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混沌理论 (查看源代码)

2020年5月18日 (一) 23:57的版本

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更便宜、更强大的计算机的出现拓宽了混沌理论的适用性。目前，混沌理论仍然是一个活跃的研究领域，<ref>{{cite journal | last1 = Motter | first1 = A. E. | last2 = Campbell | first2 = D. K. | year = 2013 | title = Chaos at fifty | url = http://www.physicstoday.org/resource/1/phtoad/v66/i5/p27_s1?bypassSSO=1 | journal = Phys. Today | volume = 66 | issue = 5| pages = 27–33 | doi=10.1063/pt.3.1977|arxiv = 1306.5777 |bibcode = 2013PhT....66e..27M }}</ref>涉及许多不同的学科(数学、拓扑学、物理学、<ref>{{cite journal|last1=Hubler|first1=A.|last2=Foster|first2=G.|last3=Phelps|first3=K.|title=Managing chaos: Thinking out of the box|journal=Complexity|volume=12|issue=3|pages=10|date=2007|doi=10.1002/cplx.20159|bibcode = 2007Cmplx..12c..10H }}</ref>社会系统、<ref>{{Cite book|title=Chaos Theory in the Social Sciences: Foundations and Applications|date=1996|publisher=University of Michigan Press|isbn=9780472106387|editor-last=Kiel|editor-first=L.|location=Ann Arbor, MI|language=en|doi=10.3998/mpub.14623|editor-last2=Elliott|editor-first2=Euel|hdl = 2027/fulcrum.d504rm03n}}</ref>人口模型、生物学、气象学、天体物理学、信息理论、计算神经科学等等)。).

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==~~Applications==~~

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==申请==

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[[File:Textile cone.JPG|thumb|left| 一个圆锥形纺织外壳，外观与第30条规则相似，一个行为混乱的细胞自动机<ref>{{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 |work=www.maths.nottingham.ac.uk |publisher=[[University of Nottingham]] |accessdate=2013-04-10}}</ref>]]

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==~~Applications==~~

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申请

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[[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 |title=The Geometry and Pigmentation of Seashells |author=Stephen Coombes |date=February 2009 |work=www.maths.nottingham.ac.uk |publisher=[[University of Nottingham]] |accessdate=2013-04-10}}</ref>]]

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~~A [[conus textile shell, similar in appearance to Rule 30, a cellular automaton with chaotic behaviour.]]~~

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~~一个[圆锥形纺织外壳，外观与第30条规则相似，一个行为混乱的细胞自动机]~~

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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]], [[microbiology]], [[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>[http://www.dspdesignline.com/218101444;jsessionid=Y0BSVTQJJTBACQSNDLOSKH0CJUNN2JVN?pgno=1 Applying Chaos Theory to Embedded Applications]</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 | 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 |url=https://www.semanticscholar.org/paper/7398a90d0d7d5b7354f6781aa03e8618e0f5e124 }}</ref> [[algorithmic trading]],<ref>{{cite book|last=Williams|first=Bill Williams, Justine|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 }}</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]], [[population dynamics]],<ref>{{cite journal |author1=Dilão, R. |author2=Domingos, T. | year = 2001 | title = Periodic and Quasi-Periodic Behavior in Resource Dependent Age Structured Population Models | journal = Bulletin of Mathematical Biology | volume = 63 |pages = 207–230|doi=10.1006/bulm.2000.0213 | issue = 2 | pmid = 11276524|url=https://www.semanticscholar.org/paper/f61a74e7be3df112bf5f8d55277c87ca68c58c31 }}</ref> [[psychology]],<ref name="SafonovTomer2002"/> 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.

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=== 密码学 ===

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=== ~~Cryptography ===~~

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~~=== Cryptography~~ ===

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~~密码学~~

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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|last=Akhavan|first=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|last=Behnia|first=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|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|url=https://www.semanticscholar.org/paper/fb51bb631b4764ed3969836ce7876e8099b29307}}</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|last=Akhavan|first=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|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 |url=https://www.semanticscholar.org/paper/0306ddee70a3f10b4eb24a42ab43ed09bedf63cd}}</ref><ref>{{Cite journal|last=Liu|first=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 }}</ref>

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=== ~~Robotics~~ ===

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===机器人学===

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~~=== Robotics ===~~

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~~=== Robotics ===~~

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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>

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===~~Biology===~~

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===生物学===

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~~===Biology~~===

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~~生物学~~

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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>

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===~~Other areas===~~

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===其他范畴===

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~~===Other areas~~===

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~~其他范畴~~

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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}}</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}}</ref>

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混沌理论已经应用于环境水循环数据(又称水文数据) ，如降雨和径流。这些研究产生了有争议的结果，因为检测混沌特征的方法往往是相对主观的。早期的研究倾向于“成功地”发现混沌，而后来的研究和元分析则对这些研究提出质疑，并解释了为什么这些数据集不可能具有低维混沌动力学。

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~~==See also==~~

==参见==

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混沌系统的例子

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===混沌系统的例子===

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* [[Tilt A Whirl]]

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其他相关话题

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===其他相关话题===

* [[Amplitude death]]

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人们

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===人们===

* [[Ralph Abraham (mathematician)|Ralph Abraham]]

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==~~References~~==

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==参考文献==

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~~==References==~~

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~~参考资料~~

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==进一步阅读==

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===文章===

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~~{{Reflist|30em}}~~

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==~~Further reading==~~

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~~==Further reading==~~

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进一步阅读

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==~~=Articles===~~

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===~~Articles~~===

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文章

* {{cite journal |first=A.N. |last=Sharkovskii |authorlink=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 }}

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* {{cite journal |authorlink1=Tien-Yien Li |last1=Li |first1=T.Y. |authorlink2=James A. Yorke |last2=Yorke |first2=J.A. |title=Period Three Implies Chaos |journal=[[American Mathematical Monthly]] |volume=82 |pages=985–92 |year=1975 |bibcode=1975AmMM...82..985L |doi=10.2307/2318254 |issue=10 |url=http://pb.math.univ.gda.pl/chaos/pdf/li-yorke.pdf |jstor=2318254 |citeseerx=10.1.1.329.5038 }}

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*{{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}}

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* {{Cite journal |date = December 1986|title=Chaos |journal=[[Scientific American]] |volume=255 |issue=6 |pages=38–49 (bibliography p.136) |bibcode = 1986SciAm.255d..38T |last2 = Tucker |last3 = Morrison |author1 = Crutchfield|author4 = J.D. Farmer|author5 = Packard|author6=N.H. |author7=Shaw |author8=R.S |authorlink1=James P. Crutchfield |authorlink5=Norman Packard |authorlink7=Robert Shaw (physicist) |author4-link=J. Doyne Farmer |doi=10.1038/scientificamerican1286-46 }} [https://web.archive.org/web/20100612040821/http://cse.ucdavis.edu/~chaos/courses/ncaso/Readings/Chaos_SciAm1986/Chaos_SciAm1986.html Online version] (Note: the volume and page citation cited for the online text differ from that cited here. The citation here is from a photocopy, which is consistent with other citations found online that don't provide article views. The online content is identical to the hardcopy text. Citation variations are related to country of publication).

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* {{cite journal |author=Kolyada, S.F. |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 |url=https://www.semanticscholar.org/paper/5834c3cdf41d979850c355b93999e785da446562 }}

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* {{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 }}

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* {{cite journal|first1=C. |last1=Strelioff |first2=A. |last2=Hübler |title=Medium-Term Prediction of Chaos |journal=Phys. Rev. Lett. |volume=96 |issue=4 |id=044101 |year=2006 |doi=10.1103/PhysRevLett.96.044101 |url=http://www.ccsr.illinois.edu/web/Techreports/2005-08/CCSR-05-4.pdf |pmid=16486826 |page=044101 |bibcode=2006PhRvL..96d4101S |url-status=dead |archiveurl=https://web.archive.org/web/20130426201635/http://www.ccsr.illinois.edu/web/Techreports/2005-08/CCSR-05-4.pdf |archivedate=2013-04-26 }}

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* {{cite journal |author1=Hübler, A. |author2=Foster, G. |author3=Phelps, K. |title=Managing Chaos: Thinking out of the Box |journal=Complexity |volume=12 |pages=10–13 |year=2007 |url=http://server17.how-why.com/blog/ManagingChaos.pdf |doi=10.1002/cplx.20159 |issue=3|bibcode = 2007Cmplx..12c..10H }}

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* {{cite journal | last1 = Motter | first1 = Adilson E. | last2 = Campbell | first2 = David K. | year = 2013 | title = Chaos at 50 | journal = Physics Today | volume = 66| issue = 5| page = 27| doi = 10.1063/PT.3.1977 |arxiv = 1306.5777 |bibcode = 2013PhT....66e..27M }}

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===教科书===

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===~~Textbooks===~~

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~~===Textbooks~~===

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~~教科书~~

* {{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 |ref=harv |url=https://books.google.com/books?id=48YHnbHGZAgC}}

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* {{cite book| author=Baker, G. L.| title=Chaos, Scattering and Statistical Mechanics| publisher=Cambridge University Press| year=1996| isbn=978-0-521-39511-3}}

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* {{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}}

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* {{cite book |editor1-last=Bunde |editor2-first=Shlomo | editor2-last=Havlin | editor2-link=Shlomo Havlin |title=Fractals and Disordered Systems |publisher=Springer |year=1996 |isbn=978-3642848704}} and {{cite book |editor1-last=Bunde |editor2-first=Shlomo | editor2-last=Havlin | editor2-link=Shlomo Havlin |title=Fractals in Science |publisher=Springer |year=1994 |isbn=978-3-540-56220-7}}

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* {{cite book| author=Collet, Pierre, and [[Jean-Pierre Eckmann|Eckmann, Jean-Pierre]]| title=Iterated Maps on the Interval as Dynamical Systems | publisher=Birkhauser | year=1980 |isbn=978-0-8176-4926-5}}

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* {{cite book |last=Devaney |first=Robert L.|authorlink= Robert L. Devaney|ref=harv |title=An Introduction to Chaotic Dynamical Systems |edition=2nd |publisher=Westview Press |year=2003 |isbn=978-0-8133-4085-2 |url=https://books.google.com/books?id=CjAnY99LwTgC}}

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* {{cite book |last=Robinson |first=Clark |ref=harv |title=Dynamical systems: Stability, symbolic dynamics, and chaos |publisher=CRC Press |year=1995 |isbn=0-8493-8493-1}}

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* {{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=http://chaos.coa.edu/index.html}}

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* {{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}}

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* {{cite book |author=Guckenheimer, John| authorlink=John Guckenheimer|author2= Holmes, Philip |authorlink2=Philip Holmes|title=Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields |publisher=Springer-Verlag |year=1983 |isbn=978-0-387-90819-9}}

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* {{cite book| author=Gulick, Denny| title=Encounters with Chaos| publisher=McGraw-Hill| year=1992| isbn=978-0-07-025203-5}}

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* {{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}}

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* {{cite book |author=Hoover, William Graham|authorlink=William G Hoover|title=Time Reversibility, Computer Simulation, and Chaos |publisher=World Scientific |origyear=1999|year=2001 |isbn=978-981-02-4073-8 |url=https://books.google.com/books?id=24kEKsdl0psC}}

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* {{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}}

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* {{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}}

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* {{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}}

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* {{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}}

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* {{cite book| author=Strogatz, Steven| authorlink=Steven Strogatz| title=Nonlinear Dynamics and Chaos| publisher=Perseus Publishing| year=2000| isbn=978-0-7382-0453-6| url=https://archive.org/details/nonlineardynamic00stro}}

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* {{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 |ref=harv |url=https://books.google.com/books?id=SEDjdjPZ158C}}

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* {{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}}

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* {{cite book| last = Teschl| given = Gerald|authorlink=Gerald Teschl| title = Ordinary Differential Equations and Dynamical Systems| publisher=[[American Mathematical Society]]| place = [[Providence, Rhode Island|Providence]]| year = 2012| isbn= 978-0-8218-8328-0| url = http://www.mat.univie.ac.at/~gerald/ftp/book-ode/}}

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* {{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}}

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* {{cite book |author1-link=Nicholas Tufillaro |last1=Tufillaro |last2=Reilly |title=An experimental approach to nonlinear dynamics and chaos |journal=American Journal of Physics |volume=61 |issue=10 |pages=[https://archive.org/details/unset0000unse_q2b7/page/958 958] |publisher=Addison-Wesley |year=1992 |isbn=978-0-201-55441-0 |bibcode=1993AmJPh..61..958T |doi=10.1119/1.17380 |url=https://archive.org/details/unset0000unse_q2b7/page/958 }}

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* {{cite book | last=Wiggins|first=Stephen | title= Introduction to Applied Dynamical Systems and Chaos | publisher= Springer | year= 2003 | isbn= 978-0-387-00177-7 }}

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* {{cite book| author=Zaslavsky, George M.| title=Hamiltonian Chaos and Fractional Dynamics| publisher=Oxford University Press| year=2005| isbn=978-0-19-852604-9}}

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+

===半技术和通俗作品===

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===~~Semitechnical and popular works===~~

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~~===Semitechnical and popular works~~===

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~~半技术和通俗作品~~

* [[Christophe Letellier]], ''Chaos in Nature'', World Scientific Publishing Company, 2012, {{ISBN|978-981-4374-42-2}}.

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* {{cite book |editor1-first=Ralph H. |editor1-last=Abraham |editor2-first=Yoshisuke |editor2-last=Ueda |title=The Chaos Avant-Garde: Memoirs of the Early Days of Chaos Theory |journal= |volume=39 |url=https://books.google.com/books?id=0E667XpBq1UC |year=2000 |publisher=World Scientific |isbn=978-981-238-647-2 |ref=harv|bibcode=2000cagm.book.....A |last1=Abraham |first1=Ralph |display-authors=et al |doi=10.1142/4510 |series=World Scientific Series on Nonlinear Science Series A }}

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* {{cite book |authorlink=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}}

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* {{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}}

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* [[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.

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* John Briggs and David Peat, ''Seven Life Lessons of Chaos: Spiritual Wisdom from the Science of Change'', Harper Perennial 2000, 224 pp.

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* {{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 }}

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* [[Predrag Cvitanović]], ''Universality in Chaos'', Adam Hilger 1989, 648 pp.

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* [[Leon Glass]] and Michael C. Mackey, ''From Clocks to Chaos: The Rhythms of Life,'' Princeton University Press 1988, 272 pp.

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* [[James Gleick]], ''[[Chaos: Making a New Science]]'', New York: Penguin, 1988. 368 pp.

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* {{cite book |author=John Gribbin |title=Deep Simplicity |series=Penguin Press Science |publisher=Penguin Books}}

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* L Douglas Kiel, Euel W Elliott (ed.), ''Chaos Theory in the Social Sciences: Foundations and Applications'', University of Michigan Press, 1997, 360 pp.

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* Arvind Kumar, ''Chaos, Fractals and Self-Organisation; New Perspectives on Complexity in Nature '', National Book Trust, 2003.

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* Hans Lauwerier, ''Fractals'', Princeton University Press, 1991.

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* [[Edward Lorenz]], ''The Essence of Chaos'', University of Washington Press, 1996.

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* {{cite book|doi=10.1142/9781860949548|title=The Unity of Nature - Wholeness and Disintegration in Ecology and Science|year=2002|last1=Marshall|first1=Alan|isbn=9781860949548}}

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* {{cite book|doi=10.1142/9781860949548|title=The Unity of Nature - Wholeness and Disintegration in Ecology and Science|year=2002|last1=Marshall|first1=Alan|isbn=9781860949548}}

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* David Peak and Michael Frame, ''Chaos Under Control: The Art and Science of Complexity'', Freeman, 1994.

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* [[Heinz-Otto Peitgen]] and [[Dietmar Saupe]] (Eds.), ''The Science of Fractal Images'', Springer 1988, 312 pp.

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* [[Clifford A. Pickover]], ''Computers, Pattern, Chaos, and Beauty: Graphics from an Unseen World '', St Martins Pr 1991.

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* [[Clifford A. Pickover]], ''Chaos in Wonderland: Visual Adventures in a Fractal World'', St Martins Pr 1994.

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* [[Ilya Prigogine]] and [[Isabelle Stengers]], ''Order Out of Chaos'', Bantam 1984.

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* {{cite book|doi=10.1007/978-3-642-61717-1|title=The Beauty of Fractals|year=1986|last1=Peitgen|first1=Heinz-Otto|last2=Richter|first2=Peter H.|isbn=978-3-642-61719-5}}

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* [[David Ruelle]], ''Chance and Chaos'', Princeton University Press 1993.

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* [[Ivars Peterson]], ''Newton's Clock: Chaos in the Solar System'', Freeman, 1993.

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* {{cite book |author1=Ian Roulstone |author2=John Norbury |title=Invisible in the Storm: the role of mathematics in understanding weather |url=https://books.google.com/?id=qnMrFEHMrWwC|year=2013 |publisher=Princeton University Press|isbn=978-0691152721 }}

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* {{cite book|doi=10.1017/CBO9780511608773|title=Chaotic Evolution and Strange Attractors|year=1989|last1=Ruelle|first1=D.|isbn=9780521362726}}

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* Manfred Schroeder, ''Fractals, Chaos, and Power Laws'', Freeman, 1991.

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* {{cite book|doi=10.1017/CBO9780511554544|title=Explaining Chaos|year=1998|last1=Smith|first1=Peter|isbn=9780511554544}}

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* [[Ian Stewart (mathematician)|Ian Stewart]], ''Does God Play Dice?: The Mathematics of Chaos '', Blackwell Publishers, 1990.

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* [[Steven Strogatz]], ''Sync: The emerging science of spontaneous order'', Hyperion, 2003.

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* Yoshisuke Ueda, ''The Road To Chaos'', Aerial Pr, 1993.

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* M. Mitchell Waldrop, ''Complexity : The Emerging Science at the Edge of Order and Chaos'', Simon & Schuster, 1992.

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* Antonio Sawaya, ''Financial Time Series Analysis : Chaos and Neurodynamics Approach'', Lambert, 2012.

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