.pl/chaos/pdf/li-yorke.pdf|author-link=Tien-Yien Li|doi=10.2307/2318254|issue=10|author2-link=James A. Yorke|bibcode=1975AmMM...82..985L|archiveurl=https://web.archive.org/web/20091229042210/http://pb.math.univ.gda.pl/chaos/pdf/li-yorke.pdf|archivedate=2009-12-29|jstor=2318254}}</ref> (1975) isboti boʻlib, uchinchi davrning muntazam sikliga ega boʻlgan har qanday uzluksiz bir oʻlchovli tizim har bir boshqa uzunlikdagi muntazam sikllarni, shuningdek, butunlay tartibsiz orbitalarni ham koʻrsatadi.

=== Noodatiy attraktorlar ===
[[Fayl:TwoLorenzOrbits.jpg|right|thumb|[[Lorenz tizimi|Lorenz attraktori]] xaotik holati koʻrsatilgan. Ushbu ikkita grafik attraktor egallagan faza fazosi sohasidagi boshlangʻich shartlarga sezgir bogʻliqlikni koʻrsatadi.]]
Baʼzi dinamik sistemalar, masalan, x → 4 x (1 – x) orqali aniqlangan bir oʻlchovli logistik xarita hamma joyda xaotik boʻladi, lekin koʻp hollarda xaotik holat faqat fazali fazoning qism toʻplamida uchraydi. Eng qiziqarli holatlar xaotik xatti-harakatlar attraktorda sodir boʻlganda paydo boʻladi, shuning uchun boshlangʻich shartlarning katta toʻplami ushbu xaotik sohaga yaqinlashuvchi orbitalarga olib keladi<ref>{{Cite journal|last=Strelioff|first1=Christopher|last2=et.|first2=al.|title=Medium-Term Prediction of Chaos|journal=Phys. Rev. Lett.|date=2006|volume=96|issue=4|pages=044101|doi=10.1103/PhysRevLett.96.044101|pmid=16486826|bibcode=2006PhRvL..96d4101S}}</ref>.

Xaotik attraktorni tasavvur qilishning eng oson usuli bu attraktorning tortish havzasidagi nuqtadan boshlash va keyin uning keyingi orbitasini chizishdir. Topologik oʻtkazuvchanlik sharti tufayli bu yakuniy attraktorning toʻliq rasmini hosil qilishi mumkin va haqiqatan ham oʻng tomondagi rasmda koʻrsatilgan ikkala orbita Lorenz attraktorining umumiy shaklini tasvirlaydi. Ushbu attraktor Lorenz ob-havo tizimining oddiy uch oʻlchovli modelidan kelib chiqadi. Lorenz attraktori, ehtimol, eng mashhur xaotik tizim diagrammalaridan biridir, chunki u nafaqat birinchilardan biri, balki eng murakkablaridan biri hamdir va shuning uchun biroz tasavvur bilan kapalak qanotlariga oʻxshagan juda qiziqarli naqshni keltirib chiqaradi.

Qoʻzgʻalmas nuqtaviy attraktorlar va limit sikllardan farqli oʻlaroq, noodatiy attraktorlar deb nomlanuvchi xaotik tizimlardan kelib chiqadigan attraktorlar katta tafsilotlarga va murakkablikka ega. Noodatiy attraktorlar ham uzluksiz dinamik tizimlarda (masalan, [[Lorenz tizimi]]), ham baʼzi [[Chekli matematika|diskret]] tizimlarda (masalan, Henon xaritasi) uchraydi. Boshqa diskret dinamik sistemalar qatʼiy nuqtalarning tortishish havzalari oʻrtasidagi chegarada hosil boʻladigan Julia toʻplami deb ataladigan itaruvchi tuzilishga ega. Juliya toʻplamlarini gʻalati repellerlar deb oʻylash mumkin. Gʻalati attraktorlar ham, Julia toʻplamlari ham odatda fraktal tuzilishga ega va ular uchun fraktal oʻlchamni hisoblash mumkin.

=== Xaotik sistemaning minimal murakkabligi ===
[[Fayl:Logistic_Map_Bifurcation_Diagram,_Matplotlib.svg|right|thumb|Logistik xaritaning bifurkatsion sxemasi x → r x (1 – x). Har bir vertikal boʻlak r ning maʼlum bir qiymati uchun attraktorni koʻrsatadi. Diagrammada r ortishi bilan davrning ikki baravar koʻpayishi va oxir-oqibat xaos hosil boʻlishi koʻrsatilgan. ]]
Diskret xaotik tizimlar. Masalan: logistika xaritasi, oʻzlarining oʻlchamidan qatʼiy nazar, noodatiy attraktorlarni namoyon qilishi mumkin. Aksincha, uzluksiz dinamik tizimlar uchun Poincaré–Bendixson teoremasi shuni koʻrsatadiki, noodatiy attraktor faqat uch yoki undan ortiq oʻlchamda paydo boʻlishi mumkin. Cheklangan oʻlchamli chiziqli tizimlar hech qachon xaotik boʻlmaydi; bir dinamik tizim xaotik xatti-harakatni koʻrsatishi uchun u noxatolik yoki cheksiz oʻlchamli boʻlishi kerak.

Poincaré–Bendixson teoremasi shuni taʼkidlaydiki, ikki oʻlchovli differensial tenglama juda muntazam xatti-harakatga ega. Quyida muhokama qilinadigan Lorenz attraktori uchta differensial tenglamadan iborat tizim tomonidan hosil boʻladi, masalan:

<math> \begin{align}
\frac{\mathrm{d}x}{\mathrm{d}t} &= \sigma y - \sigma x, \\
\frac{\mathrm{d}y}{\mathrm{d}t} &= \rho x - x z - y, \\
\frac{\mathrm{d}z}{\mathrm{d}t} &= x y - \beta z.
\end{align} </math>

bu yerda x, y va z sistema holatini tashkil qiladi, t – vaqt, σ, ρ va β tizim [[Parametr|parametrlari]] hisoblanadi. Oʻng tomondagi hadlarning beshtasi chiziqli, ikkitasi kvadratik; jami yettita had. Yana bir mashhur xaotik attraktor Rössler tenglamalari orqali hosil qilinadi, ular yettitadan faqat bitta chiziqli boʻlmagan hadga ega. Sprott<ref>{{Cite journal|last=Sprott|first1=J.C.|year=1997|title=Simplest dissipative chaotic flow|journal=[[Physics Letters A]]|volume=228|issue=4–5|pages=271–274|doi=10.1016/S0375-9601(97)00088-1|bibcode=1997PhLA..228..271S}}</ref> faqat beshta hadga ega boʻlgan uch oʻlchovli tizimni topdi, unda faqat bitta chiziqsiz had mavjud boʻlib, baʼzi parametr qiymatlari uchun tartibsizlikni namoyish etadi. Zhang va Heidel<ref>{{Cite journal|last=Fu|first1=Z.|last2=Heidel|first2=J.|year=1997|title=Non-chaotic behaviour in three-dimensional quadratic systems|journal=[[Nonlinearity (journal)|Nonlinearity]]|volume=10|issue=5|pages=1289–1303|doi=10.1088/0951-7715/10/5/014|bibcode=1997Nonli..10.1289F}}</ref><ref>{{Cite journal|last=Heidel|first1=J.|last2=Fu|first2=Z.|year=1999|title=Nonchaotic behaviour in three-dimensional quadratic systems II. The conservative case|journal=Nonlinearity|volume=12|issue=3|pages=617–633|doi=10.1088/0951-7715/12/3/012|bibcode=1999Nonli..12..617H}}</ref> koʻrsatdiki, hech boʻlmaganda dissipativ va konservativ kvadratik tizimlar uchun oʻng tomonda atigi uchta yoki toʻrtta hadga ega boʻlgan uch oʻlchovli kvadratik tizimlar tartibsiz xatti-harakatni namoyon qila olmaydi. Sababi, sodda qilib aytganda, bunday sistemalarning yechimlari ikki oʻlchovli sirtga asimptotik va shuning uchun yechimlar yaxshi tuzilgan.

Poincaré–Bendixson teoremasi, Evklid tekisligidagi uzluksiz dinamik tizim xaotik boʻla olmasligini koʻrsatsa-da, Evklid boʻlmagan geometriyaga ega boʻlgan ikki oʻlchovli uzluksiz tizimlar baʼzi xaotik xususiyatlarni koʻrsatishi mumkin<ref>{{Cite journal|last=Ulcigrai|first1=Corinna|date=2021|title=Slow chaos in surface flows|url=http://link.springer.com/10.1007/s40574-020-00267-0|journal=Bollettino dell'Unione Matematica Italiana|language=en|volume=14|issue=1|pages=231–255|arxiv=2010.06231|doi=10.1007/s40574-020-00267-0|issn=1972-6724}}</ref>. Kutilmaganda, xaos chiziqli tizimlarda ham yuzaga kelishi mumkin, agar ular cheksiz oʻlchamli boʻlsa<ref>{{Cite journal|last=Bonet|first1=J.|last2=Martínez-Giménez|first2=F.|last3=Peris|first3=A.|year=2001|title=A Banach space which admits no chaotic operator|journal=Bulletin of the London Mathematical Society|volume=33|issue=2|pages=196–8|doi=10.1112/blms/33.2.196}}</ref>. Chiziqli xaos nazariyasi funktsional tahlil deb nomlanuvchi matematik tahlil boʻlimida ishlab chiqilmoqda.

Yuqoridagi uchta oddiy differensial tenglamalarning toʻplami uch oʻlchovli Lorenz modeli deb ataladi<ref>{{Cite journal|last=Shen|first1=Bo-Wen|date=2014-05-01|title=Nonlinear Feedback in a Five-Dimensional Lorenz Model|url=https://journals.ametsoc.org/doi/10.1175/JAS-D-13-0223.1|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|issn=0022-4928}}</ref>. 1963-yildan boshlab yuqori oʻlchovli Lorenz modellari koʻplab tadqiqotlarda ishlab chiqilgan boʻlib, ular yechimning barqarorligiga taʼsir qiluvchi noaniqlik darajasining oshishi, shuningdek, isitish va energiya yoʻqotishlari bilan bogʻliq umumiy taʼsirni oʻrganishda qoʻllanilgan<ref>{{Cite journal|last=Musielak|first1=Dora E.|last2=Musielak|first2=Zdzislaw E.|last3=Kennamer|first3=Kenny S.|date=2005-03-01|title=The onset of chaos in nonlinear dynamical systems determined with a new fractal technique|url=https://www.worldscientific.com/doi/abs/10.1142/S0218348X0500274X|journal=Fractals|volume=13|issue=1|pages=19–31|doi=10.1142/S0218348X0500274X|issn=0218-348X}}</ref><ref>{{Cite journal|last=Roy|first1=D.|last2=Musielak|first2=Z. E.|date=2007-05-01|title=Generalized Lorenz models and their routes to chaos. I. Energy-conserving vertical mode truncations|url=https://www.sciencedirect.com/science/article/pii/S0960077906001937|journal=Chaos, Solitons & Fractals|language=en|volume=32|issue=3|pages=1038–1052|doi=10.1016/j.chaos.2006.02.013|bibcode=2007CSF....32.1038R|issn=0960-0779}}</ref><ref name=":4">{{Cite journal|last=Shen|first1=Bo-Wen|date=2019-03-01|title=Aggregated Negative Feedback in a Generalized Lorenz Model|journal=International Journal of Bifurcation and Chaos|volume=29|issue=3|pages=1950037–1950091|doi=10.1142/S0218127419500378|bibcode=2019IJBC...2950037S|issn=0218-1274}}</ref><ref name=":54">{{Cite journal|last=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|issn=0003-0007}}</ref>.

=== Cheksiz oʻlchamli xaritalar ===
Bogʻlangan diskret xaritalarning toʻgʻridan-toʻgʻri umumlashtirilishi<ref name="Moloney, J V 1986">{{Cite journal|title=Solitary waves as fixed points of infinite-dimensional maps for an optical bistable ring cavity: Analysis|journal=Journal of Mathematical Physics|volume=29|issue=1|pages=63|year=1988|last=Adachihara|first1=H|last2=McLaughlin|first2=D W|last3=Moloney|first3=J V|last4=Newell|first4=A C|doi=10.1063/1.528136|bibcode=1988JMP....29...63A}}</ref> fazoviy taqsimlangan xaritalar oʻrtasidagi oʻzaro taʼsirga vositachilik qiluvchi konvolyutsiya integraliga asoslanadi:

<math>\psi_{n+1}(\vec r,t)  = \int K(\vec r - \vec r^{,},t) f [\psi_{n}(\vec r^{,},t) ]d {\vec r}^{,}</math>

Bu yerda yadro <math>K(\vec r - \vec r^{,},t)</math> fizik tizimning Green funksiyasidan hosil boʻlgan tarqatuvchi hisoblanadi<ref name="Okulov, A Yu 1988">{{Kitob manbasi |chapter=Spatiotemporal dynamics of a wave packet in nonlinear medium and discrete maps |title=Proceedings of the Lebedev Physics Institute |language=ru |editor=N.G. Basov |publisher=Nauka |pages=202–222 |year=1988 |last=Okulov |first=A Yu}}</ref>. <math> f [\psi_{n}(\vec r,t) ] </math> logistik xarita singari <math> \psi \rightarrow G \psi [1 - \tanh (\psi)]</math> yoki murakkab xaritalar boʻlishi mumkin. Murakkab xaritalarga misol sifatida Julia toʻplami <math> f[\psi] = \psi^2</math> yoki Ikeda xaritasi <math> \psi_{n+1} = A + B \psi_n e^{i (|\psi_n|^2 + C)} </math> keltirilishi mumkin. Toʻlqin tarqalishi masalalarida <math>L=ct</math> masofa va <math>\lambda=2\pi/k</math> toʻlqin uzunligi hisobga olinganda, yadroviy yadro <math>K</math> [[Shredinger tenglamasi|Schrödinger tenglamasi]] uchun Green funksiyasi koʻrinishida boʻlishi mumkin<ref name="Okulov, A Yu 2000">{{Cite journal|title=Spatial soliton laser: geometry and stability|journal=Optics and Spectroscopy|volume=89|issue=1|pages=145–147|year=2000|last=Okulov|first1=A Yu|doi=10.1134/BF03356001|bibcode=2000OptSp..89..131O}}</ref><ref name="Okulov, A Yu 2020">{{Cite journal|title=Structured light entities, chaos and nonlocal maps|journal=Chaos, Solitons & Fractals|volume=133|issue=4|page=109638|year=2020|last=Okulov|first1=A Yu|doi=10.1016/j.chaos.2020.109638|arxiv=1901.09274|bibcode=2020CSF...13309638O}}</ref>:

<math> K(\vec r - \vec r^{,},L)  = \frac {ik\exp[ikL]}{2\pi L}\exp[\frac {ik|\vec r-\vec r^{,}|^2}{2 L} ]</math>

=== Jerk tizimlari ===

== Tasodifiy tartib ==
Toʻgʻri sharoitlar mavjud boʻlsa, xaos oʻz-oʻzidan tartibli naqshga aylanadi. [[Kuramoto modeli|Kuramoto modelida]], xaotik tizimda sinxronlashishni hosil qilish uchun toʻrtta shart yetarli. Misollar qatoriga [[Christiaan Huygens|Christiaan Huygensning]] pendulumlarining birikkan tebranishlari, yoritgichlar, [[Neyron (nerv hujayrasi)|neyronlar]], London Millennium koʻprigi rezonansi va [[Jozefson effekti|Josephson bogʻlanishlarining]] katta massivlari kiradi<ref>Steven Strogatz, ''Sync: The Emerging Science of Spontaneous Order'', Hyperion, 2003.</ref>.

Bundan tashqari, nazariy fizika nuqtai nazaridan dinamik xaosning oʻzi, eng umumiy koʻrinishida, oʻz-oʻzidan paydo boʻladigan tartibdir. Bu yerda mohiyati shundaki, tabiatdagi koʻpchilik tartiblar turli simmetriyalarning oʻz-oʻzidan parchalanishi natijasida paydo boʻladi. Ushbu katta hodisalar oilasiga elastiklik, oʻta oʻtkazuvchanlik, ferromagnetizm va boshqalar kiradi. [[Stoxastik dinamikaning supersimmetrik nazariyasi|Stokastik dinamikalarning supersimmetrik nazariyasiga]] koʻra, xaos yoki aniqrogʻi, uning stokastik umumlashmasi ham ushbu oilaning bir qismidir. [[Simmetriyaning o‘z-o‘zidan buzilishi|Buzilgan simmetriya]] topologik supersimmetriya boʻlib, barcha [[Stoxastik differensial tenglama|stokastik differensial tenglamalarda]] yashirin va tegishli [[Fazaviy oʻtish|tartib parametri]] kapalak effektining [[Maydon kvant nazariyasi|maydon-nazariy]] ifodasidir<ref>{{Cite journal|last=Ovchinnikov|first1=I.V.|title=Ubiquitous order known as chaos|date=2024-02-15|journal=Chaos, Solitons & Fractals|language=en|volume=181|issue=5|pages=114611|doi=10.1016/j.chaos.2024.114611|bibcode=2024CSF...18114611O|url=https://www.sciencedirect.com/science/article/abs/pii/S0960077924001620|issn=0960-0779}}</ref>.

== Tarixi ==
[[File:Barnsley_fern_plotted_with_VisSim.PNG|link=https://en.wikipedia.org/wiki/File:Barnsley_fern_plotted_with_VisSim. PNG|thumb|193x193px|Barnsley fern xaos oʻyini yordamida yaratilgan. Tabiiy shakllar (paporotniklar, bulutlar, togʻlar va boshqalar) takrorlanuvchi funksiyalar tizimi (IFS) orqali qayta yaratilishi mumkin.]]
[[James Clerk Maxwell]] birinchi boʻlib „[[Kapalak effekti|kapalak effektini]]“ taʼkidlagan va 1860—1870-yillarda xaos nazariyasini muhokama qilgan dastlabki olimlardan biri sifatida qaraladi<ref>{{Cite journal|last=Hunt|first1=Brian R.|last2=Yorke|first2=James A.|date=1993|title=Maxwell on Chaos|url=https://yorke.umd.edu/Yorke_papers_most_cited_and_post2000/1993_04_Hunt_%20Nonlin-Science-Today%20_Maxwell%20on%20Chaos.PDF|journal=Nonlinear Science Today|volume=3|issue=1}}</ref><ref>{{Veb manbasi |muallif=Everitt |ism=Francis |sana=2006-12-01 |sarlavha=James Clerk Maxwell: a force for physics |url=https://physicsworld.com/a/james-clerk-maxwell-a-force-for-physics/ |qaralgan sana=2024-12-15 |ish=Physics World |til=en-GB}}</ref><ref>{{Cite journal|last=Gardini|first1=Laura|last2=Grebogi|first2=Celso|last3=Lenci|first3=Stefano|date=2020-10-01|title=Chaos theory and applications: a retrospective on lessons learned and missed or new opportunities|journal=Nonlinear Dynamics|language=en|volume=102|issue=2|pages=643–644|doi=10.1007/s11071-020-05903-0|issn=1573-269X|bibcode=2020NonDy.102..643G}}</ref>. Xaos nazariyasining ilk tarafdori [[Henri Poincaré]] boʻldi. 1880-yillarda [[Uch jism muammosi|uch jism masalasini]] oʻrganayotganda, baʼzi orbitallar mavjud boʻlishi mumkinligini aniqladi, ular doimiy ravishda oshib ketmaydi yoki maʼlum bir nuqtaga yaqinlashmaydi<ref>{{Cite journal|last=Poincaré, Jules Henri|title=Sur le problème des trois corps et les équations de la dynamique. Divergence des séries de M. Lindstedt|journal=Acta Mathematica|volume=13|issue=1–2|pages=1–270|year=1890|doi=10.1007/BF02392506}}</ref><ref>{{Kitob manbasi |title=The three-body problem and the equations of dynamics : Poincaré's foundational work on dynamical systems theory |url=https://archive.org/details/threebodyproblem0000unse_u2y5 |last=Poincaré |first=J. Henri |publisher=Springer International Publishing |others=Popp, Bruce D. (Translator) |year=2017 |isbn=9783319528984 |location=Cham, Switzerland |oclc=987302273}}</ref><ref>{{Kitob manbasi |last=Diacu, Florin |title=Celestial Encounters: The Origins of Chaos and Stability |publisher=[[Princeton University Press]] |year=1996}}</ref>. 1898-yilda [[Adamar Jak|Jacques Hadamard]] doimiy manfiy egri chiziqqa ega yuzada ishqalanmasdan harakatlanayotgan erkin zarrachaning xaotik harakati haqida muhim bir tadqiqot eʼlon qildi, bu tadqiqot „[[Hadamard dinamik sistemasi]]“ deb ataladi<ref>{{Cite journal|first1=Jacques|last=Hadamard|year=1898|title=Les surfaces à courbures opposées et leurs lignes géodesiques|journal=Journal de Mathématiques Pures et Appliquées|volume=4|pages=27–73}}</ref>. Hadamard barcha yoʻnalishlar beqaror ekanligini koʻrsatdi, yaʼni barcha zarralar yoʻnalishlari bir-biridan eksponensial tarzda ajralib ketadi va bu jarayonda Lyapunov eksponenti musbat boʻladi.

Xaos nazariyasi ergodik nazariya sohasidan boshlangan. Keyinchalik chiziqli boʻlmagan [[Differensial tenglama|differensial tenglamalari]] mavzusida tadqiqotlar olib borilgan, bu ishlarni [[George David Birkhoff]]<ref>George D. Birkhoff, ''Dynamical Systems,'' vol.&nbsp;9 of the American Mathematical Society Colloquium Publications (Providence, Rhode Island: American Mathematical Society, 1927)</ref>, [[Andrey Kolmogorov]]<ref>{{Cite journal|last=Kolmogorov|first1=Andrey Nikolaevich|author-link=Andrey Nikolaevich Kolmogorov|year=1941|title=Local structure of turbulence in an incompressible fluid for very large Reynolds numbers|journal=[[Doklady Akademii Nauk SSSR]]|volume=30|issue=4|pages=301–5|bibcode=1941DoSSR..30..301K}} Reprinted in: {{Cite journal|journal=Proceedings of the Royal Society A|volume=434|pages=9–13|year=1991|doi=10.1098/rspa.1991.0075|title=The Local Structure of Turbulence in Incompressible Viscous Fluid for Very Large Reynolds Numbers|last=Kolmogorov|first1=A. N.|issue=1890|bibcode=1991RSPSA.434....9K}}</ref><ref>{{Cite journal|last=Kolmogorov|first1=A. N.|year=1941|title=On degeneration of isotropic turbulence in an incompressible viscous liquid|journal=Doklady Akademii Nauk SSSR|volume=31|issue=6|pages=538–540}} Reprinted in: {{Cite journal|journal=Proceedings of the Royal Society A|volume=434|pages=15–17|year=1991|doi=10.1098/rspa.1991.0076|title=Dissipation of Energy in the Locally Isotropic Turbulence|last=Kolmogorov|first1=A. N.|issue=1890|bibcode=1991RSPSA.434...15K}}</ref><ref>{{Kitob manbasi |last=Kolmogorov |first=A. N. |title=Stochastic Behavior in Classical and Quantum Hamiltonian Systems |chapter=Preservation of conditionally periodic movements with small change in the Hamilton function |pages=51–56 |doi=10.1007/BFb0021737 |series=Lecture Notes in Physics |date=1979 |isbn=978-3-540-09120-2}} Translation of ''Doklady Akademii Nauk SSSR'' (1954) 98: 527. See also [[Kolmogorov–Arnold–Moser theorem]]</ref>, Mari Cartwright, Jon Edensor Littlwood<ref>{{Cite journal|last=Cartwright|first1=Mary L.|last2=Littlewood|first2=John E.|title=On non-linear differential equations of the second order, I: The equation ''y''" + ''k''(1−''y''<sup>2</sup>)''y<nowiki>'</nowiki>'' + ''y'' = ''b''λkcos(λ''t'' + ''a''), ''k'' large|journal=Journal of the London Mathematical Society|volume=20|pages=180–9|year=1945|doi=10.1112/jlms/s1-20.3.180|issue=3}} See also: [[Van der Pol oscillator]]</ref> va [[Stephen Smale]] oʻtkazgan<ref>{{Cite journal|last=Smale, Stephen|title=Morse inequalities for a dynamical system|journal=Bulletin of the American Mathematical Society|volume=66|pages=43–49|date=January 1960|doi=10.1090/S0002-9904-1960-10386-2}}</ref>.
[[Fayl:Airplane_vortex_edit.jpg|left|thumb|233x233px|[[Samolyot]] qanotidan hosil boʻlgan uyurma uchidagi turbulentlik. Tizimda turbulentlik paydo boʻladigan kritik nuqtani oʻrganish xaos nazariyasi uchun muhim ahamiyatga ega.]]
[[Edvard Lorenz]] bu nazariyaning dastlabki tadqiqotchilaridan biri boʻlgan. Uning xaosga boʻlgan qiziqishi 1961-yilda ob-havo prognozi ustida ishlash jarayonida tasodifan yuzaga kelgan<ref name="Lorenz19612">{{Cite journal|last=Lorenz, Edward N.|title=Deterministic non-periodic flow|journal=Journal of the Atmospheric Sciences|volume=20|pages=130–141|year=1963|doi=10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2|issue=2|bibcode=1963JAtS...20..130L}}</ref>. Lorenz va uning hamkasblari [[Ellen Fetter]] va [[Margaret Hamilton]]<ref>{{Veb manbasi |muallif=Sokol |ism=Joshua |sana=May 20, 2019 |sarlavha=The Hidden Heroines of Chaos |url=https://www.quantamagazine.org/the-hidden-heroines-of-chaos-20190520/ |qaralgan sana=2022-11-09 |ish=Quanta Magazine}}</ref> ob-havo simulyatsiyalarini oʻtkazish uchun oddiy raqamli kompyuter Royal McBee LGP-30 dan foydalanganlar. Ular maʼlumotlar ketma-ketligini yana koʻrishni xohlashdi va vaqtni tejash maqsadida simulyatsiyani oʻzining oʻrtasidan boshladilar. Ular buni dastlabki simulyatsiyaning oʻrtasidagi shartlarga mos keladigan maʼlumotlarning nusxasini kiritish orqali amalga oshirdilar. Hayratlanarli tomoni shundaki, mashina bashorat qilgan ob-havo avvalgi hisoblashlardan butunlay boshqacha boʻldi. Ular buni kompyuterning chop etgan maʼlumotlariga qarab aniqlashdi. Kompyuter 6 raqamli aniqlikda ishlagan, ammo chop etilgan maʼlumotlar 3 raqamga yaxlitlangan, shuning uchun masalan, 0.506127 raqami 0.506 deb chop etilgan. Bu farq juda kichik edi va oʻsha paytda bunday kichik oʻzgarishning amaliy taʼsiri boʻlmasligi kerak deb hisoblanardi. Biroq, Lorenz kichik boshlangʻich sharoitlar oʻzgarishi uzoq muddatli natijalarda katta farqlarni keltirib chiqarishini aniqladi<ref>{{Kitob manbasi |title=Chaos: Making a New Science |last=Gleick |first=James |year=1987 |publisher=Cardinal |location=London |page=17 |isbn=978-0-434-29554-8}}</ref>. Lorenzning [[Lorenz tizimi|Lorenz attraktorlari]] kashfiyoti shuni koʻrsatdiki hatto batafsil atmosferaviy modellashtirish ham umuman uzoq muddatli ob-havo bashoratlarini aniq qilish mumkin emasligini koʻrsatdi.

1987-yilda [[Per Bak]], [[Chao Tang]] va [[Kurt Wiesenfeld]] [[Physical Review Letters]] [[Jurnal|jurnalida]] tabiatda murakkablikning yuzaga kelish mexanizmlaridan biri sifatida oʻz-oʻzini tashkil etgan kritiklik (SOC)ni birinchi marta taʼriflovchi maqola nashr etishgan<ref>{{Cite journal|last=Bak, Per|last2=Tang, Chao|last3=Wiesenfeld, Kurt|title=Self-organized criticality: An explanation of the 1/f noise|journal=Physical Review Letters|volume=59|issue=4|pages=381–4|date=27 July 1987|doi=10.1103/PhysRevLett.59.381|pmid=10035754|bibcode=1987PhRvL..59..381B}} However, the conclusions of this article have been subject to dispute. {{Veb manbasi |url=http://www.nslij-genetics.org/wli/1fnoise/1fnoise_square.html |sarlavha=? |arxivurl=https://web.archive.org/web/20071214033929/https://www.nslij-genetics.org/wli/1fnoise/1fnoise_square.html |arxivsana=2007-12-14}}. See especially: {{Cite journal|last=Laurson, Lasse|last2=Alava, Mikko J.|last3=Zapperi, Stefano|title=Letter: Power spectra of self-organized critical sand piles|journal=Journal of Statistical Mechanics: Theory and Experiment|volume=0511|id=L001|date=15 September 2005}}</ref>.

Arzonroq va kuchliroq kompyuterlarning ishlab chiqarishda xaos nazariyasining qoʻllanilish imkoniyatlari kengaytirdi<ref>{{Cite journal|last=Motter|first1=A. E.|last2=Campbell|first2=D. K.|year=2013|title=Chaos at fifty|url=http://www.physicstoday.org/resource/1/phtoad/v66/i5/p27_s1?bypassSSO=1|journal=Phys. Today|volume=66|issue=5|pages=27–33|doi=10.1063/pt.3.1977|arxiv=1306.5777|bibcode=2013PhT....66e..27M}}</ref>. Hozirgi vaqtda xaos nazariyasi [[matematika]], [[topologiya]], [[fizika]]<ref>{{Cite journal|last=Hubler|first1=A.|last2=Foster|first2=G.|last3=Phelps|first3=K.|title=Managing chaos: Thinking out of the box|journal=Complexity|volume=12|issue=3|pages=10|date=2007|doi=10.1002/cplx.20159|bibcode=2007Cmplx..12c..10H}}</ref>, ijtimoiy tizimlar, populyatsiya modellashtirish, [[biologiya]], [[meteorologiya]], axborot nazariyasi, hisoblash neyrobiologiyasi, pandemiya inqirozini boshqarish<ref name="CT-REF-202">{{Veb manbasi |url=https://www.researchgate.net/publication/340775886 |sarlavha=Covid-19 Pandemic and Chaos Theory: Applications based on a Bibliometric Analysis |muallif=Piotrowski |ism=Chris |ish=researchgate.net |qaralgan sana=2020-05-13}}</ref><ref name="CT-REF-212">{{Kitob manbasi |last=Weinberger |first=David |title=Everyday Chaos – Technology, Complexity and How We're Thriving in a New World of Possibility |publisher=Harvard Business Review Press |year=2019 |isbn=9781633693968 |url=https://books.google.com/books?id=R7V2DwAAQBAJ}}</ref> kabi turli sohalarni qamrab olgan faol tadqiqot yoʻnalishi boʻlib qolmoqda.

== Lorenzning xaotik modellashtirishga qoʻshgan ilk hissasi ==
[[Professor]] Edvard Lorenz oʻz faoliyati davomida jami 61 ta ilmiy maqola muallifi boʻlib, ulardan 58 tasi yakka mualiflikdagi maqolalar hisoblanadi<ref>{{Veb manbasi |muallif=Chen |ism=G.-R. |sana=2020-01-01 |sarlavha=Butterfly Effect and Chaos. |url=https://www.ee.cityu.edu.hk/~gchen/pdf/Lorenz_T.pdf |qaralgan sana=2024-12-15}}</ref>. 1960-yil Yaponiyada boʻlib oʻtgan konferensiyadan boshlab, Lorenz SDIC va xaotik xususiyatlarni ochib berish maqsadida turli modellarning rivojlanishiga kirishdi. 1960-yildan 2008-yilgacha boʻlgan Lorenz modelining<ref>{{Cite journal|last=Shen|first1=Bo-Wen|last2=Pielke, Sr.|first2=Roger|last3=Zeng|first3=Xubin|date=2023-08-12|title=The 50th Anniversary of the Metaphorical Butterfly Effect since Lorenz (1972): Multistability, Multiscale Predictability, and Sensitivity in Numerical Models.|journal=Atmosphere|volume=14|issue=8|pages=1279|doi=10.3390/atmos14081279|bibcode=2023Atmos..14.1279S}}</ref><ref>{{Cite journal|last=Shen|first1=Bo-Wen|last2=Pielke, Sr.|first2=Roger|last3=Zeng|first3=Xubin|date=2023-08-12|title=The 50th Anniversary of the Metaphorical Butterfly Effect since Lorenz (1972): Multistability, Multiscale Predictability, and Sensitivity in Numerical Models.|journal=Atmosphere|volume=14|issue=8|pages=1279|doi=10.3390/atmos14081279|bibcode=2023Atmos..14.1279S}}</ref> rivojlanishini yaqinda koʻrib chiqish uning xaotik hodisalarni tasvirlash uchun turli xil jismoniy tizimlardan foydalanish qobiliyatini ochib berdi. Bu tizimlar quyidagilarni oʻz ichiga oladi: kvazi-geostrofik tizimlar, Rayleigh-Bénard konveksiya tenglamalari va [[sayoz suv tenglamalari]]. Bundan tashqari, Lorenz logistik xarita orqali xaotik yechimlarni oʻrganishda birinchi boʻlib amaliyotga qoʻllagan, bu uning hamkasblaridan oldin erishgan muhim yutugʻi boʻlgan (masalan, Lorenz 1964<ref>{{Cite journal|last=Lorenz|first1=E. N.|title=The problem of deducing the climate from the governing equations.|journal=Tellus|date=1964|volume=16|issue=1|pages=1–11|doi=10.3402/tellusa.v16i1.8893|bibcode=1964Tell...16....1L}}</ref>).

1972-yilda Lorenz „kapalak effekti“ atamasini kichik bir oʻzgarishning uch oʻlchovli, tashkil etilgan va mantiqiy tuzilishga ega boʻlgan tornado hosil qilishi mumkinligini muhokama qilish uchun metafora sifatida kiritdi. Asl kapalak effekti bilan bogʻliq boʻlsa-da, uning metaforik varianti oʻziga xos farqlarga ega. Ushbu muhim voqeani yodga olish uchun, ikkala kapalak effekti haqidagi tushunchamizni chuqurlashtiradigan taklif qilingan maqolalar joylashtirilgan qayta nashr qilingan kitob rasmiy ravishda chop etildi, bu esa metaforik kapalak effektining 50 yilligini nishonlash uchun amalga oshirildi<ref>{{Kitob manbasi |url=https://www.mdpi.com/books/book/8035 |title=50th Anniversary of the Metaphorical Butterfly Effect since Lorenz (1972): Multistability, Multiscale Predictability, and Sensitivity in Numerical Models |date=2023-10-11 |publisher=MDPI |isbn=978-3-0365-8911-4 |editor=Shen |language=en |doi=10.3390/books978-3-0365-8911-4}}</ref>.

=== Xaosning ommabop, ammo noaniq analogiyasi ===
Dastlabki sharoitga sezgir bogʻliqlik (yaʼni kapalak effekti) quyidagi folklorizmlar orqali tasvirlangan<ref name=":52">{{Veb manbasi |url=https://www.fencing.uz/ru/page/1472/insoniyatning-eng-muhim-10-ta-goyasi-2006 |sarlavha=Xaos (tartibsizlik) |arxivurl=https://web.archive.org/web/20241219153641/https://www.fencing.uz/ru/page/1472/insoniyatning-eng-muhim-10-ta-goyasi-2006 |arxivsana=2024-12-19 |sana=fencing.uz |qaralgan sana=16.12.2024 |til=uz }}</ref><ref name=":8">{{Kitob manbasi |last=Gleick |first=James |title=Chaos: Making a New Science |url=https://archive.org/details/chaosmakingnewsc0000glei_r3a5 |date=August 26, 2008 |publisher=Penguin Books |isbn=978-0143113454}}</ref>:

Yuqoridagilarga asoslanib, koʻplab odamlar kichik boshlangʻich noaniqlikning taʼsiri vaqt oʻtishi bilan monoton ravishda ortib borishini va har qanday kichik noaniqlik oxir-oqibat raqamli integratsiyalarni katta taʼsir qilishiga olib kelishini notoʻgʻri tushunishadi. 2008-yilda Lorenz bu misra haqiqiy xaosni tasvirlamasligini, balki beqarorlikning oddiyroq hodisasini yaxshiroq tasvirlaydi deb aytgan<ref name=":9">{{Veb manbasi |muallif=Lorenz |ism=E. N. |sana=December 2008 |sarlavha=The butterfly effect. In Premio Felice Pietro Chisesi E Caterina Tomassoni Award Lecture; University of Rome: Rome, Italy. |url=https://eapsweb.mit.edu/sites/default/files/Tomassoni_2008.pdf |arxivurl=https://web.archive.org/web/20230610171122/https://eapsweb.mit.edu/sites/default/files/Tomassoni_2008.pdf |qaralgan sana=January 29, 2023 |ish= |arxivsana=June 10, 2023}}</ref>. Misra, shuningdek, keyingi kichik voqealar natijani teskari oʻzgartirmasligini nazarda tutadi<ref name=":73">{{Cite journal|last=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}} [[Fayl:CC-BY_icon.svg|50x50px]] Text was copied from this source, which is available under a [[ccorg:licenses/by/4.0/|Creative Commons Attribution 4.0 International License]].</ref>. Tahlilga asoslanib, bu misra faqat ajralishni bildiradi, cheklanganlikni emas. Cheklanganlik kapalak naqshining cheklangan oʻlchami uchun muhimdir<ref name=":92">{{Veb manbasi |muallif=Lorenz |ism=E. N. |sana=December 2008 |sarlavha=The butterfly effect. In Premio Felice Pietro Chisesi E Caterina Tomassoni Award Lecture; University of Rome: Rome, Italy. |url=https://eapsweb.mit.edu/sites/default/files/Tomassoni_2008.pdf |arxivurl=https://web.archive.org/web/20230610171122/https://eapsweb.mit.edu/sites/default/files/Tomassoni_2008.pdf |qaralgan sana=January 29, 2023 |ish= |arxivsana=June 10, 2023}}</ref><ref name=":72">{{Cite journal|last=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}} [[Fayl:CC-BY_icon.svg|50x50px]] Text was copied from this source, which is available under a [[ccorg:licenses/by/4.0/|Creative Commons Attribution 4.0 International License]].</ref><ref>{{Veb manbasi |muallif=Shen |ism=Bo-Wen |sarlavha=A Popular but Inaccurate Analogy for Chaos and Butterfly Effect |url=https://www.youtube.com/watch?v=M9OA3PUaUCg&t=3s |qaralgan sana=2023-02-21 |ish=YouTube |sana=20 February 2023}}</ref>. Yaqinda oʻtkazilgan bir tadqiqotda, yuqoridagi misraning xususiyati „cheklangan vaqtli sezgirlik“ deb atalgan<ref>{{Cite journal|last=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}}</ref>.

== Qoʻllanishi ==
[[Fayl:Textile_cone.JPG|left|thumb|Tashqi koʻrinishi [[rule 30]] ga oʻxshash konussimon qobigʻi<ref>{{Veb manbasi |url=https://www.maths.nottingham.ac.uk/personal/sc/pdfs/Seashells09.pdf |arxivurl=https://web.archive.org/web/20131105134513/https://www.maths.nottingham.ac.uk/personal/sc/pdfs/Seashells09.pdf |arxivsana=2013-11-05 |sarlavha=The Geometry and Pigmentation of Seashells |muallif=Stephen Coombes |sana=February 2009 |ish=www.maths.nottingham.ac.uk |nashriyot=[[University of Nottingham]] |qaralgan sana=2013-04-10}}</ref>.]]
Garchi xaos nazariyasi ob-havo qonuniyatlarini kuzatishidan paydo boʻlgan boʻlsa-da, boshqa turli vaziyatlarda qoʻllanilishi mumkin. Bugungi kunda xaos nazariyasidan foydalanayotgan baʼzi sohalar [[geologiya]], [[matematika]]{{sfn|Z.A.Ilhomov|2022|p=49}}, [[biologiya]], [[Informatika|kompyuter fanlari]], [[Iqtisodiyot|iqtisod]]<ref>{{Cite journal|last=Kyrtsou C.|last2=Labys W.|year=2006|title=Evidence for chaotic dependence between US inflation and commodity prices|url=https://archive.org/details/sim_journal-of-macroeconomics_2006-03_28_1/page/n265|journal=Journal of Macroeconomics|volume=28|issue=1|pages=256–266|doi=10.1016/j.jmacro.2005.10.019}}</ref><ref>{{Cite journal|last=Kyrtsou C., Labys W.|year=2007|title=Detecting positive feedback in multivariate time series: the case of metal prices and US inflation|doi=10.1016/j.physa.2006.11.002|journal=Physica A|volume=377|issue=1|pages=227–229|bibcode=2007PhyA..377..227K|last2=Labys}}</ref><ref>{{Kitob manbasi |last=Kyrtsou, C. |chapter=Complex dynamics in macroeconomics: A novel approach |editor=Diebolt, C. |title=New Trends in Macroeconomics |publisher=Springer Verlag |year=2005}}</ref>, [[muhandislik]]<ref>{{Cite journal|last=Hernández-Acosta|first1=M. A.|last2=Trejo-Valdez|first2=M.|last3=Castro-Chacón|first3=J. H.|last4=Miguel|first4=C. R. Torres-San|last5=Martínez-Gutiérrez|first5=H.|title=Chaotic signatures of photoconductive Cu 2 ZnSnS 4 nanostructures explored by Lorenz attractors|journal=New Journal of Physics|date=2018|volume=20|issue=2|pages=023048|doi=10.1088/1367-2630/aaad41|language=en|issn=1367-2630|bibcode=2018NJPh...20b3048H}}</ref><ref>{{Veb manbasi |url=http://www.dspdesignline.com/218101444;jsessionid=Y0BSVTQJJTBACQSNDLOSKH0CJUNN2JVN?pgno=1 |sarlavha=Applying Chaos Theory to Embedded Applications |arxivurl=https://archive.today/20110809025913/http://www.dspdesignline.com/218101444;jsessionid=Y0BSVTQJJTBACQSNDLOSKH0CJUNN2JVN?pgno=1 |arxivsana=9 August 2011}}</ref>, [[moliya]]<ref>{{Cite journal|last=Hristu-Varsakelis, D.|last2=Kyrtsou, C.|title=Evidence for nonlinear asymmetric causality in US inflation, metal and stock returns|journal=Discrete Dynamics in Nature and Society|id=138547|year=2008|doi=10.1155/2008/138547|volume=2008|pages=1–7}}</ref><ref>{{Cite journal|doi=10.1023/A:1023939610962|last=Kyrtsou, C.|last2=M. Terraza|year=2003|title=Is it possible to study chaotic and ARCH behaviour jointly? Application of a noisy Mackey-Glass equation with heteroskedastic errors to the Paris Stock Exchange returns series|url=https://archive.org/details/sim_computational-economics_2003-06_21_3/page/n66|journal=Computational Economics|volume=21|issue=3|pages=257–276}}</ref><ref>{{Kitob manbasi |first=Justine |last=Gregory-Williams |title=Trading Chaos: Maximize Profits with Proven Technical Techniques |url=https://archive.org/details/tradingchaosmaxi0000greg |year=2004 |publisher=Wiley |location=New York |isbn=9780471463085 |edition=2nd}}</ref><ref>{{Kitob manbasi |last=Peters |first=Edgar E. |title=Fractal market analysis : applying chaos theory to investment and economics |url=https://archive.org/details/fractalmarketana0000pete |year=1994 |publisher=Wiley |location=New York u.a. |isbn=978-0471585244 |edition=2. print.}}</ref><ref>{{Kitob manbasi |last=Peters |first=/ Edgar E. |title=Chaos and order in the capital markets : a new view of cycles, prices, and market volatility |url=https://archive.org/details/chaosorderincapi0002pete |year=1996 |publisher=John Wiley & Sons |location=New York |isbn=978-0471139386 |edition=2nd}}</ref>, [[meteorologiya]], [[falsafa]], [[antropologiya]]<ref name=":02">{{Kitob manbasi |title=On the order of chaos. Social anthropology and the science of chaos |last=Mosko M.S., Damon F.H. (Eds.) |publisher=Berghahn Books |year=2005 |location=Oxford}}</ref>, [[fizika]]<ref>{{Cite journal|last=Hubler|first1=A.|last2=Phelps|first2=K.|title=Guiding a self-adjusting system through chaos|journal=Complexity|volume=13|issue=2|pages=62|date=2007|doi=10.1002/cplx.20204|bibcode=2007Cmplx..13b..62W}}</ref><ref>{{Cite journal|last=Gerig|first1=A.|title=Chaos in a one-dimensional compressible flow|journal=Physical Review E|volume=75|issue=4|pages=045202|date=2007|doi=10.1103/PhysRevE.75.045202|pmid=17500951|arxiv=nlin/0701050|bibcode=2007PhRvE..75d5202G}}</ref><ref>{{Cite journal|last=Wotherspoon|first1=T.|last2=Hubler|first2=A.|title=Adaptation to the Edge of Chaos in the Self-Adjusting Logistic Map|journal=The Journal of Physical Chemistry A|volume=113|issue=1|pages=19–22|date=2009|doi=10.1021/jp804420g|pmid=19072712|bibcode=2009JPCA..113...19W}}</ref>, [[siyosat]]<ref>{{Cite journal|last=Borodkin|first1=Leonid I.|title=Challenges of Instability: The Concepts of Synergetics in Studying the Historical Development of Russia|journal=Ural Historical Journal|date=2019|volume=63|issue=2|pages=127–136|doi=10.30759/1728-9718-2019-2(63)-127-136}}</ref><ref>{{Kitob manbasi |last=Progonati |first=E |title=Chaos, complexity and leadership 2018 explorations of chaotic and complexity theory |date=2018 |publisher=Springer |isbn=978-3-030-27672-0 |chapter=Brexit in the Light of Chaos Theory and Some Assumptions About the Future of the European Union}}</ref>, [[aholi dinamikasi]]<ref>{{Cite journal|last=Dilão, R.|last2=Domingos, T.|year=2001|title=Periodic and Quasi-Periodic Behavior in Resource Dependent Age Structured Population Models|journal=Bulletin of Mathematical Biology|volume=63|pages=207–230|doi=10.1006/bulm.2000.0213|issue=2|pmid=11276524}}</ref>, va [[robototexnika]] kiradi. Quyida bir nechta toifalar misollar bilan keltirilgan, ammo bu toʻliq roʻyxat emas, chunki yangi qoʻllanadigan sohalar ham paydo boʻlmoqda.

=== Kriptografiya ===
{{Asosiy|Xaotik kriptologiya}}
Xaos nazariyasi [[Kriptografiya|kriptografiyada]] koʻp yillardan beri qoʻllanib kelinmoqda. Soʻnggi oʻn yilliklarda xaos va nolinellar dinamikasi yuzlab [[Kriptografik primitivi|kriptografik primitivalarni]] loyihalashda ishlatilgan. Bu algoritmlar tasvirni shifrlash algoritmlari, xesh-funksiyalar, xavfsiz pseudo-tasodifiy sonlar generatorlari, oqim shifrlar, raqamli suv belgilari qoʻyish va steganografiya kabi texnikalarni oʻz ichiga oladi<ref name="Akhavan 1797–1813">{{Cite journal|last=Akhavan|first1=A.|last2=Samsudin|first2=A.|last3=Akhshani|first3=A.|date=2011-10-01|title=A symmetric image encryption scheme based on combination of nonlinear chaotic maps|journal=Journal of the Franklin Institute|volume=348|issue=8|pages=1797–1813|doi=10.1016/j.jfranklin.2011.05.001}}</ref>.

=== Robototexnika ===
Robototexnika sohasi soʻnggi paytlarda xaos nazariyasidan foydalangan yana bir soha hisoblanadi. Xaos nazariyasi robotlar atrof-muhit bilan oʻzaro taʼsir qilish uchun sinovdan oʻtkazish va xatolarni yaxshilash oʻrniga, [[Bashoratli model|bashoratli modelni]] yaratish uchun xaos nazariyasidan foydalangan<ref>{{Cite journal|last=Nehmzow|first1=Ulrich|date=Dec 2005|title=Quantitative description of robot–environment interaction using chaos theory|journal=Robotics and Autonomous Systems|volume=53|issue=3–4|pages=177–193|doi=10.1016/j.robot.2005.09.009|last2=Keith Walker|url=http://cswww.essex.ac.uk/staff/udfn/ftp/ecmrw3.pdf|access-date=2024-12-15|archiveurl=https://web.archive.org/web/20170812003513/http://cswww.essex.ac.uk/staff/udfn/ftp/ecmrw3.pdf|archivedate=2017-08-12|seperator=}}</ref>. Xaotik dinamika [[Passiv dinamika|passiv yuradigan]] ikki oyoqli robotlar tomonidan namoyish etilgan<ref>{{Cite journal|last=Goswami|first1=Ambarish|year=1998|title=A Study of the Passive Gait of a Compass-Like Biped Robot: Symmetry and Chaos|url=https://archive.org/details/sim_international-journal-of-robotics-research_1998-12_17_12/page/n43|journal=The International Journal of Robotics Research|volume=17|issue=12|pages=1282–1301|doi=10.1177/027836499801701202|last2=Thuilot, Benoit|last3=Espiau, Bernard}}</ref>.

=== Biologiya ===
Yuz yildan ortiq vaqt davomida [[biologlar]] turli xil turlar populyatsiyalarini oʻrganib, [[Populyatsiya modeli|populyatsiya modellari]] orqali kuzatib kelishgan. Koʻplab modellari [[Uzluksiz funksiya|uzluksiz]], lekin soʻnggi paytlarda olimlar baʼzi populyatsiyalarni xaotik modellarda amalga oshirishga muvaffaq boʻlishdi<ref>{{Cite journal|last=Eduardo|first1=Liz|last2=Ruiz-Herrera, Alfonso|title=Chaos in discrete structured population models|journal=[[SIAM Journal on Applied Dynamical Systems]]|year=2012|volume=11|issue=4|pages=1200–1214|doi=10.1137/120868980}}</ref>. Misol uchun, Kanada lynxlarining populyatsiya oʻsishini oʻrganish boʻyicha olib borilgan tadqiqotlar populyatsiya oʻsishida xaotik xatti-harakatlarni koʻrsatdi<ref>{{Cite journal|last=Lai|first1=Dejian|title=Comparison study of AR models on the Canadian lynx data: a close look at BDS statistic|url=https://archive.org/details/sim_computational-statistics-data-analysis_1996-08-10_22_4/page/n68|journal=[[Computational Statistics & Data Analysis]]|year=1996|volume=22|issue=4|pages=409–423|doi=10.1016/0167-9473(95)00056-9}}</ref>. Xaos ekologik tizimlarda ham uchraydi, masalan, [[gidrologiya]] sohasida. Gidrologiya uchun xaotik modelning baʼzi kamchiliklari boʻlsa-da, maʼlumotlarga xaos nazariyasi nuqtai nazaridan qarash orqali koʻplab yangi maʼlumotlarni olish mumkin<ref>{{Cite journal|last=Sivakumar|first1=B|title=Chaos theory in hydrology: important issues and interpretations|journal=[[Journal of Hydrology]]|date=31 January 2000|volume=227|issue=1–4|pages=1–20|bibcode=2000JHyd..227....1S|doi=10.1016/S0022-1694(99)00186-9}}</ref>. Yana bir biologik qoʻllanma [[Kardiotokografiya|kardiotokografiyada]] uchraydi. Homila holatini kuzatishda aniq maʼlumot olish va iloji boricha noinvaziv usullardan foydalanish oʻrtasidagi nozik muvozanatdir. Xaotik modellashtirish orqali homila gipoksiyasining ogohlantiruvchi belgilarini aniqlashning yanada samarali modellari yaratilishi mumkin<ref>{{Cite journal|last=Bozóki|first1=Zsolt|title=Chaos theory and power spectrum analysis in computerized cardiotocography|journal=[[European Journal of Obstetrics & Gynecology and Reproductive Biology]]|date=February 1997|volume=71|issue=2|pages=163–168|doi=10.1016/s0301-2115(96)02628-0|pmid=9138960}}</ref>.

Perryning taʼkidlashicha, ekologiyada xaotik [[Vaqt ketma-ketligi|vaqt ketma-ketliklarini]] [[modellashtirish]] cheklovlar bilan yordam beradi<ref name="realdata">{{Kitob manbasi |year=2000 |edition=1 |publisher=[[Springer Science+Business Media Dordrecht]] |first=Joe |last=Perry |editor=Perry |pages=xii+226 |title=Chaos in Real Data : The Analysis of Non-Linear Dynamics from Short Ecological Time Series |series=[[Population and Community Biology Series]] |doi=10.1007/978-94-011-4010-2 |isbn=978-94-010-5772-1}}</ref>. Har doim haqiqiy xaosni faqat modeldagi xaosdan ajratish qiyinligi mavjud<ref name="realdata2">{{Kitob manbasi |year=2000 |edition=1 |publisher=[[Springer Science+Business Media Dordrecht]] |first=Joe |last=Perry |editor=Perry |pages=xii+226 |title=Chaos in Real Data : The Analysis of Non-Linear Dynamics from Short Ecological Time Series |series=[[Population and Community Biology Series]] |doi=10.1007/978-94-011-4010-2 |isbn=978-94-010-5772-1}}</ref>. Shunday qilib, modelga qoʻshimcha cheklovlar kiritish yoki taqqoslash uchun oʻxshash vaqt qatorlari maʼlumotlaridan foydalanish modelni haqiqiylikka yaqinroq qilishda yordam beradi. Masalan, Perry va Wall 1984-yil tadqiqoti<ref name="realdata3">{{Kitob manbasi |year=2000 |edition=1 |publisher=[[Springer Science+Business Media Dordrecht]] |first=Joe |last=Perry |editor=Perry |pages=xii+226 |title=Chaos in Real Data : The Analysis of Non-Linear Dynamics from Short Ecological Time Series |series=[[Population and Community Biology Series]] |doi=10.1007/978-94-011-4010-2 |isbn=978-94-010-5772-1}}</ref>.

=== Iqtisodiyot ===
Iqtisodiy modellarga ham xaos nazariyasini qoʻllash orqali yaxshilanishi mumkin, lekin iqtisodiy tizimning holatini va unga eng koʻp taʼsir qiladigan omillarni bashorat qilish juda murakkab vazifa hisoblanadi<ref>{{Cite journal|last=Juárez|first1=Fernando|title=Applying the theory of chaos and a complex model of health to establish relations among financial indicators|journal=Procedia Computer Science|year=2011|volume=3|pages=982–986|doi=10.1016/j.procs.2010.12.161}}</ref>. Iqtisodiy va moliyaviy tizimlar tabiiy fanlardagi tizimlardan nisbatan farq qiladi, chunki ular tabiatan stoxastik boʻlib, odamlar oʻrtasidagi oʻzaro taʼsirlar natijasida yuzaga keladi. Shuning uchun sof deterministik modellar maʼlumotlarning aniq tasvirini taqdim etishi ehtimoli kam<ref>{{Cite journal|last=Brooks|first1=Chris|author-link=Chris Brooks (academic)|title=Chaos in foreign exchange markets: a sceptical view|journal=Computational Economics|year=1998|volume=11|issue=3|pages=265–281|issn=1572-9974|doi=10.1023/A:1008650024944|url=http://centaur.reading.ac.uk/35988/1/35988.pdf|archiveurl=https://web.archive.org/web/20170809072731/http://centaur.reading.ac.uk/35988/1/35988.pdf|archivedate=2017-08-09}}</ref>.

Xaos iqtisodiyotda takroriylikni miqdoriy tahlil qilish orqali aniqlanishi mumkin. Masalan, Orlando „takroriylikni miqdoriy oʻlchash indeksi“dan foydalangan holda vaqt qatorlaridagi yashirin oʻzgarishlarni aniqlashga muvaffaq boʻlishgan<ref>{{Cite journal|last=Orlando|first1=Giuseppe|last2=Zimatore|first2=Giovanna|title=RQA correlations on real business cycles time series|journal=Indian Academy of Sciences – Conference Series|date=18 December 2017|volume=1|issue=1|pages=35–41|doi=10.29195/iascs.01.01.0009}}</ref>. Keyinchalik ushbu usul laminar (muntazam) fazalardan turbulent (xaotik) fazalarga oʻtish jarayonlarini, makroiqtisodiy oʻzgaruvchilar orasidagi farqlarni aniqlash va iqtisodiy dinamikadagi yashirin xususiyatlarni yoritishda qoʻllanilgan<ref>{{Cite journal|last=Orlando|first1=Giuseppe|last2=Zimatore|first2=Giovanna|title=Recurrence quantification analysis of business cycles|journal=Chaos, Solitons & Fractals|date=1 May 2018|volume=110|pages=82–94|doi=10.1016/j.chaos.2018.02.032|bibcode=2018CSF...110...82O|url=https://www.sciencedirect.com/science/article/abs/pii/S0960077918300924|language=en|issn=0960-0779}}</ref>. Nihoyat, xaos nazariyasi iqtisodiyot qanday ishlashini modellashtirishda va [[COVID-19 pandemiyasi|COVID-19]] kabi tashqi hodisalarning taʼsirini modellashtirishda yordam berishi mumkin<ref>{{Cite journal|last=Orlando|first1=Giuseppe|last2=Zimatore|first2=Giovanna|title=Business cycle modeling between financial crises and black swans: Ornstein–Uhlenbeck stochastic process vs Kaldor deterministic chaotic model|url=https://aip.scitation.org/doi/10.1063/5.0015916|journal=Chaos: An Interdisciplinary Journal of Nonlinear Science|pages=083129|doi=10.1063/5.0015916|date=1 August 2020|volume=30|issue=8|pmid=32872798|bibcode=2020Chaos..30h3129O}}</ref>.

=== AIni kengaytirilgan modellash tizimi ===
[[Sunʼiy intellekt|Sunʼiy intellektga]] asoslangan katta til modellarida javoblar formatdagi oʻzgarishlar va soʻrovlardagi farqlari kabi omillarga sezgir boʻlishi mumkin. Ushbu sezgirliklar kapalak effektlariga oʻxshaydi<ref>{{ArXiv manbasi|last=Salinas|first1=Abel|last2=Morstatter|first2=Fred|date=2024-01-01|title=The Butterfly Effect of Altering Prompts: How Small Changes and Jailbreaks Affect Large Language Model Performance|class=cs.CL|eprint=2401.03729}}</ref>. AI bilan ishlaydigan katta til modellarini klassik deterministik xaotik tizimlar sifatida tasniflash mavjud boʻlsa-da, xaosdan ilhomlangan yondashuvlar va usullar (masalan, ansambl modellashtirish) ushbu keng koʻlamli til modellaridan ishonchli maʼlumotlarni olish uchun ishlatilishi mumkin (yana qarang: [[Ommaviy madaniyatda kapalak effekti#AI bilan ishlaydigan katta til modellari|Ommaviy madaniyatda kapalak effekti]]).

=== Boshqa sohalar ===
[[File:BML_N=200_P=32.png|link=https://en.wikipedia.org/wiki/File:BML_N=200_P=32.png|right|352x352px|The red cars and blue cars take turns to move; the red ones only move upwards, and the blue ones move rightwards. Every time, all the cars of the same colour try to move one step if there is no car in front of it. Here, the model has self-organized in a somewhat geometric pattern where there are some traffic jams and some areas where cars can move at top speed.]]
Kimyoda gazlarning eruvchanligini bashorat qilish [[polimerlar]] ishlab chiqarishda muhim ahamiyatga ega, ammo [[Zarralar uyumini optimallashtirish|zarrachalar uyumini optimallashtirish]] (PSO) usulidan foydalangan holdagi modellarda natijalar notoʻgʻri nuqtalarga kelib qolishga moyil boʻladi. Simulyatsiyalarning bunday holatlarda toʻxtab qolishini oldini olish uchun PSO ning takomillashtirilgan versiyasi yaratilgan boʻlib, unga xaos nazariyasi joriy etilgan<ref>{{Cite journal|last=Li|first1=Mengshan|last2=Xingyuan Huanga|last3=Hesheng Liua|last4=Bingxiang Liub|last5=Yan Wub|last6=Aihua Xiongc|last7=Tianwen Dong|title=Prediction of gas solubility in polymers by back propagation artificial neural network based on self-adaptive particle swarm optimization algorithm and chaos theory|journal=Fluid Phase Equilibria|date=25 October 2013|volume=356|pages=11–17|doi=10.1016/j.fluid.2013.07.017|bibcode=2013FlPEq.356...11L}}</ref>. [[Samoviy mexanika|Osmon mexanikasida]], ayniqsa asteroidlarga oid kuzatuvlar qilishda, xaos nazariyasini qoʻllash, bu obyektlar Yer va boshqa sayyoralar bilan yaqinlashishi vaqtini yanada aniqroq bashorat qilishga yordam beradi. [[Plutonning yo'ldoshlari|Plutonning beshta yoʻldoshidan]] toʻrttasi tartibsiz aylanadi. [[Kvant mexanika|Kvant fizikasi]] va [[Elektrotexnika|elektrotexnikada]] [[Jozefson effekti|Jozefson tutashmalarining]] katta massivlarini oʻrganishda xaos nazariyasidan foydalangan<ref>Steven Strogatz, ''Sync: The Emerging Science of Spontaneous Order'', Hyperion, 2003</ref>. Uyga yaqinroq boʻlgan koʻmir konlari har doim tabiiy gazning tez-tez sizib chiqishi koʻplab odamlarning oʻlimiga olib keladigan xavfli joylar boʻlgan. Yaqin-yaqingacha ularning qachon sodir boʻlishini bashorat qilishning ishonchli usuli yoʻq edi. Ammo bu gaz sizib chiqishi xaotik tendensiyaga ega boʻlib, agar toʻgʻri modellashtirilsa, uni yetarlicha aniq bashorat qilish mumkin<ref>{{Cite journal|last=Dingqi|first1=Li|last2=Yuanping Chenga|last3=Lei Wanga|last4=Haifeng Wanga|last5=Liang Wanga|last6=Hongxing Zhou|title=Prediction method for risks of coal and gas outbursts based on spatial chaos theory using gas desorption index of drill cuttings|journal=Mining Science and Technology|date=May 2011|volume=21|issue=3|pages=439–443|doi=10.1016/j.mstc.2011.05.010|bibcode=2011MiSTC..21..439L}}</ref>.

Redington va Reidbord (1992) odamning yuragi xaotik xususiyatlarga ega boʻlishi mumkinligini koʻrsatishga harakat qilishgan. Ular bir nafar psixoterapiya bemorining yurak urishlari orasidagi vaqt intervallarini kuzatgan, bemor psixoterapiya jarayonida turli hissiy intensivlik davrlaridan oʻtgan paytda. Natijalar aniq boʻlmagan. Olimlar tomonidan xaotik dinamikani koʻrsatishga harakat qilgan grafiklar (spektral tahlil, faza trajektoriyasi va avtokorrelyatsiya grafiklari) oʻrtasida noaniqliklar bor edi, shuningdek, Lyapunov eksponentini hisoblashga urinishlari ham muvaffaqiyatsiz boʻldi. Olimlar bu ishni ishonchli bajarishmaganini aniqladilar<ref>{{Cite journal|last=Redington|first1=D. J.|last2=Reidbord|first2=S. P.|title=Chaotic dynamics in autonomic nervous system activity of a patient during a psychotherapy session|url=https://archive.org/details/sim_biological-psychiatry_1992-05-15_31_10/page/n24|journal=Biological Psychiatry|date=1992|volume=31|issue=10|pages=993–1007|pmid=1511082|doi=10.1016/0006-3223(92)90093-F}}</ref>.

== Yana qarang ==
{{Portal|Matematika|Systems science}}
'''Xaotik sistemalarga misollar'''
{{Div col|colwidth=18em}}
* [[Kontur adveksiyasi]]
* [[Arnoldning mushuk xaritasi]]
* [[Bifurkatsiya nazariyasi]]
* [[Muvozanatlashgan to‘p]]
* [[Chua zanjiri]]
* [[Kliodinamika]]
* [[Birlashtirilgan xarita panjarasi]]
* [[Juft mayatnik]]
* [[Duffing tenglamasi]]
* [[Dinamik bilyardlar]]
* [[Iqtisodiy pufak]]
* [[Xaotik sochilish#Gaspard-Rice tizimi|Gaspard-Rice tizimi]]
* [[Hénon xaritasi]]
* [[Xaotik xaritalar ro‘yxati]]
* [[Rössler attraktor]]
* [[Standard xarita]]
* [[Tilt A Whirl]]
{{Div col end}}
'''Aloqador mavzular'''

{{Div col|colwidth=18em}}
* [[Amplituda o‘limi]]
* [[Anosov diffeomorfizmi]]
* [[Halokat nazariyasi]]
* [[Sabablilik]]
* [[Stoxastik dinamikaning supersimmetrik nazariyasi]]
* [[Xaos mashinasi]]
* [[Xaotik aralashtirish]]
* [[Determinizm]]
{{Div col end}}

'''Odamlar'''
{{Div col|colwidth=18em}}
* [[Ralph Abraham (matemetik)|Ralph Abraham]]
* [[Michael Berry (fizik)|Michael Berry]]
* [[Leon O. Chua]]
* [[Ivar Ekeland]]
* [[Doyne Farmer]]
* [[Martin Gutzwiller]]
* [[Brosl Hasslacher]]
* [[Michel Hénon]]
* [[Aleksandr Lyapunov]]
* [[Norman Packard]]
* [[Otto Rössler]]
* [[David Ruelle]]
* [[Oleksandr Mikolaiovich Sharkovsky]]
* [[Robert Shaw (fizik)|Robert Shaw]]
* [[Floris Takens]]
* [[James A. Yorke]]
* [[George M. Zaslavsky]]
{{Div col end}}

== Manbalar ==
{{manbalar}}
== Adabiyotlar ==
=== Maqolalar ===
* {{cite journal |first=A.N. |last=Sharkovskii |author-link=Oleksandr Mykolaiovych Sharkovsky |title=Co-existence of cycles of a continuous mapping of the line into itself |journal=Ukrainian Math. J. |volume=16 |pages=61–71 |year=1964 }}
* {{cite journal |author-link1=Tien-Yien Li |last1=Li |first1=T.Y. |author-link2=James A. Yorke |last2=Yorke |first2=J.A. |title=Period Three Implies Chaos |journal=[[American Mathematical Monthly]] |volume=82 |pages=985–92 |year=1975 |bibcode=1975AmMM...82..985L |doi=10.2307/2318254 |issue=10 |url=http://pb.math.univ.gda.pl/chaos/pdf/li-yorke.pdf |jstor=2318254 |citeseerx=10.1.1.329.5038 |access-date=2009-08-12 |archive-date=2009-12-29 |archive-url=https://web.archive.org/web/20091229042210/http://pb.math.univ.gda.pl/chaos/pdf/li-yorke.pdf |url-status=dead |archivedate=2009-12-29 |archiveurl=https://web.archive.org/web/20091229042210/http://pb.math.univ.gda.pl/chaos/pdf/li-yorke.pdf }}
* {{cite journal|last1=Alemansour|first1=Hamed|last2=Miandoab|first2=Ehsan Maani|last3=Pishkenari|first3=Hossein Nejat|title=Effect of size on the chaotic behavior of nano resonators|journal=Communications in Nonlinear Science and Numerical Simulation|date=March 2017|volume=44|pages=495–505|doi=10.1016/j.cnsns.2016.09.010|bibcode=2017CNSNS..44..495A}}
* {{Cite journal |date = December 1986|title=Chaos |url = https://archive.org/details/sim_scientific-american_1986-12_255_6/page/n39|journal=[[Scientific American]] |volume=255 |issue=6 |pages=38–49 (bibliography p.136) |bibcode = 1986SciAm.255d..38T |last2 = Tucker |last3 = Morrison |author1 = Crutchfield|author4 = J.D. Farmer|author5 = Packard|author6=N.H. |author7=Shaw |author8=R.S |author-link1=James P. Crutchfield |author-link5=Norman Packard |author-link7=Robert Shaw (physicist) |author4-link=J. Doyne Farmer |doi=10.1038/scientificamerican1286-46 }} [https://web.archive.org/web/20100612040821/http://cse.ucdavis.edu/~chaos/courses/ncaso/Readings/Chaos_SciAm1986/Chaos_SciAm1986.html Online version] (Note: the volume and page citation cited for the online text differ from that cited here. The citation here is from a photocopy, which is consistent with other citations found online that donʼt provide article views. The online content is identical to the hardcopy text. Citation variations are related to country of publication).
* {{cite journal |author=Kolyada, S.F. |s2cid=207251437 |title=Li-Yorke sensitivity and other concepts of chaos |journal=Ukrainian Math. J. |volume=56 |pages=1242–57 |year=2004 |doi=10.1007/s11253-005-0055-4 |issue=8 }}
* {{cite journal | last1 = Day| first1 = R.H. | last2 = Pavlov| first2 = O.V. | year = 2004| title = Computing Economic Chaos | ssrn = 806124| journal = Computational Economics | volume = 23 | issue = 4 | pages = 289–301 | doi = 10.1023/B:CSEM.0000026787.81469.1f | arxiv = 2211.02441 | s2cid = 119972392 }}
* {{cite journal |first1=C. |last1=Strelioff |first2=A. |last2=Hübler |title=Medium-Term Prediction of Chaos |journal=Phys. Rev. Lett. |volume=96 |issue=4 |id=044101 |year=2006 |doi=10.1103/PhysRevLett.96.044101 |url=http://www.ccsr.illinois.edu/web/Techreports/2005-08/CCSR-05-4.pdf |pmid=16486826 |page=044101 |bibcode=2006PhRvL..96d4101S |url-status=dead |archive-url=https://web.archive.org/web/20130426201635/http://www.ccsr.illinois.edu/web/Techreports/2005-08/CCSR-05-4.pdf |archive-date=2013-04-26 |access-date=2024-12-13 |archivedate=2013-04-26 |archiveurl=https://web.archive.org/web/20130426201635/http://www.ccsr.illinois.edu/web/Techreports/2005-08/CCSR-05-4.pdf }}
* {{cite journal |author1=Hübler, A. |author2=Foster, G. |author3=Phelps, K. |title=Managing Chaos: Thinking out of the Box |journal=Complexity |volume=12 |pages=10–13 |year=2007 |url=http://server17.how-why.com/blog/ManagingChaos.pdf |doi=10.1002/cplx.20159 |issue=3 |bibcode=2007Cmplx..12c..10H |access-date=2011-07-17 |archive-url=https://web.archive.org/web/20121030165633/http://server17.how-why.com/blog/ManagingChaos.pdf |archive-date=2012-10-30 |url-status=dead }}
* {{cite journal | last1 = Motter | first1 = Adilson E. | last2 = Campbell | first2 = David K. | year = 2013 | title = Chaos at 50 | journal = Physics Today | volume = 66| issue = 5| page = 27| doi = 10.1063/PT.3.1977 |arxiv = 1306.5777 |bibcode = 2013PhT....66e..27M | s2cid = 54005470 }}

=== Darsliklar ===
* {{Kitob manbasi|author=Z.A.Ilhomov|title=“Tarix o‘qitish metodikasi|date=2022 |publisher=|location=Toshkent|url=https://lib.bimm.uz/items/download/8730 |pages=115 |isbn=|ref=Z.A.Ilhomov}}
* {{Kitob manbasi|author=Alimova.D.A|title=Tarix fani metodologiyasi|date=2018 |publisher=„O‘zbekiston faylasuflari milliy jamiyati“ |location=Toshkent|url=http://ilmiy.bmti.uz/blib/files/86/Tarix%20fani%20metodologiyasi.pdf |pages=324 |isbn=978-9943-5488-3-1|ref=Alimova.D.A}}
* {{cite book |last1=Alligood |first1=K.T. |last2=Sauer |first2=T. |last3=Yorke |first3=J.A. |title=Chaos: an introduction to dynamical systems |publisher=Springer-Verlag |year=1997 |isbn=978-0-387-94677-1 |url=https://books.google.com/books?id=48YHnbHGZAgC}}
* {{cite book| author=Baker, G. L.| title=Chaos, Scattering and Statistical Mechanics| publisher=Cambridge University Press| year=1996| isbn=978-0-521-39511-3}}
* {{cite book |author1=Badii, R. |author2=Politi A. |title=Complexity: hierarchical structures and scaling in physics |publisher=Cambridge University Press |year=1997 |isbn=978-0-521-66385-4 |url=http://www.cambridge.org/gb/academic/subjects/physics/statistical-physics/complexity-hierarchical-structures-and-scaling-physics}}
* {{cite book |author1=Collet, Pierre |author2=[[Jean-Pierre Eckmann|Eckmann, Jean-Pierre]] |title=Iterated Maps on the Interval as Dynamical Systems |url=https://archive.org/details/iteratedmapsonin0000coll |publisher=Birkhauser |year=1980 |isbn=978-0-8176-4926-5}}
* {{cite book |last=Devaney |first=Robert L. |author-link=Robert L. Devaney |title=An Introduction to Chaotic Dynamical Systems |url=https://archive.org/details/introductiontoch0000deva_b6r9 |edition=2nd |publisher=Westview Press |year=2003 |isbn=978-0-8133-4085-2 }}
* {{cite book |last=Robinson |first=Clark |title=Dynamical systems: Stability, symbolic dynamics, and chaos |publisher=CRC Press |year=1995 |isbn=0-8493-8493-1}}
* {{cite book |author1=Feldman, D. P. |title=Chaos and Fractals: An Elementary Introduction |publisher=Oxford University Press |year=2012 |isbn=978-0-19-956644-0 |url=https://dpfeldman.github.io/Chaos/index.html |access-date=2016-12-29 |archive-date=2019-12-31 |archive-url=https://web.archive.org/web/20191231030717/http://chaos.coa.edu/index.html |url-status=live}}
* {{cite book |author1=Gollub, J. P. |author2=Baker, G. L. |title=Chaotic dynamics |publisher=Cambridge University Press |year=1996 |isbn=978-0-521-47685-0 |url=https://books.google.com/books?id=n1qnekRPKtoC}}
* {{cite book |author=Guckenheimer, John| author-link=John Guckenheimer|author2= Holmes, Philip |author-link2=Philip Holmes|title=Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields |publisher=Springer-Verlag |year=1983 |isbn=978-0-387-90819-9}}
* {{cite book| author=Gulick, Denny| title=Encounters with Chaos| url=https://archive.org/details/encounterswithch0000unse| publisher=McGraw-Hill| year=1992| isbn=978-0-07-025203-5}}
* {{cite book |author=Gutzwiller, Martin |title=Chaos in Classical and Quantum Mechanics |publisher=Springer-Verlag |year=1990 |isbn=978-0-387-97173-5 |url=https://books.google.com/books?id=fnO3XYYpU54C}}
* {{cite book |author=Hoover, William Graham|author-link=William G Hoover|title=Time Reversibility, Computer Simulation, and Chaos |publisher=World Scientific |orig-year=1999|year=2001 |isbn=978-981-02-4073-8 |url=https://books.google.com/books?id=24kEKsdl0psC}}
* {{cite book |author=Kautz, Richard |title=Chaos: The Science of Predictable Random Motion |publisher=Oxford University Press |year=2011 |isbn=978-0-19-959458-0 |url=https://books.google.com/books?id=x5YbNZjulN0C}}
* {{cite book |author1=Kiel, L. Douglas |author2=Elliott, Euel W. |title=Chaos Theory in the Social Sciences |publisher=Perseus Publishing |year=1997 |isbn=978-0-472-08472-2 |url=https://books.google.com/books?id=K46kkMXnKfcC}}
* {{cite book |author=Moon, Francis |title=Chaotic and Fractal Dynamics |publisher=Springer-Verlag |year=1990 |isbn=978-0-471-54571-2 |url=https://books.google.com/books?id=Ddz-CI-nSKYC}}
* {{cite book|last1=Orlando|first1=Giuseppe|author-link1=Giuseppe Orlando|last2=Pisarchick|first2=Alexander|last3=Stoop|first3=Ruedi|title=Nonlinearities in Economics |chapter=|series=Dynamic Modeling and Econometrics in Economics and Finance|year=2021|volume=29|url=https://link.springer.com/book/10.1007/978-3-030-70982-2#editorsandaffiliations|language=en-gb|doi=10.1007/978-3-030-70982-2|isbn=978-3-030-70981-5|s2cid=239756912}}
* {{cite book |author=Ott, Edward |title=Chaos in Dynamical Systems |publisher=Cambridge University Press |year=2002 |isbn=978-0-521-01084-9 |url=https://books.google.com/books?id=nOLx--zzHSgC}}
* {{cite book| author=Strogatz, Steven| author-link=Steven Strogatz| title=Nonlinear Dynamics and Chaos| publisher=Perseus Publishing| year=2000| isbn=978-0-7382-0453-6| url=https://archive.org/details/nonlineardynamic00stro}}
* {{cite book |last=Sprott |first=Julien Clinton |title=Chaos and Time-Series Analysis |publisher=Oxford University Press |year=2003 |isbn=978-0-19-850840-3 |url=https://books.google.com/books?id=SEDjdjPZ158C}}
* {{cite book |author1=Tél, Tamás |author2=Gruiz, Márton |title=Chaotic dynamics: An introduction based on classical mechanics |publisher=Cambridge University Press |year=2006 |isbn=978-0-521-83912-9 |url=https://books.google.com/books?id=P2JL7s2IvakC}}
* {{cite book| last = Teschl| given = Gerald|author-link=Gerald Teschl| title = Ordinary Differential Equations and Dynamical Systems| publisher=[[American Mathematical Society]]| place = [[Providence, Rhode Island|Providence]]| year = 2012| isbn= 978-0-8218-8328-0| url = https://www.mat.univie.ac.at/~gerald/ftp/book-ode/}}
* {{cite book|vauthors=Thompson JM, Stewart HB | title=Nonlinear Dynamics And Chaos| publisher=John Wiley and Sons Ltd| year=2001| isbn=978-0-471-87645-8}}
* {{cite book |author1-link=Nicholas Tufillaro |last1=Tufillaro |last2=Reilly |title=An experimental approach to nonlinear dynamics and chaos |series=American Journal of Physics |volume=61 |issue=10 |pages=958 |publisher=Addison-Wesley |year=1992 |isbn=978-0-201-55441-0 |bibcode=1993AmJPh..61..958T |doi=10.1119/1.17380 |url=https://archive.org/details/unset0000unse_q2b7 |url-access=limited }}
* {{cite book | last=Wiggins|first=Stephen | title= Introduction to Applied Dynamical Systems and Chaos | url=https://archive.org/details/springer_10.1007-b97481| publisher= Springer | year= 2003 | isbn= 978-0-387-00177-7 }}
* {{cite book| author=Zaslavsky, George M.| title=Hamiltonian Chaos and Fractional Dynamics| publisher=Oxford University Press| year=2005| isbn=978-0-19-852604-9}}

=== Yarim texnik va ommabop asarlar ===
* [[Christophe Letellier]], ''Chaos in Nature'', World Scientific Publishing Company, 2012, {{ISBN|978-981-4374-42-2}}.
* {{cite book |editor1-first=Ralph H. |editor1-last=Abraham |editor2-first=Yoshisuke |editor2-last=Ueda |title=The Chaos Avant-Garde: Memoirs of the Early Days of Chaos Theory |journal=<!--Citation bot --> |volume=39 |url=https://books.google.com/books?id=0E667XpBq1UC |year=2000 |publisher=World Scientific |isbn=978-981-238-647-2 |bibcode=2000cagm.book.....A |doi=10.1142/4510 |series=World Scientific Series on Nonlinear Science Series A }}
* {{cite book |author-link=Michael F. Barnsley |first=Michael F. |last=Barnsley |title=Fractals Everywhere |url=https://books.google.com/books?id=oh7NoePgmOIC |year=2000 |publisher=Morgan Kaufmann |isbn=978-0-12-079069-2}}
* {{cite book |first=Richard J. |last=Bird |title=Chaos and Life: Complexity and Order in Evolution and Thought |url=https://books.google.com/books?id=fv3sltQBS54C |year=2003 |publisher=Columbia University Press |isbn=978-0-231-12662-5}}
* [[John Briggs (author)|John Briggs]] and David Peat, ''Turbulent Mirror: : An Illustrated Guide to Chaos Theory and the Science of Wholeness'', Harper Perennial 1990, 224 pp.
* John Briggs and David Peat, ''Seven Life Lessons of Chaos: Spiritual Wisdom from the Science of Change'', Harper Perennial 2000, 224 pp.
* {{cite journal |author=Cunningham, Lawrence A. |title=From Random Walks to Chaotic Crashes: The Linear Genealogy of the Efficient Capital Market Hypothesis |journal=George Washington Law Review |volume=62 |page=546 |year=1994 }}
* [[Predrag Cvitanović]], ''Universality in Chaos'', Adam Hilger 1989, 648 pp.
* [[Leon Glass]] and Michael C. Mackey, ''From Clocks to Chaos: The Rhythms of Life,'' Princeton University Press 1988, 272 pp.
* [[James Gleick]], ''[[Chaos: Making a New Science]]'', New York: Penguin, 1988. 368 pp.
* {{cite book |author=John Gribbin |title=Deep Simplicity |date=2005-01-27 |url=https://archive.org/details/deepsimplicity0000john |series=Penguin Press Science |publisher=Penguin Books}}
* L Douglas Kiel, Euel W Elliott (ed.), ''Chaos Theory in the Social Sciences: Foundations and Applications'', University of Michigan Press, 1997, 360 pp.
* Arvind Kumar, ''Chaos, Fractals and Self-Organisation; New Perspectives on Complexity in Nature '', National Book Trust, 2003.
* Hans Lauwerier, ''Fractals'', Princeton University Press, 1991.
* [[Edward Lorenz]], ''The Essence of Chaos'', University of Washington Press, 1996.
* {{cite book|doi=10.1142/9781860949548|title=The Unity of Nature - Wholeness and Disintegration in Ecology and Science|year=2002|last1=Marshall|first1=Alan|isbn=9781860949548}}
* David Peak and Michael Frame, ''Chaos Under Control: The Art and Science of Complexity'', Freeman, 1994.
* [[Heinz-Otto Peitgen]] and [[Dietmar Saupe]] (Eds.), ''The Science of Fractal Images'', Springer 1988, 312 pp.
* [[Nuria Perpinya]], ''Caos, virus, calma. La Teoría del Caos aplicada al desórden artístico, social y político'', Páginas de Espuma, 2021.
* [[Clifford A. Pickover]], ''Computers, Pattern, Chaos, and Beauty: Graphics from an Unseen World '', St Martins Pr 1991.
* [[Clifford A. Pickover]], ''Chaos in Wonderland: Visual Adventures in a Fractal World'', St Martins Pr 1994.
* [[Ilya Prigogine]] and [[Isabelle Stengers]], ''Order Out of Chaos'', Bantam 1984.
* {{cite book|doi=10.1007/978-3-642-61717-1|title=The Beauty of Fractals|url=https://archive.org/details/beautyoffractals0000peit|url-access=registration|year=1986|last1=Peitgen|first1=Heinz-Otto|last2=Richter|first2=Peter H.|isbn=978-3-642-61719-5}}
* [[David Ruelle]], ''Chance and Chaos'', Princeton University Press 1993.
* [[Ivars Peterson]], ''Newtonʼs Clock: Chaos in the Solar System'', Freeman, 1993.
* {{cite book |author1=Ian Roulstone |author2=John Norbury |title=Invisible in the Storm: the role of mathematics in understanding weather |url=https://books.google.com/books?id=qnMrFEHMrWwC|year=2013 |publisher=Princeton University Press|isbn=978-0691152721 }}
* {{cite book|doi=10.1017/CBO9780511608773|title=Chaotic Evolution and Strange Attractors|url=https://archive.org/details/chaoticevolution0000ruel|url-access=registration|year=1989|last1=Ruelle|first1=D.|isbn=9780521362726}}
* Manfred Schroeder, ''Fractals, Chaos, and Power Laws'', Freeman, 1991.
* {{cite book|doi=10.1017/CBO9780511554544|title=Explaining Chaos|year=1998|last1=Smith|first1=Peter|isbn=9780511554544}}
* [[Ian Stewart (mathematician)|Ian Stewart]], ''Does God Play Dice?: The Mathematics of Chaos '', Blackwell Publishers, 1990.
* [[Steven Strogatz]], ''Sync: The emerging science of spontaneous order'', Hyperion, 2003.
* Yoshisuke Ueda, ''The Road To Chaos'', Aerial Pr, 1993.
* M. Mitchell Waldrop, ''Complexity : The Emerging Science at the Edge of Order and Chaos'', Simon & Schuster, 1992.
* Antonio Sawaya, ''Financial Time Series Analysis : Chaos and Neurodynamics Approach'', Lambert, 2012.

== Havolalar ==
{{Commons category}}
* [https://web.archive.org/web/20160310065017/http://lagrange.physics.drexel.edu/ Nonlinear Dynamics Research Group] with Animations in Flash
* [http://www.chaos.umd.edu/ The Chaos group at the University of Maryland]
* [https://hypertextbook.com/chaos/ The Chaos Hypertextbook]. An introductory primer on chaos and fractals
* [https://chaosbook.org/ ChaosBook.org] An advanced graduate textbook on chaos (no fractals)
* [https://www.societyforchaostheory.org/ Society for Chaos Theory in Psychology & Life Sciences]
* [https://web.archive.org/web/20130425220329/http://www.csdc.unifi.it/mdswitch.html?newlang=eng Nonlinear Dynamics Research Group at CSDC], [[Florence]], [[Italy]]
* [http://www.creatingtechnology.org/papers/chaos.htm Nonlinear dynamics: how science comprehends chaos], talk presented by Sunny Auyang, 1998.
* [https://www.egwald.ca/nonlineardynamics/ Nonlinear Dynamics]. Models of bifurcation and chaos by Elmer G. Wiens
* [https://around.com/books/chaos/ Gleick’s ''Chaos'' (excerpt)] {{Webarchive|url=https://web.archive.org/web/20070202075958/http://www.around.com/chaos.html |date=2007-02-02 }}
* [https://web.archive.org/web/20070428110552/http://www.eng.ox.ac.uk/samp/ Systems Analysis, Modelling and Prediction Group] at the University of Oxford
* [https://web.archive.org/web/20090307094012/http://www.mgix.com/snippets/?MackeyGlass A page about the Mackey-Glass equation]
* [https://www.youtube.com/watch?v=5pKrKdNclYs High Anxieties – The Mathematics of Chaos] (2008) BBC documentary directed by [[David Malone (independent filmmaker)|David Malone]]
* [https://www.newscientist.com/article/mg20827821.000-the-chaos-theory-of-evolution.html The chaos theory of evolution] – article published in Newscientist featuring similarities of evolution and non-linear systems including fractal nature of life and chaos.
* Jos Leys, [[Étienne Ghys]] et Aurélien Alvarez, [https://www.chaos-math.org/en.html ''Chaos, A Mathematical Adventure'']. Nine films about dynamical systems, the butterfly effect and chaos theory, intended for a wide audience.
* [https://www.bbc.co.uk/programmes/p00548f6 „Chaos Theory“], BBC Radio 4 discussion with Susan Greenfield, David Papineau & Neil Johnson (''In Our Time'', May 16, 2002)
* [https://www.youtube.com/watch?v=fDek6cYijxI Chaos: The Science of the Butterfly Effect] (2019) an explanation presented by [[Derek Muller]]

== Mualliflik huquqi eslatmasi ==
* Ushbu maqola bepul kontent asaridan olingan matnni oʻz ichiga oladi. CC-BY litsenziyasi asosida ruxsat berilgan ([https://creativecommons.org/licenses/by/4.0/ litsenziya]). Matn [https://www.mdpi.com/2673-8392/2/3/84 Lorenz modellari doirasidagi uch xil kapalak effekti] nomli maqoladan olingan boʻlib, mualliflar: Bo-Wen Shen, Roger A. Pielke, Sr., Xubin Zeng, Jialin Cui, Sara Faghih-Naini, Wei Paxson, va Robert Atlas, MDPI. Encyclopedia.
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