Editing Chaos theory (section)

== Applications ==
[[File:Textile cone.JPG|thumb|left|A [[conus textile]] shell, similar in appearance to [[Rule 30]], a [[cellular automaton]] with chaotic behaviour<ref>{{cite web |url=https://www.maths.nottingham.ac.uk/personal/sc/pdfs/Seashells09.pdf |archive-url=https://web.archive.org/web/20131105134513/https://www.maths.nottingham.ac.uk/personal/sc/pdfs/Seashells09.pdf |archive-date=2013-11-05 |url-status=live |title=The Geometry and Pigmentation of Seashells |author=Stephen Coombes |date=February 2009 |work=www.maths.nottingham.ac.uk |publisher=[[University of Nottingham]] |access-date=2013-04-10}}</ref>]]

Although chaos theory was born from observing weather patterns, it has become applicable to a variety of other situations. Some areas benefiting from chaos theory today are [[geology]], [[mathematics]], [[biology]], [[computer science]], [[economics]],<ref>{{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>{{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 | last2 = Labys }}</ref><ref>{{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> [[engineering]],<ref>{{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 |pages=023048 |doi=10.1088/1367-2630/aaad41 |language=en |issn=1367-2630|bibcode=2018NJPh...20b3048H |doi-access=free }}</ref><ref>{{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 |url-status=dead}}</ref> [[finance]],<ref>{{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 |doi-access=free }}</ref><ref>{{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>{{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=9780471463085|edition=2nd }}</ref><ref>{{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=978-0471585244|edition=2. print.}}</ref><ref>{{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=978-0471139386|edition=2nd }}</ref> [[meteorology]], [[philosophy]], [[anthropology]],<ref name=":0" /> [[physics]],<ref>{{cite journal|last1=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|last1=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 |s2cid=45804559}}</ref><ref>{{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> [[politics]],<ref>{{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>{{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> [[population dynamics]],<ref>{{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> and [[BEAM robotics|robotics]]. A few categories are listed below with examples, but this is by no means a comprehensive list as new applications are appearing.

=== Cryptography ===
{{Main|Chaotic cryptology}}

Chaos theory has been used for many years in [[cryptography]]. In the past few decades, chaos and nonlinear dynamics have been used in the design of hundreds of [[cryptographic primitive]]s. These algorithms include image [[encryption algorithms]], [[hash functions]], [[Cryptographically secure pseudorandom number generator|secure pseudo-random number generators]], [[stream ciphers]], [[Digital watermarking|watermarking]], and [[steganography]].<ref name="Akhavan 1797–1813">{{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> The majority of these algorithms are based on uni-modal chaotic maps and a big portion of these algorithms use the control parameters and the initial condition of the chaotic maps as their keys.<ref>{{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> From a wider perspective, without loss of generality, the similarities between the chaotic maps and the cryptographic systems is the main motivation for the design of chaos based cryptographic algorithms.<ref name="Akhavan 1797–1813"/> One type of encryption, secret key or [[symmetric key]], relies on [[diffusion and confusion]], which is modeled well by chaos theory.<ref>{{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> Another type of computing, [[DNA computing]], when paired with chaos theory, offers a way to encrypt images and other information.<ref>{{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> Many of the DNA-Chaos cryptographic algorithms are proven to be either not secure, or the technique applied is suggested to be not efficient.<ref>{{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>{{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|pages=15|doi=10.1007/s13319-017-0126-y|issn=2092-6731|bibcode = 2017TDR.....8..126X }}</ref><ref>{{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>

=== Robotics ===
Robotics is another area that has recently benefited from chaos theory. Instead of robots acting in a trial-and-error type of refinement to interact with their environment, chaos theory has been used to build a [[Predictive modelling|predictive model]].<ref>{{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|url-status=dead|citeseerx=10.1.1.105.9178}}</ref>
Chaotic dynamics have been exhibited by [[Passive dynamics|passive walking]] biped robots.<ref>{{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>

===Biology===
For over a hundred years, biologists have been keeping track of populations of different species with [[population model]]s. Most models are [[continuous function|continuous]], but recently scientists have been able to implement chaotic models in certain populations.<ref>{{cite journal|last=Eduardo|first=Liz|author2=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> For example, a study on models of [[Canada lynx|Canadian lynx]] showed there was chaotic behavior in the population growth.<ref>{{cite journal|last=Lai|first=Dejian|title=Comparison study of AR models on the Canadian lynx data: a close look at BDS statistic|journal=[[Computational Statistics & Data Analysis]]|year=1996|volume=22|issue=4|pages=409–423|doi=10.1016/0167-9473(95)00056-9}}</ref> Chaos can also be found in ecological systems, such as [[hydrology]]. While a chaotic model for hydrology has its shortcomings, there is still much to learn from looking at the data through the lens of chaos theory.<ref>{{cite journal|last=Sivakumar|first=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> Another biological application is found in [[cardiotocography]]. Fetal surveillance is a delicate balance of obtaining accurate information while being as noninvasive as possible. Better models of warning signs of [[Intrauterine hypoxia|fetal hypoxia]] can be obtained through chaotic modeling.<ref>{{cite journal|last=Bozóki|first=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>

As Perry points out, [[mathematical model|modeling]] of chaotic [[time series]] in [[theoretical ecology|ecology]] is helped by constraint.<ref name=realdata/>{{rp|176,177}} There is always potential difficulty in distinguishing real chaos from chaos that is only in the model.<ref name=realdata/>{{rp|176,177}} Hence both constraint in the model and or duplicate time series data for comparison will be helpful in constraining the model to something close to the reality, for example Perry & Wall 1984.<ref name=realdata>{{cite book|year=2000|edition=1|publisher=[[Springer Science+Business Media Dordrecht]]|first4=David|first3=Ian|first2=Robert|first1=Joe|last4=Morse|last2=Smith|last1=Perry|last3=Woiwod|editor-first1=Joe N|editor-first2=Robert H|editor-first3=Ian P|editor-first4=David R|editor-last1=Perry|editor-last2=Smith|editor-last3=Woiwod|editor-last4=Morse|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|s2cid=37855255}}</ref>{{rp|176,177}} [[Gene-for-gene]] co-evolution sometimes shows chaotic dynamics in [[allele frequencies]].<ref name = "GFG" /> Adding variables exaggerates this: Chaos is more common in [[evolutionary model|models]] incorporating additional variables to reflect additional facets of real populations.<ref name = "GFG" /> [[Robert M. May]] himself did some of these foundational crop co-evolution studies, and this in turn helped shape the entire field.<ref name = "GFG" >
 {{ Cite journal |
       language=en|
           year=1992|
      publisher=[[Nature Publishing Group]]|
         volume=360|
        journal=[[Nature (journal)|Nature]]|
           issn=0028-0836|
          eissn=1476-4687|
         first2=Jeremy|
         first1=John|
     department=Review Article|
          pages=121–125|
          last1=Thompson|
          last2=Burdon|
          title=Gene-for-gene coevolution between plants and parasites|
 issue=6400|
            doi=10.1038/360121a0|
 bibcode=1992Natur.360..121T|
 s2cid=4346920}}
</ref> Even for a steady environment, merely combining one [[crop]] and one [[pathogen]] may result in [[quasiperiodicity|quasi-periodic-]] or [[chaotic oscillation|chaotic-]] oscillations in pathogen [[statistical population|population]].<ref name = "Epidemiology" >
 {{ Cite book |
       language=en|
           year=1998|
        edition=1|
      publisher=[[Springer Science+Business Media Dordrecht]]|
          first=Gareth|
           last=Jones|
 editor-first1=D. Gareth|
 editor-last1=Jones|
          pages=xvi + 460 + 26{{NBSP}}b/w{{NBSP}}{{NBSP}}ill. + 33{{NBSP}}color{{NBSP}}ill|
          title=The Epidemiology of Plant Diseases|
            doi=10.1007/978-94-017-3302-1|
          s2cid=1793087|
           isbn=978-94-017-3302-1}}
</ref>{{ RP |page=169}}

===Economics===
It is possible that economic models can also be improved through an application of chaos theory, but predicting the health of an economic system and what factors influence it most is an extremely complex task.<ref>{{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> Economic and financial systems are fundamentally different from those in the classical natural sciences since the former are inherently stochastic in nature, as they result from the interactions of people, and thus pure deterministic models are unlikely to provide accurate representations of the data. The empirical literature that tests for chaos in economics and finance presents very mixed results, in part due to confusion between specific tests for chaos and more general tests for non-linear relationships.<ref>{{cite journal |last=Brooks |first=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|s2cid=118329463 |url=http://centaur.reading.ac.uk/35988/1/35988.pdf |archive-url=https://web.archive.org/web/20170809072731/http://centaur.reading.ac.uk/35988/1/35988.pdf |archive-date=2017-08-09 |url-status=live }}</ref>

Chaos could be found in economics by the means of [[recurrence quantification analysis]]. In fact, Orlando et al.<ref>{{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> by the means of the so-called recurrence quantification correlation index were able to detect hidden changes in time series. Then, the same technique was employed to detect transitions from laminar (regular) to turbulent (chaotic) phases as well as differences between macroeconomic variables and highlight hidden features of economic dynamics.<ref>{{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 |url=https://www.sciencedirect.com/science/article/abs/pii/S0960077918300924 |language=en |issn=0960-0779}}</ref> Finally, chaos theory could help in modeling how an economy operates as well as in embedding shocks due to external events such as COVID-19.<ref>{{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 |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 |s2cid=235909725 }}</ref>

=== Finite predictability in weather and climate ===
Due to the sensitive dependence of solutions on initial conditions (SDIC), also known as the butterfly effect, chaotic systems like the Lorenz 1963 model imply a finite predictability horizon. This means that while accurate predictions are possible over a finite time period, they are not feasible over an infinite time span. Considering the nature of Lorenz's chaotic solutions, the committee led by Charney et al. in 1966<ref>{{Cite book |date=1966-01-01 |title=The Feasibility of a Global Observation and Analysis Experiment |url=http://dx.doi.org/10.17226/21272 |doi=10.17226/21272|isbn=978-0-309-35922-1 }}</ref> extrapolated a doubling time of five days from a general circulation model, suggesting a predictability limit of two weeks. This connection between the five-day doubling time and the two-week predictability limit was also recorded in a 1969 report by the Global Atmospheric Research Program (GARP).<ref>{{Cite journal |last=GARP |title=GARP topics |journal=Bull. Am. Meteorol. Soc. |volume=50 |pages=136–141}}</ref> To acknowledge the combined direct and indirect influences from the Mintz and Arakawa model and Lorenz's models, as well as the leadership of Charney et al., Shen et al.<ref>{{Cite journal |last1=Shen |first1=Bo-Wen |last2=Pielke |first2=Roger A. |last3=Zeng |first3=Xubin |last4=Zeng |first4=Xiping |date=2024-07-16 |title=Exploring the Origin of the Two-Week Predictability Limit: A Revisit of Lorenz's Predictability Studies in the 1960s |journal=Atmosphere |language=en |volume=15 |issue=7 |pages=837 |doi=10.3390/atmos15070837 |doi-access=free |bibcode=2024Atmos..15..837S |issn=2073-4433}}</ref> refer to the two-week predictability limit as the "Predictability Limit Hypothesis," drawing an analogy to Moore's Law.

=== AI-extended modeling framework ===
In AI-driven large language models, responses can exhibit sensitivities to factors like alterations in formatting and variations in prompts. These sensitivities are akin to butterfly effects.<ref>{{Cite arXiv|last1=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> Although classifying AI-powered large language models as classical deterministic chaotic systems poses challenges, chaos-inspired approaches and techniques (such as ensemble modeling) may be employed to extract reliable information from these expansive language models (see also "[[Butterfly effect in popular culture#AI-powered Large Language Models|Butterfly Effect in Popular Culture]]").

===Other areas===
In chemistry, predicting gas solubility is essential to manufacturing [[polymers]], but models using [[particle swarm optimization]] (PSO) tend to converge to the wrong points. An improved version of PSO has been created by introducing chaos, which keeps the simulations from getting stuck.<ref>{{cite journal|last=Li|first=Mengshan|author2=Xingyuan Huanga|author3=Hesheng Liua|author4=Bingxiang Liub|author5=Yan Wub|author6=Aihua Xiongc|author7=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> In [[celestial mechanics]], especially when observing asteroids, applying chaos theory leads to better predictions about when these objects will approach Earth and other planets.<ref>{{cite journal|last=Morbidelli|first=A.|title=Chaotic diffusion in celestial mechanics|journal=Regular & Chaotic Dynamics |year=2001|volume=6|issue=4|pages=339–353|doi=10.1070/rd2001v006n04abeh000182}}</ref> Four of the five [[moons of Pluto]] rotate chaotically. In [[quantum physics]] and [[electrical engineering]], the study of large arrays of [[Josephson junctions]] benefitted greatly from chaos theory.<ref>Steven Strogatz, ''Sync: The Emerging Science of Spontaneous Order'', Hyperion, 2003</ref> Closer to home, coal mines have always been dangerous places where frequent natural gas leaks cause many deaths. Until recently, there was no reliable way to predict when they would occur. But these gas leaks have chaotic tendencies that, when properly modeled, can be predicted fairly accurately.<ref>{{cite journal|last=Dingqi|first=Li|author2=Yuanping Chenga|author3=Lei Wanga|author4=Haifeng Wanga|author5=Liang Wanga|author6=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>

Chaos theory can be applied outside of the natural sciences, but historically nearly all such studies have suffered from lack of reproducibility; poor external validity; and/or inattention to cross-validation, resulting in poor predictive accuracy (if out-of-sample prediction has even been attempted). Glass<ref>{{cite book | last1 = Glass | first1 = L |editor1-first=C |editor1-last= Grebogi |editor2-first=J. A. | editor2-last=Yorke |title= The impact of chaos on science and society|publisher= United Nations University Press |year=1997 |chapter= Dynamical disease: The impact of nonlinear dynamics and chaos on cardiology and medicine }}</ref> and Mandell and Selz<ref>{{cite book | last1 = Mandell |first1= A. J. | last2 = Selz |first2= K. A. |editor1-first=C |editor1-last= Grebogi |editor2-first=J. A. | editor2-last=Yorke |title= The impact of chaos on science and society|publisher= United Nations University Press |year=1997 |chapter= Is the EEG a strange attractor? }}</ref> have found that no EEG study has as yet indicated the presence of strange attractors or other signs of chaotic behavior.

Redington and Reidbord (1992) attempted to demonstrate that the human heart could display chaotic traits. They monitored the changes in between-heartbeat intervals for a single psychotherapy patient as she moved through periods of varying emotional intensity during a therapy session. Results were admittedly inconclusive. Not only were there ambiguities in the various plots the authors produced to purportedly show evidence of chaotic dynamics (spectral analysis, phase trajectory, and autocorrelation plots), but also when they attempted to compute a Lyapunov exponent as more definitive confirmation of chaotic behavior, the authors found they could not reliably do so.<ref>{{cite journal|last1=Redington|first1=D. J.|last2=Reidbord|first2=S. P.|s2cid=214722|title=Chaotic dynamics in autonomic nervous system activity of a patient during a psychotherapy session|journal=Biological Psychiatry|date=1992|volume=31|issue=10|pages=993–1007|pmid=1511082|doi=10.1016/0006-3223(92)90093-F}}</ref>

In their 1995 paper, Metcalf and Allen<ref>{{cite book | last1 = Metcalf |first1= B. R. | last2 = Allen |first2= J. D. |editor1-first=F. D. |editor1-last= Abraham |editor2-first=A. R. | editor2-last=Gilgen |title= Chaos theory in psychology |publisher= Greenwood Press |year=1995 |chapter= In search of chaos in schedule-induced polydipsia }}</ref> maintained that they uncovered in animal behavior a pattern of period doubling leading to chaos. The authors examined a well-known response called schedule-induced polydipsia, by which an animal deprived of food for certain lengths of time will drink unusual amounts of water when the food is at last presented. The control parameter (r) operating here was the length of the interval between feedings, once resumed. The authors were careful to test a large number of animals and to include many replications, and they designed their experiment so as to rule out the likelihood that changes in response patterns were caused by different starting places for r.

Time series and first delay plots provide the best support for the claims made, showing a fairly clear march from periodicity to irregularity as the feeding times were increased. The various phase trajectory plots and spectral analyses, on the other hand, do not match up well enough with the other graphs or with the overall theory to lead inexorably to a chaotic diagnosis. For example, the phase trajectories do not show a definite progression towards greater and greater complexity (and away from periodicity); the process seems quite muddied. Also, where Metcalf and Allen saw periods of two and six in their spectral plots, there is room for alternative interpretations. All of this ambiguity necessitate some serpentine, post-hoc explanation to show that results fit a chaotic model.

By adapting a model of career counseling to include a chaotic interpretation of the relationship between employees and the job market, Amundson and Bright found that better suggestions can be made to people struggling with career decisions.<ref>{{cite journal|last=Pryor|first=Robert G. L.|author2=Norman E. Amundson|author3=Jim E. H. Bright|title=Probabilities and Possibilities: The Strategic Counseling Implications of the Chaos Theory of Careers|author3-link=Jim Bright (psychologist)|journal=The Career Development Quarterly|date=June 2008|volume=56|issue=4|pages=309–318|doi=10.1002/j.2161-0045.2008.tb00096.x}}</ref> Modern organizations are increasingly seen as open [[complex adaptive system]]s with fundamental natural nonlinear structures, subject to internal and external forces that may contribute chaos. For instance, [[team building]] and [[group development]] is increasingly being researched as an inherently unpredictable system, as the uncertainty of different individuals meeting for the first time makes the trajectory of the team unknowable.<ref>{{Cite journal|last1=Thompson|first1=Jamie|last2=Johnstone|first2=James|last3=Banks|first3=Curt|date=2018|title=An examination of initiation rituals in a UK sporting institution and the impact on group development|journal=European Sport Management Quarterly|volume=18|issue=5|pages=544–562|doi=10.1080/16184742.2018.1439984|s2cid=149352680}}</ref>

[[File:BML N=200 P=32.png|400px|right|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.]]

Traffic forecasting may benefit from applications of chaos theory. Better predictions of when a congestion will occur would allow measures to be taken to disperse it before it would have occurred. Combining chaos theory principles with a few other methods has led to a more accurate short-term prediction model (see the plot of the [[BML traffic model]] at right).<ref>{{cite journal|last=Wang|first=Jin|author2=Qixin Shi|title=Short-term traffic speed forecasting hybrid model based on Chaos–Wavelet Analysis-Support Vector Machine theory|journal=Transportation Research Part C: Emerging Technologies|date=February 2013|volume=27|pages=219–232|doi=10.1016/j.trc.2012.08.004|bibcode=2013TRPC...27..219W }}</ref>

Chaos theory has been applied to environmental [[water cycle]] data (also [[hydrological]] data), such as rainfall and streamflow.<ref>{{Cite web|url=http://pasternack.ucdavis.edu/research/projects/chaos-hydrology/|title=Dr. Gregory B. Pasternack – Watershed Hydrology, Geomorphology, and Ecohydraulics :: Chaos in Hydrology|website=pasternack.ucdavis.edu|language=en|access-date=2017-06-12}}</ref> These studies have yielded controversial results, because the methods for detecting a chaotic signature are often relatively subjective. Early studies tended to "succeed" in finding chaos, whereas subsequent studies and meta-analyses called those studies into question and provided explanations for why these datasets are not likely to have low-dimension chaotic dynamics.<ref>{{Cite journal|last=Pasternack|first=Gregory B.|date=1999-11-01|title=Does the river run wild? Assessing chaos in hydrological systems|journal=Advances in Water Resources|volume=23|issue=3|pages=253–260|doi=10.1016/s0309-1708(99)00008-1|bibcode = 1999AdWR...23..253P }}</ref>