7,129
个编辑
更改
跳到导航
跳到搜索
第254行：
第254行： − 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.+ − − 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, engineering, finance, algorithmic trading, meteorology, philosophy, anthropology, politics, population dynamics, psychology, and robotics. A few categories are listed below with examples, but this is by no means a comprehensive list as new applications are appearing. − − 虽然混沌理论诞生于观测天气模式，但它已经适用于各种其他情况。今天受益于混沌理论的领域包括地质学、数学、微生物学、生物学、计算机科学、经济学、工程学、金融学、算法贸易、气象学、哲学、人类学、政治学、族群动态、心理学和机器人学。下面列出了一些类别和示例，但这绝不是一个全面的清单，因为新的应用程序正在出现。 − 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>+ − − 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 primitives. These algorithms include image encryption algorithms, hash functions, secure pseudo-random number generators, stream ciphers, watermarking and steganography. 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. 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. Another type of computing, DNA computing, when paired with chaos theory, offers a way to encrypt images and other information. Many of the DNA-Chaos cryptographic algorithms are proven to be either not secure, or the technique applied is suggested to be not efficient. − − 混沌理论在密码学中已应用多年。在过去的几十年中，混沌和非线性动力学被用于数百种密码原语的设计。这些算法包括图像加密算法，散列函数，安全伪随机数生成器，流密码，水印和隐写术。这些算法大多基于单模态混沌映射，其中很大一部分以控制参数和混沌映射的初始条件为关键。从更广泛的角度来看，混沌映射和密码系统之间的相似性是设计基于混沌的密码算法的主要不失一般性。另一种类型的计算，DNA 计算，与混沌理论相结合，提供了一种加密图像和其他信息的方法。许多 dna- 混沌密码算法被证明是不安全的，或者应用的技术是不高效的。 − − − − 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>+ − − 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 model. − − 机器人学是最近受益于混沌理论的另一个领域。混沌理论已经被用来建立一个预测模型，而不是机器人通过反复试验来改进与环境的相互作用。 − − 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}}</ref> − − Chaotic dynamics have been exhibited by passive walking biped robots. − − 被动行走的两足机器人展示了混沌动力学。 − − − − 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>+ − − For over a hundred years, biologists have been keeping track of populations of different species with population models. Most models are continuous, but recently scientists have been able to implement chaotic models in certain populations. For example, a study on models of Canadian lynx showed there was chaotic behavior in the population growth. 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. 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 fetal hypoxia can be obtained through chaotic modeling. − − 一百多年来，生物学家一直在用种群模型跟踪不同物种的种群。大多数模型是连续的，但是最近科学家已经能够在某些种群中实现混沌模型。例如，一项关于加拿大猞猁模型的研究表明，其种群增长存在混乱行为。混乱也可以发现在生态系统，如水文学。虽然水文学的混沌模型有其自身的缺点，但是从混沌理论的角度来看数据还有很多值得学习的地方。另一个生物学应用是发现在心血管造影术。胎儿监护是在尽可能无创的情况下获得准确信息的人海万花筒(电影)。通过混沌建模可以获得较好的胎儿缺氧预警信号模型。 − − − − 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>+ − − 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. In celestial mechanics, especially when observing asteroids, applying chaos theory leads to better predictions about when these objects will approach Earth and other planets. 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. 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. − − 在化学方面，预测气体的溶解度对于聚合物的制造是至关重要的，但是使用微粒群算法(PSO)的模型往往会收敛到错误的粒子群优化。通过引入混沌，改进了粒子群优化算法，避免了仿真陷入僵局。在21天体力学，特别是在观测小行星时，应用混沌理论可以更好地预测这些天体何时会接近地球和其他行星。冥王星的五个卫星中有四个以混乱的方式旋转。在量子物理和电子工程中，混沌理论对约瑟夫森结大阵列的研究有很大的帮助。离家更近的地方，煤矿一直是危险的地方，频繁的天然气泄漏导致许多人死亡。直到最近，还没有可靠的方法来预测它们何时会发生。但是这些天然气泄漏有混乱的趋势，如果正确地建模，可以相当准确地预测。 − − − − − − 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. − − 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 and Mandell and Selz have found that no EEG study has as yet indicated the presence of strange attractors or other signs of chaotic behavior. − − 混沌理论可以应用于自然科学之外的领域，但是从历史上看，几乎所有这类研究都存在缺乏可重复性、外部效度不足和 / 或对交叉验证缺乏关注等问题，从而导致预测准确性差(如果尝试过样本外预测)。格拉斯、曼德尔和塞尔兹发现，迄今为止，没有任何脑电图研究表明存在奇怪吸引子或其他混沌行为的迹象。 − − − − − − Researchers have continued to apply chaos theory to psychology. For example, in modeling group behavior in which heterogeneous members may behave as if sharing to different degrees what in [[Wilfred Bion]]'s theory is a basic assumption, researchers have found that the group dynamic is the result of the individual dynamics of the members: each individual reproduces the group dynamics in a different scale, and the chaotic behavior of the group is reflected in each member.<ref name="Dal FornoMerlone2013">{{cite journal | last1 = Dal Forno | first1 = Arianna | last2 = Merlone | first2 = Ugo | title = Nonlinear dynamics in work groups with Bion's basic assumptions | journal = Nonlinear Dynamics, Psychology, and Life Sciences | volume = 17| issue=2 | year = 2013 | pages = 295–315 | issn = 1090-0578 }}</ref> − − Researchers have continued to apply chaos theory to psychology. For example, in modeling group behavior in which heterogeneous members may behave as if sharing to different degrees what in Wilfred Bion's theory is a basic assumption, researchers have found that the group dynamic is the result of the individual dynamics of the members: each individual reproduces the group dynamics in a different scale, and the chaotic behavior of the group is reflected in each member. − − 研究人员继续将混沌理论应用于心理学。例如，在模拟群体行为中，异质成员可能表现为不同程度的共享，威尔弗雷德 · 比昂的理论是一个基本假设，研究人员发现，群体动态是成员个人动态的结果: 每个个人在不同的尺度上再现群体动态，群体的混沌行为反映在每个成员。 − − − − − − 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|last=Redington|first=D. J.|last2=Reidbord|first2=S. P.|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|url=https://www.semanticscholar.org/paper/3873365d697901d3422df5ab8930c6221f4f4c05}}</ref> − − 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. − − 雷丁顿和 Reidbord (1992)试图证明人类的心脏可以表现出混乱的特征。他们监测了一位心理治疗患者在治疗过程中经历不同情绪强度时的心跳间隔时间的变化。结果无可否认是不确定的。不仅在作者制作的各种图表中存在模糊性，据称显示了混沌动力学的证据(频谱分析、相轨迹和自相关图) ，而且当他们试图计算李亚普诺夫指数作为更确定的混沌行为的确认时，作者发现他们不能可靠地这样做。 − − + − 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. − In their 1995 paper, Metcalf and Allen 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.+ − 在他们1995年的论文中，梅特卡夫和艾伦坚持认为他们在动物行为中发现了一种周期加倍导致混乱的模式。作者们研究了一种众所周知的反应，称为时间表诱发的多饮，通过这种方法，一只动物在一定时间内缺乏食物，当食物最终呈现时，它会喝下不寻常数量的水。这里的控制参数(r)是恢复喂食间隔的长度。作者小心翼翼地测试了大量的动物并进行了许多复制实验，他们设计实验的目的是为了排除反应模式的改变是由不同的起始位置引起的可能性。 + + − − 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. − − 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, Aniundson 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. Aniundson|author3=Jim E. H. Bright|title=Probabilities and Possibilities: The Strategic Counseling Implications of the Chaos Theory of Careers|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|last=Thompson|first=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}}</ref> − − By adapting a model of career counseling to include a chaotic interpretation of the relationship between employees and the job market, Aniundson and Bright found that better suggestions can be made to people struggling with career decisions. Modern organizations are increasingly seen as open complex adaptive systems 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. − − 通过调整职业咨询的模型，包括对雇员和就业市场之间关系的混乱解释，安尼森和布莱特发现，对于那些在职业决策中挣扎的人们，可以提出更好的建议。现代组织越来越多地被视为具有基本的自然非线性结构的开放复杂适应系统，受到可能导致混乱的内部和外部力量的影响。例如，团队建设和团队发展作为一个内在的不可预测的系统正在越来越多地被研究，因为不同的个体第一次见面的不确定性使得团队的轨迹不可知。 − − − − − − Some say the chaos metaphor—used in verbal theories—grounded on mathematical models and psychological aspects of human behavior − − Some say the chaos metaphor—used in verbal theories—grounded on mathematical models and psychological aspects of human behavior − − 有人说混沌隐喻是以数学模型和人类行为的心理方面为基础的语言理论 − − provides helpful insights to describing the complexity of small work groups, that go beyond the metaphor itself.<ref>{{cite book | last1 = Dal Forno | first1 = Arianna | last2 = Merlone | first2 = Ugo |editor1-first=Gian Italo |editor1-last=Bischi |editor2-first=Carl | editor2-last=Chiarella |editor3-first=Irina | editor3-last=Shusko |title=Global Analysis of Dynamic Models in Economics and Finance |publisher=Springer-Verlag |year=2013 |pages=185–204 |chapter= Chaotic Dynamics in Organization Theory |isbn= 978-3-642-29503-4}}</ref> − − provides helpful insights to describing the complexity of small work groups, that go beyond the metaphor itself. − − 为描述小型工作组的复杂性提供了有益的见解，超越了比喻本身。 − − − − − − [[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.]] − − 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. − − 红色的车和蓝色的车轮流行驶，红色的只向上行驶，蓝色的向右行驶。每一次，如果前面没有车，所有颜色相同的车都试图移动一步。在这里，模型有自我组织在一个有点几何图形模式，其中有一些交通堵塞和一些地区的汽车可以移动在最高速度。 − − − − − − 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|bibcode=2010ProCS...1.1119G|arxiv=1005.5384}}</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 |authorlink=Chris Brooks (academic)|title=Chaos in foreign exchange markets: a sceptical view |journal=Computational Economics|year=1998 |volume=11 |issue=3 |pages=265–281 |issn=1572-9974 |doi=10.1023/A:1008650024944|url=http://centaur.reading.ac.uk/35988/1/35988.pdf }}</ref> − − 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. 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. − − 通过混沌理论的应用也可以改进经济模型，但是预测一个经济系统的健康状况以及什么因素对其影响最大是一个极其复杂的任务。经济和金融系统与古典自然科学中的系统有着根本的不同，因为前者本质上是随机的，因为它们来自于人们之间的相互作用，因此纯粹的确定性模型不可能提供准确的数据表示。检验经济学和金融学中混沌的实证文献呈现出非常复杂的结果，部分原因是混沌的具体检验与非线性关系的更一般检验之间的混淆。 − − − − − − Traffic forecasting may benefit from applications of chaos theory. Better predictions of when traffic 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}}</ref> − − Traffic forecasting may benefit from applications of chaos theory. Better predictions of when traffic 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). − − 应用混沌理论进行交通量预测具有重要意义。更好地预测交通将在何时发生将允许采取措施在它发生之前驱散它。将混沌理论原理与其他一些方法相结合，得到了一个更精确的短期预测模型(见右边 BML 流量模型的图)。 − + + + − Chaos theory has been applied to environmental [[water cycle]] data (aka 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> − Chaos theory has been applied to environmental water cycle data (aka hydrological data), such as rainfall and streamflow. 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.+ − 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。 +
→申请
[[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>]]
[[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>]]
虽然混沌理论诞生于观测天气模式，但它已经适用于各种其他情况。今天受益于混沌理论的领域包括地质学、数学、微生物学、生物学、计算机科学、经济学、<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>工程学、<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>金融学、<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>算法贸易、<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>气象学、哲学、人类学、<ref name=":0" /> 政治学、<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>族群动态、<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> 心理学<ref name="SafonovTomer2002"/>和机器人学。下面列出了一些类别和示例，但这绝不是一个全面的清单，因为新的应用程序正在出现。
=== 密码学 ===
=== 密码学 ===
混沌理论在密码学中已应用多年。在过去的几十年中，混沌和非线性动力学被用于数百种密码原语的设计。这些算法包括图像加密算法，散列函数，安全伪随机数生成器，流密码，水印和隐写术。<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> 这些算法大多基于单模态混沌映射，其中很大一部分以控制参数和混沌映射的初始条件为关键。<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>从更广泛的角度来看，混沌映射和密码系统之间的相似性是设计基于混沌的密码算法的主要不失一般性。<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>另一种类型的计算，DNA 计算，与混沌理论相结合，提供了一种加密图像和其他信息的方法。<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> 许多DNA-混沌密码算法被证明是不安全的，或者应用的技术是不高效的。<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>
===机器人学===
===机器人学===
机器人学是最近受益于混沌理论的另一个领域。混沌理论已经被用来建立一个预测模型，而不是机器人通过反复试验来改进与环境的相互作用。<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>被动行走的两足机器人展示了混沌动力学。<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}}</ref>
===生物学===
===生物学===
一百多年来，生物学家一直在用种群模型跟踪不同物种的种群。大多数模型是连续的，但是最近科学家已经能够在某些种群中实现混沌模型。<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>例如，一项关于加拿大猞猁模型的研究表明，其种群增长存在混乱行为。<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>混乱也可以发现在生态系统，如水文学。虽然水文学的混沌模型有其自身的缺点，但是从混沌理论的角度来看数据还有很多值得学习的地方。<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> 另一个生物学应用是发现在心血管造影术。胎儿监护是在尽可能无创的情况下获得准确信息的人海万花筒(电影)。通过混沌建模可以获得较好的胎儿缺氧预警信号模型。<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>
===其他范畴===
===其他范畴===
在化学方面，预测气体的溶解度对于聚合物的制造是至关重要的，但是使用微粒群算法(PSO)的模型往往会收敛到错误的粒子群优化。通过引入混沌，改进了粒子群优化算法，避免了仿真陷入僵局。<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>在21天体力学，特别是在观测小行星时，应用混沌理论可以更好地预测这些天体何时会接近地球和其他行星。<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>冥王星的五个卫星中有四个以混乱的方式旋转。在量子物理和电子工程中，混沌理论对约瑟夫森结大阵列的研究有很大的帮助。<ref>Steven Strogatz, ''Sync: The Emerging Science of Spontaneous Order, Hyperion, 2003</ref>离家更近的地方，煤矿一直是危险的地方，频繁的天然气泄漏导致许多人死亡。直到最近，还没有可靠的方法来预测它们何时会发生。但是这些天然气泄漏有混乱的趋势，如果正确地建模，可以相当准确地预测。<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>
混沌理论可以应用于自然科学之外的领域，但是从历史上看，几乎所有这类研究都存在缺乏可重复性、外部效度不足和 / 或对交叉验证缺乏关注等问题，从而导致预测准确性差(如果尝试过样本外预测)。格拉斯 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>、曼德尔 Mandell和塞尔兹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>发现，迄今为止，没有任何脑电图研究表明存在奇怪吸引子或其他混沌行为的迹象。
研究人员继续将混沌理论应用于心理学。例如，在模拟群体行为中，异质成员可能表现为不同程度的共享，威尔弗雷德·比昂 Wilfred Bion的理论是一个基本假设，研究人员发现，群体动态是成员个人动态的结果: 每个个人在不同的尺度上再现群体动态，群体的混沌行为反映在每个成员。<ref name="Dal FornoMerlone2013">{{cite journal | last1 = Dal Forno | first1 = Arianna | last2 = Merlone | first2 = Ugo | title = Nonlinear dynamics in work groups with Bion's basic assumptions | journal = Nonlinear Dynamics, Psychology, and Life Sciences | volume = 17| issue=2 | year = 2013 | pages = 295–315 | issn = 1090-0578 }}</ref>
雷丁顿 Redington和 Reidbord (1992)试图证明人类的心脏可以表现出混乱的特征。他们监测了一位心理治疗患者在治疗过程中经历不同情绪强度时的心跳间隔时间的变化。结果无可否认是不确定的。不仅在作者制作的各种图表中存在模糊性，据称显示了混沌动力学的证据(频谱分析、相轨迹和自相关图) ，而且当他们试图计算李亚普诺夫指数作为更确定的混沌行为的确认时，作者发现他们不能可靠地这样做。<ref>{{cite journal|last=Redington|first=D. J.|last2=Reidbord|first2=S. P.|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|url=https://www.semanticscholar.org/paper/3873365d697901d3422df5ab8930c6221f4f4c05}}</ref>
在他们1995年的论文中，梅特卡夫 Metcalf和艾伦 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>坚持认为他们在动物行为中发现了一种周期加倍导致混乱的模式。作者们研究了一种众所周知的反应，称为时间表诱发的多饮，通过这种方法，一只动物在一定时间内缺乏食物，当食物最终呈现时，它会喝下不寻常数量的水。这里的控制参数(r)是恢复喂食间隔的长度。作者小心翼翼地测试了大量的动物并进行了许多复制实验，他们设计实验的目的是为了排除反应模式的改变是由不同的起始位置引起的可能性。
时间序列和第一延迟图为所提出的要求提供了最好的支持，随着喂食时间的增加，从周期性到不规则性的发展变化相当明显。另一方面，各种相轨迹图和谱分析与其他图形或整体理论不匹配，不可避免地导致混沌诊断。例如，相轨迹并没有显示一个朝着越来越复杂的方向发展的确切过程(并且远离周期性) ; 这个过程看起来相当混乱。此外，梅特卡夫和艾伦在他们的光谱图中看到了两个和六个周期，这里也有其他解释的空间。所有这些模糊性都需要一些曲折的、事后的解释，以表明结果符合混沌模型。
时间序列和第一延迟图为所提出的要求提供了最好的支持，随着喂食时间的增加，从周期性到不规则性的发展变化相当明显。另一方面，各种相轨迹图和谱分析与其他图形或整体理论不匹配，不可避免地导致混沌诊断。例如，相轨迹并没有显示一个朝着越来越复杂的方向发展的确切过程(并且远离周期性) ; 这个过程看起来相当混乱。此外，梅特卡夫和艾伦在他们的光谱图中看到了两个和六个周期，这里也有其他解释的空间。所有这些模糊性都需要一些曲折的、事后的解释，以表明结果符合混沌模型。
通过调整职业咨询的模型，包括对雇员和就业市场之间关系的混乱解释，安尼森和布莱特发现，对于那些在职业决策中挣扎的人们，可以提出更好的建议。<ref>{{cite journal|last=Pryor|first=Robert G. L.|author2=Norman E. Aniundson|author3=Jim E. H. Bright|title=Probabilities and Possibilities: The Strategic Counseling Implications of the Chaos Theory of Careers|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>现代组织越来越多地被视为具有基本的自然非线性结构的开放复杂适应系统，受到可能导致混乱的内部和外部力量的影响。例如，团队建设和团队发展作为一个内在的不可预测的系统正在越来越多地被研究，因为不同的个体第一次见面的不确定性使得团队的轨迹不可知。<ref>{{Cite journal|last=Thompson|first=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}}</ref>
有人说混沌隐喻是以数学模型和人类行为的心理方面为基础的语言理论，为描述小型工作组的复杂性提供了有益的见解，超越了比喻本身。<ref>{{cite book | last1 = Dal Forno | first1 = Arianna | last2 = Merlone | first2 = Ugo |editor1-first=Gian Italo |editor1-last=Bischi |editor2-first=Carl | editor2-last=Chiarella |editor3-first=Irina | editor3-last=Shusko |title=Global Analysis of Dynamic Models in Economics and Finance |publisher=Springer-Verlag |year=2013 |pages=185–204 |chapter= Chaotic Dynamics in Organization Theory |isbn= 978-3-642-29503-4}}</ref>
[[File:BML N=200 P=32.png|400px|right|红色的车和蓝色的车轮流行驶，红色的只向上行驶，蓝色的向右行驶。每一次，如果前面没有车，所有颜色相同的车都试图移动一步。在这里，模型有自我组织在一个有点几何图形模式，其中有一些交通堵塞和一些地区的汽车可以移动在最高速度。
]]
通过混沌理论的应用也可以改进经济模型，但是预测一个经济系统的健康状况以及什么因素对其影响最大是一个极其复杂的任务。<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|bibcode=2010ProCS...1.1119G|arxiv=1005.5384}}</ref>经济和金融系统与古典自然科学中的系统有着根本的不同，因为前者本质上是随机的，因为它们来自于人们之间的相互作用，因此纯粹的确定性模型不可能提供准确的数据表示。检验经济学和金融学中混沌的实证文献呈现出非常复杂的结果，部分原因是混沌的具体检验与非线性关系的更一般检验之间的混淆。<ref>{{cite journal |last=Brooks |first=Chris |authorlink=Chris Brooks (academic)|title=Chaos in foreign exchange markets: a sceptical view |journal=Computational Economics|year=1998 |volume=11 |issue=3 |pages=265–281 |issn=1572-9974 |doi=10.1023/A:1008650024944|url=http://centaur.reading.ac.uk/35988/1/35988.pdf }}</ref>
应用混沌理论进行交通量预测具有重要意义。更好地预测交通将在何时发生将允许采取措施在它发生之前驱散它。将混沌理论原理与其他一些方法相结合，得到了一个更精确的短期预测模型(见右边 BML 流量模型的图)。<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}}</ref>
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。<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>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。<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>
==参见==
==参见==