TXT

{{Short description|Field of mathematics and science based on non-linear systems and initial conditions}}
{{Author|Harold Foppele}}
{{Mathematics}}
{{learning project}}
<!-- Hidden references block at the top -->
<div style="display:none;">
<ref name="01C">{{Cite journal|last1=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|hdl=1721.1/5961|s2cid=12209977|hdl-access=free}}</ref>
<ref name="02C">''Sync: The Emerging Science of Spontaneous Order'', Steven Strogatz, Hyperion, New York, 2003, pages 189–190.</ref>
<ref name="03C">{{cite web |url=https://www.britannica.com/science/chaos-theory |title=Chaos Theory: Definition & Facts |website=Encyclopedia Britannica |language=en |access-date=2019-11-24}}</ref>
<ref name="04C">{{cite web |url=https://fractalfoundation.org/resources/what-is-chaos-theory/ |title=What Is Chaos Theory? – Fractal Foundation |website=Fractal Foundation |language=en-US |access-date=2019-11-24}}</ref>
<ref name="05C">{{cite web |url=http://mathworld.wolfram.com/Chaos.html |title=Chaos |last=Weisstein |first=Eric W. |website=MathWorld |publisher=Wolfram Research |language=en |access-date=2019-11-24}}</ref>
<ref name="06C">{{cite web |url=https://geoffboeing.com/2015/03/chaos-theory-logistic-map/ |title=Chaos Theory and the Logistic Map |last=Boeing |first=Geoff |date=2015-03-26 |access-date=2020-05-17}}</ref>
<ref name="07C">{{Cite book |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 name="08C">{{Cite journal |last1=Shen |first1=Bo-Wen |last2=Pielke |first2=Roger A. |last3=Zeng |first3=Xubin |last4=Cui |first4=Jialin |last5=Faghih-Naini |first5=Sara |last6=Paxson |first6=Wei |last7=Atlas |first7=Robert |date=2022-07-04 |title=Three Kinds of Butterfly Effects within Lorenz Models |journal=Encyclopedia |volume=2 |issue=3 |pages=1250–1259 |doi=10.3390/encyclopedia2030084 |issn=2673-8392|doi-access=free}} [[File:CC-BY icon.svg|50px]] Text adapted from this source, available under a [https://creativecommons.org/licenses/by/4.0/ Creative Commons Attribution 4.0 International License].</ref>
<ref name="09C">{{cite book |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 |ref=harv}}</ref>
<ref name="10C">{{Citation|last=Bishop|first=Robert|title=Chaos|date=2017|url=https://plato.stanford.edu/archives/spr2017/entries/chaos/|encyclopedia=The Stanford Encyclopedia of Philosophy|editor-last=Zalta|editor-first=Edward N.|edition=Spring 2017|publisher=Metaphysics Research Lab, Stanford University|access-date=2019-11-24}}</ref>

<ref name="11C">{{cite book |last=Kellert |first=Stephen H. |title=In the Wake of Chaos: Unpredictable Order in Dynamical Systems |publisher=University of Chicago Press |year=1993 |isbn=978-0-226-42976-2 |page=56 |url=https://archive.org/details/inwakeofchaosunp0000kell/page/56}}</ref>

<ref name="12C">{{cite book |last=Kellert |first=Stephen H. |title=In the Wake of Chaos: Unpredictable Order in Dynamical Systems |publisher=University of Chicago Press |year=1993 |isbn=978-0-226-42976-2 |page=62 |url=https://archive.org/details/inwakeofchaosunp0000kell/page/62}}</ref>

<ref name="13C">{{cite journal |author = Werndl, Charlotte |title = What are the New Implications of Chaos for Unpredictability? |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 |s2cid = 354849 }}</ref>
<ref name="14C">{{cite web |url = http://mpe.dimacs.rutgers.edu/2013/03/17/chaos-in-an-atmosphere-hanging-on-a-wall/ |title = Chaos in an Atmosphere Hanging on a Wall |last1 = Danforth |first1 = Christopher M. |date = April 2013 |work = Mathematics of Planet Earth 2013 |access-date = 12 June 2018 }}</ref>
<ref name="15C">{{cite book |last = Ivancevic |first = Vladimir G. |title = Complex nonlinearity: chaos, phase transitions, topology change, and path integrals |year = 2008 |publisher = Springer |isbn = 978-3-540-79356-4 |author2 = Tijana T. Ivancevic }}</ref>
<ref name="16C">{{Cite book|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>
<ref name="17C">{{cite web |url=https://www.researchgate.net/publication/340775886 |title=Covid-19 Pandemic and Chaos Theory: Applications based on a Bibliometric Analysis |last=Piotrowski |first=Chris |website=researchgate.net |access-date=2020-05-13}}</ref>
<ref name="18C">{{cite book |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=978-1-63369-396-8 |url=https://books.google.com/books?id=R7V2DwAAQBAJ}}</ref>
<ref name="19C">{{Cite web|url=https://www.dictionary.com/browse/chaos|title=Definition of chaos {{!}} Dictionary.com|website=www.dictionary.com|language=en|access-date=2019-11-24}}</ref>
<ref name="20C">{{cite book|title=A First Course in Dynamics: With a Panorama of Recent Developments|last=Hasselblatt|first=Boris|author2=Anatole Katok|year=2003|publisher=Cambridge University Press|isbn=978-0-521-58750-1}}</ref>

<ref name="21C">{{cite book |author=Elaydi, Saber N. |title=Discrete Chaos |publisher=Chapman & Hall/CRC |year=1999 |isbn=978-1-58488-002-8 |page=137 }}</ref>
<ref name="22C">{{cite book |author=Basener, William F. |title=Topology and its applications |publisher=Wiley |year=2006 |isbn=978-0-471-68755-9 |page=42 }}</ref>
<ref name="23C">{{cite journal | author1=Banks | author2=Brooks | author3=Cairns | author4=Davis | author5=Stacey | title= On Devaney's definition of chaos | journal=The American Mathematical Monthly | volume=99|issue=4|date=1992| pages=332–334 | doi=10.1080/00029890.1992.11995856 }}</ref>
<ref name="24C">{{cite journal |author1=Vellekoop, Michel |author2=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>
<ref name="25C">{{cite book |author1=Medio, Alfredo |author2=Lines, Marji |title=Nonlinear Dynamics: A Primer |url=https://archive.org/details/nonlineardynamic00medi |url-access=limited |publisher=Cambridge University Press |year=2001 |isbn=978-0-521-55874-7 |page=[https://archive.org/details/nonlineardynamic00medi/page/n175 165] }}</ref>
<ref name="26C">{{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>
<ref name="27C">{{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 |page=753 |doi=10.3390/atmos13050753 |bibcode=2022Atmos..13..753S |issn=2073-4433|doi-access=free }}</ref>
<ref name="28C">{{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>
<ref name="29C">{{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>
<ref name="30C">{{cite web | title=Science: Mathematician discovers a more complex form of chaos | url=https://www.newscientist.com/article/mg12617232-800-science-mathematician-discovers-a-more-complex-form-of-chaos/ }}</ref>
<ref name="31C">{{cite web | title='Next-Level' Chaos Traces the True Limit of Predictability | date=7 March 2025 | url=https://www.quantamagazine.org/next-level-chaos-traces-the-true-limit-of-predictability-20250307/ }}</ref>
<ref name="32C">{{Cite journal |last1=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 name="33C">{{Cite web |last=Everitt |first=Francis |date=2006-12-01 |title=James Clerk Maxwell: a force for physics |url=https://physicsworld.com/a/james-clerk-maxwell-a-force-for-physics/ |access-date=2023-11-03 |website=Physics World |language=en-GB}}</ref>
<ref name="34C">{{Cite journal |last1=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 |s2cid=225246631 |issn=1573-269X|doi-access=free |bibcode=2020NonDy.102..643G |hdl=2164/17003 |hdl-access=free }}</ref>
<ref name="35C">{{cite journal |author=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 |doi-access=free }}</ref>
<ref name="36C">{{Cite book|title=The three-body problem and the equations of dynamics: Poincaré's foundational work on dynamical systems theory|last=Poincaré|first=J. Henri|publisher=Springer International Publishing|others=Popp, Bruce D. (Translator)|year=2017|isbn=978-3-319-52898-4|location=Cham, Switzerland|oclc=987302273}}</ref>
<ref name="37C">{{cite book |author1=Diacu, Florin |author2=Holmes, Philip |title=Celestial Encounters: The Origins of Chaos and Stability |publisher=Princeton University Press |year=1996 |isbn=978-0-691-02743-2}}</ref>
<ref name="38C">{{cite journal|first = 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|url=https://www.numdam.org/item/JMPA_1898_5_4__27_0.pdf}}</ref>
<ref name="39C">{{cite journal|first1=R.|last1=Aurich|first2=M.|last2=Sieber|first3=F.|last3=Steiner|title=Quantum Chaos of the Hadamard–Gutzwiller Model|journal=Physical Review Letters|date=1 August 1988|pages=483–487|volume=61|issue=5|doi=10.1103/PhysRevLett.61.483|pmid=10039347|bibcode=1988PhRvL..61..483A|url=http://bib-pubdb1.desy.de/record/324018/files/PhysRevLett61483.pdf}}</ref>
<ref name="40C">W:George D. Birkhoff, ''Dynamical Systems,'' vol. 9 of the American Mathematical Society Colloquium Publications (Providence, Rhode Island: American Mathematical Society, 1927)</ref>

<ref name="41C">{{cite journal |last1=Kolmogorov |first1=A. N. |year=1941 |title=The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers |journal=Doklady Akademii Nauk SSSR |volume=30 |issue=4 |pages=301–305 |bibcode=1941DoSSR..30..301K |language=ru |trans-title=Local structure of turbulence in an incompressible fluid for very large Reynolds numbers}} {{cite journal |last1=Kolmogorov |first1=A. N. |year=1991 |title=The Local Structure of Turbulence in Incompressible Viscous Fluid for Very Large Reynolds Numbers |journal=Proceedings of the Royal Society A |volume=434 |issue=1890 |pages=9–13 |doi=10.1098/rspa.1991.0075 |bibcode=1991RSPSA.434....9K |s2cid=123612939}}</ref>

<ref name="42C">{{cite journal| last=Kolmogorov | first=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 |last1=Kolmogorov |first1=A. N. |s2cid=122060992 |issue=1890 |bibcode=1991RSPSA.434...15K}}</ref>
<ref name="43C">{{cite book| 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| bibcode=1979LNP....93...51K| doi=10.1007/BFb0021737| series=Lecture Notes in Physics| date=1979 | volume=93 | isbn=978-3-540-09120-2}} Translation of ''Doklady Akademii Nauk SSSR'' (1954) 98: 527.</ref>
<ref name="44C">{{cite journal |last1=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/>'' + ''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 }}</ref>
<ref name="45C">{{cite journal |author=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 |doi-access=free }}</ref>
<ref name="46C">{{cite web |url=https://www.maths.nottingham.ac.uk/personal/sc/pdfs/Seashells09.pdf |title=The Geometry and Pigmentation of Seashells |author=Stephen Coombes |date=February 2009 |publisher=University of Nottingham |access-date=2013-04-10 |archive-url=https://web.archive.org/web/20131105134513/https://www.maths.nottingham.ac.uk/personal/sc/pdfs/Seashells09.pdf |archive-date=2013-11-05}}</ref>
<ref name="47C">{{cite journal |author1=Kyrtsou C. |author2=Labys W. | year = 2006 | title = Evidence for chaotic dependence between US inflation and commodity prices | journal = Journal of Macroeconomics | volume = 28 | issue = 1| pages = 256–266 |doi=10.1016/j.jmacro.2005.10.019 }}</ref>
<ref name="48C">{{cite journal | author = 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 }}</ref>
<ref name="49C">{{cite book |author1=Kyrtsou, C. |author2=Vorlow, C. |chapter=Complex dynamics in macroeconomics: A novel approach |editor1=Diebolt, C. |editor2=Kyrtsou, C. |title=New Trends in Macroeconomics |publisher=Springer Verlag |year=2005 }}</ref>
<ref name="50C">{{cite journal |last1=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 |page=023048 |doi=10.1088/1367-2630/aaad41 |language=en |issn=1367-2630|bibcode=2018NJPh...20b3048H |doi-access=free }}</ref>

<ref name="51C">{{cite web |url = http://www.dspdesignline.com/218101444;jsessionid=Y0BSVTQJJTBACQSNDLOSKH0CJUNN2JVN?pgno=1 |title = Applying Chaos Theory to Embedded Applications |archive-url=https://archive.today/20110809025913/http://www.dspdesignline.com/218101444;jsessionid=Y0BSVTQJJTBACQSNDLOSKH0CJUNN2JVN?pgno=1 |archive-date=9 August 2011 }}</ref>
<ref name="52C">{{cite journal |author1=Hristu-Varsakelis, D. |author2=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 |article-number=138547 |doi-access=free }}</ref>
<ref name="53C">{{Cite journal | doi = 10.1023/A:1023939610962 |author1=Kyrtsou, C. |author2=M. Terraza |s2cid=154202123 | 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 | journal = Computational Economics | volume = 21 | issue = 3| pages = 257–276 }}</ref>
<ref name="54C">{{cite book |first1=Justine |last1=Gregory-Williams |first2=Bill |last2=Williams |title=Trading Chaos: Maximize Profits with Proven Technical Techniques |year=2004|publisher=Wiley|location=New York|isbn=978-0-471-46308-5|edition=2nd }}</ref>
<ref name="55C">{{cite book|last=Peters|first=Edgar E.|title=Fractal market analysis: applying chaos theory to investment and economics|year=1994|publisher=Wiley|location=New York u.a.|isbn=978-0-471-58524-4|edition=2. print.}}</ref>
<ref name="56C">{{cite book|last=Peters|first= Edgar E.|title=Chaos and order in the capital markets: a new view of cycles, prices, and market volatility|year=1996|publisher=John Wiley & Sons|location=New York|isbn=978-0-471-13938-6|edition=2nd }}</ref>
<ref name="57C">{{cite journal|last1=Hubler|first1=A.|last2=Phelps|first2=K.|title=Guiding a self-adjusting system through chaos|journal=Complexity|volume=13|issue=2|page=62|date=2007|doi=10.1002/cplx.20204|bibcode = 2007Cmplx..13b..62W }}</ref>
<ref name="58C">{{cite journal|last1=Gerig|first1=A.|title=Chaos in a one-dimensional compressible flow|journal=Physical Review E |volume=75 |issue=4|article-number=045202|date=2007|doi=10.1103/PhysRevE.75.045202|pmid=17500951|arxiv=nlin/0701050|bibcode = 2007PhRvE..75d5202G |s2cid=45804559}}</ref>
<ref name="59C">{{cite journal|last1=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>
<ref name="60C">{{cite journal |last1=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|doi-access=free }}</ref>
<ref name="61C">{{cite book |last1=Progonati |first1=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>
<ref name="62C">{{cite journal |author1=Dilão, R. |author2=Domingos, T. |s2cid=697164 | 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>
<ref name="63C">{{Cite journal|last1=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>
<ref name="64C">{{Cite journal|last1=Behnia|first1=S.|last2=Akhshani|first2=A.|last3=Mahmodi|first3=H.|last4=Akhavan|first4=A.|date=2008-01-01|title=A novel algorithm for image encryption based on mixture of chaotic maps|journal=Chaos, Solitons & Fractals|volume=35|issue=2|pages=408–419|doi=10.1016/j.chaos.2006.05.011|bibcode = 2008CSF....35..408B }}</ref>
<ref name="65C">{{cite journal|last=Wang|first=Xingyuan|year=2012|title=An improved key agreement protocol based on chaos|journal=Commun. Nonlinear Sci. Numer. Simul.|volume=15|issue=12|pages=4052–4057|bibcode=2010CNSNS..15.4052W|doi=10.1016/j.cnsns.2010.02.014|author2=Zhao, Jianfeng}}</ref>
<ref name="66C">{{cite journal|last=Babaei|first=Majid|s2cid=18407251|year=2013|title=A novel text and image encryption method based on chaos theory and DNA computing|journal=Natural Computing |volume=12|issue=1|pages=101–107|doi=10.1007/s11047-012-9334-9}}</ref>
<ref name="67C">{{Cite journal|last1=Akhavan|first1=A.|last2=Samsudin|first2=A.|last3=Akhshani|first3=A.|date=2017-10-01|title=Cryptanalysis of an image encryption algorithm based on DNA encoding|journal=Optics & Laser Technology|volume=95|pages=94–99|doi=10.1016/j.optlastec.2017.04.022|bibcode = 2017OptLT..95...94A }}</ref>
<ref name="68C">{{Cite journal|last=Xu|first=Ming|s2cid=125169427|date=2017-06-01|title=Cryptanalysis of an Image Encryption Algorithm Based on DNA Sequence Operation and Hyper-chaotic System|journal=3D Research|language=en|volume=8|issue=2|article-number=15|doi=10.1007/s13319-017-0126-y|issn=2092-6731|bibcode = 2017TDR.....8..126X }}</ref>
<ref name="69C">{{Cite journal |last1=Liu|first1=Yuansheng|last2=Tang|first2=Jie|last3=Xie|first3=Tao|date=2014-08-01|title=Cryptanalyzing a RGB image encryption algorithm based on DNA encoding and chaos map|journal=Optics & Laser Technology|volume=60|pages=111–115|doi=10.1016/j.optlastec.2014.01.015|arxiv=1307.4279|bibcode = 2014OptLT..60..111L |s2cid=18740000}}</ref>
<ref name="70C">{{cite journal|last=Nehmzow|first=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|author2=Keith Walker|url=http://cswww.essex.ac.uk/staff/udfn/ftp/ecmrw3.pdf|access-date=2017-10-25|archive-url=https://web.archive.org/web/20170812003513/http://cswww.essex.ac.uk/staff/udfn/ftp/ecmrw3.pdf|archive-date=2017-08-12|citeseerx=10.1.1.105.9178}}</ref>
<ref name="71C">{{cite journal|last=Goswami|first=Ambarish|year=1998|title=A Study of the Passive Gait of a Compass-Like Biped Robot: Symmetry and Chaos|journal=The International Journal of Robotics Research|volume=17|issue=12|pages=1282–1301|doi=10.1177/027836499801701202|author2=Thuilot, Benoit|author3=Espiau, Bernard|citeseerx=10.1.1.17.4861|s2cid=1283494}}</ref>
<ref name="72C">{{cite journal |last1=Eduardo |first1=Liz |last2=Ruiz-Herrera |first2=Alfonso |year=2012 |title=Chaos in discrete structured population models |journal=SIAM Journal on Applied Dynamical Systems |volume=11 |issue=4 |pages=1200–1214 |doi=10.1137/120868980}}</ref>

<ref name="73C">{{cite journal |last=Lai |first=Dejian |year=1996 |title=Comparison study of AR models on the Canadian lynx data: a close look at BDS statistic |journal=Computational Statistics & Data Analysis |volume=22 |issue=4 |pages=409–423 |doi=10.1016/0167-9473(95)00056-9}}</ref>

<ref name="74C">{{cite journal |last=Sivakumar |first=B. |date=2000-01-31 |title=Chaos theory in hydrology: important issues and interpretations |journal=Journal of Hydrology |volume=227 |issue=1–4 |pages=1–20 |doi=10.1016/S0022-1694(99)00186-9 |bibcode=2000JHyd..227....1S}}</ref>

<ref name="75C">{{cite journal |last=Bozóki |first=Zsolt |date=1997-02 |title=Chaos theory and power spectrum analysis in computerized cardiotocography |journal=European Journal of Obstetrics & Gynecology and Reproductive Biology |volume=71 |issue=2 |pages=163–168 |doi=10.1016/s0301-2115(96)02628-0 |pmid=9138960}}</ref>

<ref name="76C">{{cite book |last1=Perry |first1=Joe N. |last2=Smith |first2=Robert H. |last3=Woiwod |first3=Ian P. |last4=Morse |first4=David R. |year=2000 |title=Chaos in Real Data: The Analysis of Non-Linear Dynamics from Short Ecological Time Series |series=Population and Community Biology Series |volume=22 |edition=1 |publisher=Springer |doi=10.1007/978-94-011-4010-2 |isbn=978-94-010-5772-1}}</ref>

<ref name="77C">{{cite journal |last1=Thompson |first1=John N. |last2=Burdon |first2=Jeremy J. |year=1992 |title=Gene-for-gene coevolution between plants and parasites |journal=Nature |volume=360 |issue=6400 |pages=121–125 |doi=10.1038/360121a0 |bibcode=1992Natur.360..121T |s2cid=4346920}}</ref>

<ref name="78C">{{cite book |last=Jones |first=Gareth |year=1998 |title=The Epidemiology of Plant Diseases |publisher=Springer |doi=10.1007/978-94-017-3302-1 |isbn=978-94-017-3302-1}}</ref><ref name="79C">{{cite journal|last=Juárez|first=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|doi-access=free}}</ref>
<ref name="80C">{{cite journal |last=Brooks |first=Chris |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|s2cid=118329463 }}</ref>
<ref name="81C">{{cite journal |last1=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|doi-access=free }}</ref>
<ref name="82C">{{cite journal |last1=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 |s2cid=85526993 }}</ref>
<ref name="83C">{{cite journal |last1=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 |journal=Chaos: An Interdisciplinary Journal of Nonlinear Science |page=083129 |doi=10.1063/5.0015916 |date=1 August 2020|volume=30 |issue=8 |pmid=32872798 |bibcode=2020Chaos..30h3129O |s2cid=235909725 }}</ref>

</div>
[[File:Lorenz attractor yb.svg|thumb|right|Plot of the [[W:Lorenz attractor|Lorenz attractor]] with parameters <math>r = 28, \;\sigma = 10, \;b = \frac{8}{3}</math>.]]
[[File:3D Lorenz Chaotic Attactor.png|thumb|A 3D visualization of the [[W:Lorenz attractor|Lorenz attractor]].]]
[[File:Double-compound-pendulum.gif||thumb|Animation of a [[W:double pendulum|double-rod pendulum]] showing chaotic motion. Even tiny changes in the [[W:initial condition|initial conditions]] lead to a dramatically different [[W:trajectory|trajectory]]. The double pendulum is one of the simplest systems that displays chaotic behavior.]]
Chaos theory studies deterministic systems that can be predicted only for a limited period before their behavior begins to appear random. The length of time over which a chaotic system can be forecast reliably depends on three factors: the amount of uncertainty that can be tolerated in the prediction, the precision with which the system’s current state can be measured, and a characteristic time scale determined by the system’s dynamics, called the [[W:Lyapunov time|Lyapunov time]]. Examples of Lyapunov times include chaotic electrical circuits, around 1 millisecond; weather systems, a few days (though this is unproven); and the inner solar system, approximately 4 to 5 million years.<ref name="01C" />
In chaotic systems, forecast uncertainty grows [[W:Exponential growth|exponentially]] over time. Mathematically, doubling the forecast period increases the relative uncertainty by more than a factor of four. Consequently, in practice, reliable predictions cannot be made over periods longer than two or three Lyapunov times. When forecasts lose meaningful accuracy, the system’s behavior effectively appears random.<ref name="02C" />
===Chaos in Everyday Life===
Something “chaotic” in life, just means it seems random, uncontrollable, or impossible to predict. It’s that feeling that a tiny event can blow up into something bigger.
==== Everyday Examples of Chaos ====
* Traffic jams: One driver hits the brakes, and the whole highway slows to a crawl. A hesitation spreads and becomes a massive block.
* Weather: A small change, like the temperature being 0.1 °C higher in one spot, can shift the atmosphere enough to change next week’s weather.
* People and relationships: One off timed remark can throw off a whole conversation or ruin a relationship. Being just five minutes late might mean missing someone who could have changed your life.
We meet this chaos constantly: misplacing your keys, spilling a drink, markets swinging because of a rumor, a funny video exploding online, a virus spreading fast, or a toddler switching moods. Chaos is the difference between our wish to control things and how complicated the world actually is.
=== Chaos in the Physical World ===
The universe follows exact laws, but chaos shows up everywhere, from the tiniest particles to everyday physics.
* Molecules in a Glass of Water A single glass of water contains ± <math> 10^{23} </math> molecules zipping around and colliding billions of times per second. Even if you somehow knew every molecule’s exact location and speed, the smallest measurement error would grow so quickly that predictions would fall apart almost immediately. That’s why physicists don’t follow each particle, they use statistical mechanics instead.
=== Turbulent Flow ===
If you turn a glass gently, the water flows smoothly. Turn it up, and the flow suddenly becomes messy and unpredictable. Smoke behaves the same way: a steady stream at first, then chaotic twisting. Fluid equations are completely deterministic, but nearly identical starting conditions can lead to wildly different results, classic chaos.
=== A Classic Example: The Double Pendulum ===
Set up two double pendulums with almost the same starting angle and speed, and within seconds they’ll be swinging in totally different patterns. It’s a simple mechanical device, but its behavior becomes unpredictable very quickly.
=== Fractals Everywhere ===
Coastlines, rivers, mountains, trees, blood vessels, broccoli—so many natural shapes repeat similar patterns at different scales. These fractal-like forms usually come from chaotic processes running over and over. Even heartbeats and brain signals show healthy chaotic variation; when they become too perfectly regular, that’s usually a sign that something is wrong.
=== Chaos on Cosmic Scales ===
Even the universe at its largest scales is shaped by chaos.
=== The Early Universe ===
Right after the Big Bang, tiny quantum fluctuations created random differences in density. When the universe expanded rapidly during inflation, these tiny “wiggles” were stretched out and became the seeds of all future structure. The enormous web of galaxies and empty regions we see today grew out of that early chaotic noise.
=== How Galaxies Form ===
Start with an almost even cloud of gas, add a few small density differences, and let gravity act for billions of years. With complications like explosions, black holes, and collisions, those tiny early variations decide whether a galaxy ends up as a smooth spiral or a jumbled merger full of starbursts.
=== Star Clusters ===
In dense star clusters, each star feels the pull of countless neighbors. The orbits are so sensitive that after just a few crossings, predictions break down. Stars get flung outward or drawn into the center, and the whole cluster slowly loses members because of this chaotic behavior.
=== Galaxy Collisions ===
When galaxies run into each other, stars almost never physically collide, but gravity throws them onto completely new paths. Long tidal tails, stretched bridges of stars, and distorted shapes appear. It’s chaotic but often stunning to look at.
Yet even with all this underlying chaos, the universe still settles into recognizable patterns—spiral arms, predictable relations between galaxy properties, the smoothness of the cosmic microwave background, and more.
=== Bottom Line ===
Chaos isn’t the absence of order, it’s what helps create the most interesting kinds of order. From the swirls in a cup of coffee to massive cosmic structures spanning hundreds of millions of light-years, tiny differences can lead to huge outcomes. Perfect prediction will always be out of reach, and that fundamental unpredictability is part of what makes the universe active, complex, and far more than a simple clockwork machine.
==='''Chaos theory'''===
is an [[W:interdisciplinary|interdisciplinary]] field of study and a branch of [[W:mathematics|mathematics]] that examines how deterministic [[W:dynamical system|dynamical systems]] can produce highly unpredictable behavior. Although such systems follow exact rules, they can respond extremely sensitively to their starting conditions, a property once mistaken for pure randomness.<ref name="03C" />
Chaos theory shows that chaotic [[W:chaotic complex system|complex systems]] are not simply disordered. Hidden within their irregular behavior are repeating structures: patterns, interconnections, ongoing [[W:feedback loops|feedback loops]], forms of [[W:self-similarity|self-similarity]], [[W:fractals|fractal]] organization, and various kinds of [[W:self-organization|self-organization]].<ref name="04C" />
A key concept is the [[W:butterfly effect|butterfly effect]]: in a deterministic [[W:nonlinear system|nonlinear system]], a tiny shift in initial conditions can amplify into major changes later on.<ref name="05C" /> This is often illustrated with the metaphor of a butterfly’s wings in [[W:Brazil|Brazil]] influencing the development of a tornado in [[W:Texas|Texas]].<ref name="06C" /><ref name="07C" />{{rp|181–184}}<ref name="08C" />
Small variations in starting conditions—whether from measurement inaccuracies or rounding in [[W:numerical analysis|numerical computation]]—can cause systems of this kind to evolve in dramatically different ways, making reliable long-term forecasts of their behavior generally impossible.<ref name="09C" /> This occurs even though these systems are [[W:deterministic system (mathematics)|deterministic]]—their future evolution is uniquely fixed by their initial state<ref name="10C" /> and contains no [[W:randomness|random]] ingredients.<ref name="11C" /> Thus, even though such systems follow deterministic rules, this does not guarantee that they are predictable.<ref name="12C" /><ref name="13C" /> This phenomenon is referred to as '''deterministic chaos''', or simply '''chaos'''. [[W:Edward Lorenz|Edward Lorenz]] summarized this idea as follows:<ref name="14C" />
{{Blockquote|Chaos: When the present determines the future but the approximate present does not approximately determine the future.}}
Chaotic dynamics appear in many natural contexts, such as fluid turbulence, irregular heart rhythms, and the behavior of weather and climate systems.<ref name="15C" /><ref name="10C" /> Chaos can also arise spontaneously in systems involving human-made components, like [[W:road traffic|road traffic]]. Researchers analyze this behavior using chaotic [[W:mathematical model|mathematical models]] and tools such as [[W:recurrence plot|recurrence plots]] and [[W:Poincaré map|Poincaré maps]]. Chaos theory plays a role in diverse fields including [[W:meteorology|meteorology]],<ref name="10C" /> [[W:anthropology|anthropology]],<ref name="16C" /> [[W:sociology|sociology]], [[W:environmental science|environmental science]], [[W:computer science|computer science]], [[W:engineering|engineering]], [[W:economics|economics]], [[W:ecology|ecology]], and [[W:pandemic|pandemic]] [[W:crisis management|crisis management]].<ref name="17C" /><ref name="18C" /> The theory has helped shape areas such as [[W:dynamical systems|complex dynamical systems]], [[W:edge of chaos|edge of chaos]] theory, and [[W:self-assembly|self-assembly]] processes.
==Chaotic dynamics==
[[File:Chaos Sensitive Dependence.svg|thumb|The [[W:Map (mathematics)|map]] defined by <span style="white-space: nowrap;">''x'' → 4 ''x'' (1 – ''x'')</span> and <span style="white-space: nowrap;">''y'' → (''x'' + ''y'') [[W:Modulo operation|mod]] 1</span> shows sensitivity to initial x positions. Two series of ''x'' and ''y'' values, starting from a very small difference, diverge rapidly over time.]]
In everyday language, "chaos" refers to "a state of disorder".<ref name="19C" /> In chaos theory, however, the term is used in a more precise sense. While there is no universally accepted mathematical definition of chaos, a widely used formulation by [[W:Robert L. Devaney|Robert L. Devaney]] states that a dynamical system is considered chaotic if it satisfies the following conditions:<ref name="20C" />
it exhibits [[W:sensitive dependence on initial conditions|sensitivity to initial conditions]],
it is [[W:Mixing (mathematics)#Topological mixing|topologically transitive]],
it possesses [[W:dense set|dense]] [[W:periodic orbit|periodic orbits]].
In some situations, the second and third properties can actually imply sensitivity to initial conditions.<ref name="21C" /><ref name="22C" /> For discrete-time systems, this holds for all [[W:Continuous function|continuous]] [[W:Map (mathematics)|maps]] on [[W:metric space|metric spaces]].<ref name="23C" /> In these cases, although sensitivity to initial conditions is often the most practically relevant property, it does not need to be explicitly stated in the definition.
When attention is restricted to [[W:Interval (mathematics)|intervals]], the second property alone implies the other two.<ref name="24C" /> A slightly weaker alternative definition of chaos uses only the first two conditions above.<ref name="25C" />
===Sensitivity to initial conditions===
{{Main|W:Butterfly effect|l1=Butterfly effect}}
[[File:SensInitCond.gif|thumb|Lorenz equations used to generate plots for the <math>y</math> variable. The initial conditions for <math>x</math> and <math>z</math> were kept the same, while those for <math>y</math> were slightly varied between <math>1.001</math>, <math>1.0001</math>, and <math>1.00001</math>. The parameters were <math>\rho = 45.91</math>, <math>\sigma = 16</math>, and <math>\beta = 4</math>. The graph shows that even minimal differences in initial values produce substantial divergence after about 12 seconds.]]
'''Sensitivity to initial conditions''' refers to the property that in a chaotic system, points that are initially very close can follow drastically different trajectories over time. Even an imperceptibly small change in the starting state can lead to markedly different outcomes.
This phenomenon is popularly known as the "[[W:butterfly effect|butterfly effect]]", named after a 1972 presentation by [[W:Edward Lorenz|Edward Lorenz]] to the [[W:American Association for the Advancement of Science|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 name="26C" /> In this metaphor, the butterfly’s wing represents a tiny change in the initial conditions, which can trigger a cascade of events that prevents reliable prediction of large-scale behavior. Had the butterfly not flapped, the system’s trajectory could have been entirely different.
As Lorenz emphasized in his 1993 book ''The Essence of Chaos'',<ref name="07C"/>{{rp|8}} "sensitive dependence can serve as an acceptable definition of chaos". He 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="07C"/>{{rp|23}} This description matches the concept of sensitive dependence on initial conditions (SDIC).
Lorenz illustrated SDIC with an idealized skiing model, showing how small changes in starting positions affect time-dependent paths.<ref name="07C"/>{{rp|189–204}} A predictability horizon can be estimated before SDIC fully manifests, i.e., before initially close trajectories diverge significantly.<ref name="27C" />
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="28C" /> 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 [[W:geologic era|geologic era]]), but we cannot predict exactly which day will have the hottest temperature of the year.
In more mathematical terms, the [[W:Lyapunov exponent|Lyapunov exponent]] measures the sensitivity to initial conditions, in the form of rate of exponential divergence from the perturbed initial conditions.<ref name="29C" /> More specifically, given two starting [[W:trajectory|trajectories]] in the [[W:phase space|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="10C" />
In addition to the above property, other properties related to sensitivity of initial conditions also exist. These include, for example, [[W:Measure (mathematics)|measure-theoretical]] [[W:Mixing (mathematics)|mixing]] (as discussed in [[W:ergodicity|ergodic]] theory) and properties of a [[W:Kolmogorov automorphism|K-system]].<ref name="13C" />
===Non-periodicity===
A chaotic system may have sequences of values for the evolving variable that exactly repeat themselves, giving periodic behavior starting from any point in that sequence. However, such periodic sequences are repelling rather than attracting, meaning that if the evolving variable is outside the sequence, however close, it will not enter the sequence and in fact, will diverge from it. Thus for [[W:almost all|almost all]] initial conditions, the variable evolves chaotically with non-periodic behavior. A consequence of sensitivity to initial conditions is that when we know a system’s state only approximately, which is the usual situation, its behavior becomes unpredictable after some finite time. Weather is a well-known example: forecasts are generally reliable for only about a week.<ref name="28C" /> This does not mean that nothing can be said about the distant future; rather, only broad constraints apply.
Other mathematical notions related to sensitivity to initial conditions include measure-theoretic [[W:Mixing (mathematics)|mixing]] from [[W:ergodicity|ergodic theory]] and properties associated with [[W:Kolmogorov automorphism|K-systems]].<ref name="13C" />
Chaotic systems may contain strictly periodic orbits, but these are repelling rather than attracting. A trajectory starting even slightly off such an orbit will move away from it over time. Consequently, for [[W:almost all|almost all]] initial conditions, the system evolves in a non-periodic, chaotic manner.
===Combinatorial (or complex) chaos===
Some definitions of chaos do not rely on sensitivity to initial conditions. One example is combinatorial chaos, which arises when a discrete combinatorial rule is applied repeatedly.<ref name="30C" /> This kind of behavior is closely related to the dynamics seen in [[W:cellular automata|cellular automata]]. It is significant because systems exhibiting this form of chaos can be computationally universal: they can simulate a [[W:Turing machine|Turing machine]], meaning that the [[W:halting problem|halting problem]] becomes undecidable within their evolution. As a result, certain computational processes within such systems may never terminate. This represents a fundamentally different pathway to unpredictability.<ref name="31C" />
==History==
[[File:Barnsley fern plotted with VisSim.PNG|thumb|upright|[[W:Barnsley fern|Barnsley fern]] created using the [[W:chaos game|chaos game]]. Natural forms (ferns, clouds, mountains, etc.) may be recreated through an [[W:iterated function system|iterated function system]] (IFS).]]
[[W:James Clerk Maxwell|James Clerk Maxwell]] was one of the first scientists to emphasize the importance of initial conditions, and he is considered an early contributor to chaos theory, with work in the 1860s and 1870s.<ref name="32C" /><ref name="33C" /><ref name="34C" />
In the 1880s, while studying the [[W:three-body problem|three-body problem]], [[W:Henri Poincaré|Henri Poincaré]] discovered that certain orbits can be nonperiodic, yet neither diverge to infinity nor approach a fixed point.<ref name="35C" /><ref name="36C" /><ref name="37C" />
In 1898, [[W:Jacques Hadamard|Jacques Hadamard]] published a study of a free particle moving frictionlessly on a surface of constant negative curvature, known as "[[W:Hadamard's billiards|Hadamard's billiards]]". He showed that all trajectories are unstable,<ref name="38C" /> with particle trajectories diverging exponentially, corresponding to a positive [[W:Lyapunov exponent|Lyapunov exponent]].<ref name="39C" />
Further work on nonlinear [[W:differential equation|differential equations]] was conducted by [[W:George David Birkhoff|George David Birkhoff]],<ref name="40C" /> [[W:Andrey Nikolaevich Kolmogorov|Andrey Nikolaevich Kolmogorov]],<ref name="41C" /><ref name="42C" /><ref name="43C" /> [[W:Mary Lucy Cartwright|Mary Lucy Cartwright]] and [[W:John Edensor Littlewood|John Edensor Littlewood]],<ref name="44C" /> and [[W:Stephen Smale|Stephen Smale]].<ref name="45C" />
Experimentalists and mathematicians had observed turbulence in fluid motion, chaotic behaviour in society and economy, nonperiodic oscillation in radio circuits, and fractal patterns in nature long before a formal theory existed.
== A popular but inaccurate analogy for chaos ==
The sensitive dependence on initial conditions (i.e., the butterfly effect) has often been illustrated through the well-known piece of folklore:
<poem style="margin-left: 2em;"> For want of a nail, the shoe was lost.
For want of a shoe, the horse was lost.
For want of a horse, the rider was lost.
For want of a rider, the battle was lost.
For want of a battle, the kingdom was lost.
And all for the want of a horseshoe nail. </poem>
Because of this verse, many readers incorrectly assume that the effect of a tiny initial perturbation must increase monotonically with time, or that any arbitrarily small change will inevitably produce a large impact in numerical integrations. In 2008, however, Lorenz argued that the verse does not describe true chaos, but instead illustrates the simpler notion of instability. The rhyme also suggests an irreversible cascade of consequences, whereas chaotic systems often exhibit later events that can partially offset earlier divergences. In this sense, the verse indicates divergence but omits the requirement of boundedness, which is necessary for the finite extent of a butterfly-shaped attractor. The behaviour described by the rhyme is therefore better characterized as “finite-time sensitive dependence.”
== Applications ==
[[File:Textile cone.JPG|thumb|left|A [[W:conus textile|conus textile]] shell, similar in appearance to [[W:Rule 30|Rule 30]], a [[W:cellular automaton|cellular automaton]] with chaotic behaviour<ref name="46C" />]]
Although chaos theory originated from the study of weather, it has since found application in a broad range of fields. These include [[W:geology|geology]], [[W:mathematics|mathematics]], [[W:biology|biology]], [[W:computer science|computer science]], [[W:economics|economics]],<ref name="47C" /><ref name="48C" /><ref name="49C" /> [[W:engineering|engineering]],<ref name="50C" /><ref name="51C" /> [[W:finance|finance]],<ref name="52C" /><ref name="53C" /><ref name="54C" /><ref name="55C" /><ref name="56C" /> [[W:meteorology|meteorology]], [[W:philosophy|philosophy]], [[W:anthropology|anthropology]],<ref name="16C" /> [[W:physics|physics]],<ref name="57C" /><ref name="58C" /><ref name="59C" /> [[W:politics|politics]],<ref name="60C" /><ref name="61C" /> [[W:population dynamics|population dynamics]],<ref name="62C" /> and [[W:BEAM robotics|robotics]]. The following subsections provide examples but are not exhaustive, as new applications continue to emerge.
=== Cryptography ===
{{Main|W:Chaotic cryptology|l1 = Chaotic cryptology}}
Chaos theory has been used in [[W:cryptography|cryptography]] for decades. In recent years, chaos and nonlinear dynamics have inspired hundreds of [[W:cryptographic primitive|cryptographic primitives]], including image [[W:encryption algorithm|encryption algorithms]], [[W:hash function|hash functions]], [[W:Cryptographically secure pseudorandom number generator|secure pseudorandom number generators]], [[W:stream cipher|stream ciphers]], [[W:Digital watermarking|watermarking]], and [[W:steganography|steganography]].<ref name="63C" /> Most such algorithms use uni-modal chaotic maps, with the control parameters and initial conditions serving as cryptographic keys.<ref name="64C" /> The conceptual similarity between chaotic maps and cryptographic systems is a major motivation for chaos-based design.<ref name="63C" />
Symmetric-key cryptography relies on [[W:diffusion and confusion|diffusion and confusion]], which can be modeled effectively by chaotic dynamics.<ref name="65C" /> In addition, the combination of chaos theory with [[W:DNA computing|DNA computing]] has been explored for image and data encryption,<ref name="66C" /> although many DNA–chaos encryption schemes have later been shown insecure or inefficient.<ref name="67C" /><ref name="68C" /><ref name="69C" />
=== Robotics ===
Robotics has also benefited from chaos theory. Instead of relying solely on trial-and-error exploration, chaotic models can be used to build [[W:Predictive modelling|predictive models]] of robot–environment interaction.<ref name="70C" />
Chaotic behaviour has also been observed in passive-dynamics [[W:passive dynamics|biped robots]], which can exhibit complex gait patterns.<ref name="71C" />
=== Biology ===
For more than a century, biologists have modeled species populations using [[W:population model|population models]], many of them continuous. More recently, chaotic models have been applied to certain discrete populations.<ref name="72C" /> For example, time-series models of [[W:Canada lynx|Canadian lynx]] populations have displayed evidence of chaotic dynamics.<ref name="73C" />
Chaos is also investigated in ecological systems such as [[W:hydrology|hydrology]]. Although hydrological models may face limitations, analyzing them from a chaotic perspective can still provide insight.<ref name="74C" />
In [[W:cardiotocography|cardiotocography]], chaos-based modeling has been used to develop more sensitive indicators of [[W:Intrauterine hypoxia|fetal hypoxia]] while maintaining non-invasiveness.<ref name="75C" />
As Perry notes, modeling chaotic [[W:time series|time series]] in [[W:theoretical ecology|ecology]] benefits from appropriate constraints.<ref name="76C" />{{rp|176,177}} Distinguishing genuine chaos from model-induced chaos can be difficult, so constrained models or duplicate time series help ensure realism, for instance in Perry & Wall 1984.<ref name="76C" />{{rp|176,177}}
In evolutionary biology, [[W:Gene-for-gene|gene-for-gene]] co-evolution may exhibit chaotic dynamics in [[W:allele frequency|allele frequencies]].<ref name="77C" /> Adding variables, reflecting more realistic population structure, often increases the likelihood of chaos.<ref name="77C" /> Foundational co-evolutionary studies by [[W:Robert M. May|Robert M. May]] helped establish this line of research.<ref name="77C" />
Even in constant environments, a single [[W:crop|crop]] interacting with a single [[W:pathogen|pathogen]] may generate [[W:quasiperiodicity|quasi-periodic]] or [[W:chaotic oscillation|chaotic]] oscillations in pathogen [[W:statistical population|population]].<ref name="78C" />
=== Economics ===
Economic models may also benefit from ideas from chaos theory, though assessing the stability of an economic system and identifying the most influential factors remains highly complex.<ref name="79C" /> Unlike classical physical systems, economic and financial systems are fundamentally stochastic, emerging from interactions among people. Purely deterministic models therefore tend to fall short in representing economic data. Empirical attempts to test for chaos in economics and finance have produced mixed results, partly because studies sometimes confuse tests for genuine chaos with more general tests for nonlinear structure.<ref name="80C" />
Chaos in economic time series can be investigated using [[W:recurrence quantification analysis|recurrence quantification analysis]] (RQA). Orlando et al.<ref name="81C" /> used the recurrence quantification correlation index to detect subtle structural changes in time-series data. The same technique has been applied to identify transitions from laminar (regular) to turbulent (chaotic) behavior, and to distinguish dynamical differences among macroeconomic variables, thereby exposing hidden features of economic evolution.<ref name="82C" /> More recently, chaos-based approaches have been explored in modeling economic activity and incorporating shocks from external events such as COVID-19.<ref name="83C" />
==See also==
'''Examples of chaotic systems'''
{{Div col|colwidth=18em}}
* [[W:Contour advection|Advected contours]]
* [[W:Arnold's cat map|Arnold's cat map]]
* [[W:Bifurcation theory|Bifurcation theory]]
* [[W:Bouncing ball dynamics|Bouncing ball dynamics]]
* [[W:Chua's circuit|Chua's circuit]]
* [[W:Cliodynamics|Cliodynamics]]
* [[W:Coupled map lattice|Coupled map lattice]]
* [[W:Double pendulum|Double pendulum]]
* [[W:Duffing equation|Duffing equation]]
* [[W:Dynamical billiards|Dynamical billiards]]
* [[W:Economic bubble|Economic bubble]]
* [[W:Chaotic scattering#Gaspard–Rice system|Gaspard-Rice system]]
* [[W:Logistic map|Logistic map]]
* [[W:Hénon map|Hénon map]]
* [[W:Horseshoe map|Horseshoe map]]
* [[W:List of chaotic maps|List of chaotic maps]]
* [[W:Rössler attractor|Rössler attractor]]
* [[W:Standard map|Standard map]]
* [[W:Swinging Atwood's machine|Swinging Atwood's machine]]
* [[W:Tilt A Whirl|Tilt A Whirl]]
{{Div col end}}
'''Other related topics'''
{{Div col|colwidth=18em}}
* [[W:Amplitude death|Amplitude death]]
* [[W:Anosov diffeomorphism|Anosov diffeomorphism]]
* [[W:Catastrophe theory|Catastrophe theory]]
* [[W:Causality|Causality]]
* [[W:Supersymmetric theory of stochastic dynamics|Chaos as topological supersymmetry breaking]]
* [[W:Chaos machine|Chaos machine]]
* [[W:Chaotic mixing|Chaotic mixing]]
* [[W:Chaotic scattering|Chaotic scattering]]
* [[W:Control of chaos|Control of chaos]]
* [[W:Determinism|Determinism]]
* [[W:Edge of chaos|Edge of chaos]]
* [[W:Emergence|Emergence]]
* [[W:Mandelbrot set|Mandelbrot set]]
* [[W:Kolmogorov–Arnold–Moser theorem|Kolmogorov–Arnold–Moser theorem]]
* [[W:Ill-conditioning|Ill-conditioning]]
* [[W:Ill-posedness|Ill-posedness]]
* [[W:Nonlinear system|Nonlinear system]]
* [[W:Patterns in nature|Patterns in nature]]
* [[W:Predictability|Predictability]]
* [[W:Quantum chaos|Quantum chaos]]
* [[W:Santa Fe Institute|Santa Fe Institute]]
* [[W:Shadowing lemma|Shadowing lemma]]
* [[W:Synchronization of chaos|Synchronization of chaos]]
* [[W:Unintended consequence|Unintended consequence]]
{{Div col end}}
'''People'''
{{Div col|colwidth=18em}}
* [[W:Ralph Abraham (mathematician)|Ralph Abraham]]
* [[W:Michael Berry (physicist)|Michael Berry]]
* [[W:Leon O. Chua|Leon O. Chua]]
* [[W:Ivar Ekeland|Ivar Ekeland]]
* [[W:Doyne Farmer|Doyne Farmer]]
* [[W:Martin Gutzwiller|Martin Gutzwiller]]
* [[W:Brosl Hasslacher|Brosl Hasslacher]]
* [[W:Michel Hénon|Michel Hénon]]
* [[W:Aleksandr Lyapunov|Aleksandr Lyapunov]]
* [[W:Norman Packard|Norman Packard]]
* [[W:Otto Rössler|Otto Rössler]]
* [[W:David Ruelle|David Ruelle]]
* [[W:Oleksandr Mikolaiovich Sharkovsky|Oleksandr Mikolaiovich Sharkovsky]]
* [[W:Greg Sams|Greg Sams]]
* [[W:Robert Shaw (physicist)|Robert Shaw]]
* [[W:Floris Takens|Floris Takens]]
* [[W:James A. Yorke|James A. Yorke]]
* [[W:George M. Zaslavsky|George M. Zaslavsky]]
{{Div col end}}
==Further reading==
===Articles===
* {{cite journal |first=A.N. |last=Sharkovskii |author-link=W: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=[[W:American Mathematical Monthly|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 }}
* {{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 |journal=[[W:Scientific American|Scientific American]] |volume=255 |issue=6 |pages=38–49 (bibliography p.136) |bibcode = 1986SciAm.255d..38T |last2 = Tucker |last3 = Morrison |author1 = Crutchfield|author5 = Packard|author6=N.H. |author7=Shaw |author8=R.S |author-link1=W:James P. Crutchfield |author-link5=Norman Packard |author-link7=Robert Shaw (physicist) |author4-link=J. Doyne Farmer |doi=10.1038/scientificamerican1286-46 }}
* {{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 |pmid=16486826 |article-number=044101 |bibcode=2006PhRvL..96d4101S }}
* {{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 |doi=10.1002/cplx.20159 |issue=3 |bibcode=2007Cmplx..12c..10H }}
* {{cite journal | last1 = Motter | first1 = Adilson E. | last2 = Campbell | first2 = David K. | year = 2013 | title = Chaos at fifty | journal = Physics Today | volume = 66| issue = 5| page = 27| doi = 10.1063/PT.3.1977 |arxiv = 1306.5777 |bibcode = 2013PhT....66e..27M | s2cid = 54005470 }}
===Textbooks===
* {{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=[[W:Jean-Pierre Eckmann|Jean-Pierre Eckmann]] |title=Iterated Maps on the Interval as Dynamical Systems |publisher=Birkhauser |year=1980 |isbn=978-0-8176-4926-5}}
* {{cite book |last=Devaney |first=Robert L. |author-link=W:Robert L. Devaney|title=An Introduction to Chaotic Dynamical Systems |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 }}
* {{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=W: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| 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=W:William G Hoover|title=Time Reversibility, Computer Simulation, and Chaos |publisher=World Scientific |orig-date=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 |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|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=W: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=W:Gerald Teschl| title = Ordinary Differential Equations and Dynamical Systems| publisher=[[W:American Mathematical Society|American Mathematical Society]]| place = [[W: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=W:Nicholas Tufillaro |last1=Tufillaro |last2=Reilly |title=An experimental approach to nonlinear dynamics and chaos |series=American Journal of Physics |volume=61 |issue=10 |page=958 |publisher=Addison-Wesley |year=1992 |isbn=978-0-201-55441-0 |bibcode=1993AmJPh..61..958T |doi=10.1119/1.17380 }}
* {{cite book | last=Wiggins|first=Stephen | title= Introduction to Applied Dynamical Systems and Chaos | 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}}
===Semitechnical and popular works===
* [[W:Christophe Letellier|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 |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=W: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}}
* [[W: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 }}
* [[W:Predrag Cvitanović|Predrag Cvitanović]], ''Universality in Chaos'', Adam Hilger 1989, 648 pp.
* [[W:Leon Glass|Leon Glass]] and Michael C. Mackey, ''From Clocks to Chaos: The Rhythms of Life,'' Princeton University Press 1988, 272 pp.
* [[W:James Gleick|James Gleick]], ''[[W:Chaos: Making a New Science|Chaos: Making a New Science]]'', New York: Penguin, 1988. 368 pp.
* {{cite book |author=W:John Gribbin|title=Deep Simplicity |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.
* [[W:Edward Lorenz|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=978-1-86094-954-8}}
* David Peak and Michael Frame, ''Chaos Under Control: The Art and Science of Complexity'', Freeman, 1994.
* [[W:Heinz-Otto Peitgen|Heinz-Otto Peitgen]] and [[W:Dietmar Saupe|Dietmar Saupe]] (Eds.), ''The Science of Fractal Images'', Springer 1988, 312 pp.
* [[W:Nuria Perpinya|Nuria Perpinya]], ''Caos, virus, calma. La Teoría del Caos aplicada al desórden artístico, social y político'', Páginas de Espuma, 2021.
* [[W:Clifford A. Pickover|Clifford A. Pickover]], ''Computers, Pattern, Chaos, and Beauty: Graphics from an Unseen World '', St Martins Pr 1991.
* [[W:Clifford A. Pickover|Clifford A. Pickover]], ''Chaos in Wonderland: Visual Adventures in a Fractal World'', St Martins Pr 1994.
* [[W:Ilya Prigogine|Ilya Prigogine]] and [[W:Isabelle Stengers|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}}
* [[W:David Ruelle|David Ruelle]], ''Chance and Chaos'', Princeton University Press 1993.
* [[W:Ivars Peterson|Ivars Peterson]], ''Newton's Clock: Chaos in the Solar System'', Freeman, 1993.
* {{cite book |author1=W:Ian Roulstone |author2=W: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=978-0-691-15272-1 }}
* {{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=978-0-521-36272-6}}
* 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=978-0-511-55454-4}}
* [[W:Ian Stewart (mathematician)|Ian Stewart]], ''Does God Play Dice?: The Mathematics of Chaos '', Blackwell Publishers, 1990.
* [[W:Steven Strogatz|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.
==External links==
* [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], [[W:Florence|Florence]], [[W:Italy|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 [[W: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, [[W:Étienne Ghys|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 [[W:Derek Muller|Derek Muller]]
'''Attribution'''
* License = CC-BY
==References==
{{Authority control}}