更改

::混乱：当当前决定未来时，但是近似当前并不能近似确定未来。

::混沌：当当前决定未来时，但是近似当前并不能近似确定未来。

常见的混沌理论忽略了拓扑混合，认为混沌只是对初始条件敏感。然而，对初始条件的敏感依赖本身并不会造成混乱。例如，考虑一下通过反复将初始值加倍而产生的简单的动力系统。这个系统对任何地方的初始条件都有敏感的依赖关系，因为任何一对邻近的点最终都会变得广泛分离。然而，这个例子没有拓扑混合，因此没有混沌。事实上，它的行为极其简单: 除了0以外的所有点都趋向于正或负无穷。

常见的混沌理论忽略了拓扑混合，认为混沌只是对初始条件敏感。然而，对初始条件的敏感依赖本身并不会造成混沌。例如，考虑一下通过反复将初始值加倍而产生的简单的动力系统。这个系统对任何地方的初始条件都有敏感的依赖关系，因为任何一对邻近的点最终都会变得广泛分离。然而，这个例子没有拓扑混合，因此没有混沌。事实上，它的行为极其简单: 除了0以外的所有点都趋向于正或负无穷。

==应用==

==应用==

[[File:Textile cone.jpg|thumb| 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| 一个圆锥形纺织外壳，外观与第30条规则相似，一个行为混沌的细胞自动机<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">{{Cite book|title=On the order of chaos. Social anthropology and the science of chaos|last=Mosko M.S., Damon F.H. (Eds.)|publisher=Berghahn Books|year=2005|isbn=|location=Oxford|pages=}}</ref> 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">{{cite journal |last1 = Safonov |first1 = Leonid A. |last2 = Tomer |first2 = Elad |last3 = Strygin |first3 = Vadim V. |last4 = Ashkenazy |first4 = Yosef |last5 = Havlin |first5 = Shlomo |title = Multifractal chaotic attractors in a system of delay-differential equations modeling road traffic |journal = Chaos: An Interdisciplinary Journal of Nonlinear Science |volume = 12 |issue = 4 |pages = 1006–1014 |year = 2002 |issn = 1054-1500 |doi = 10.1063/1.1507903 |pmid = 12779624 |bibcode = 2002Chaos..12.1006S }}</ref>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。

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">{{Cite book|title=On the order of chaos. Social anthropology and the science of chaos|last=Mosko M.S., Damon F.H. (Eds.)|publisher=Berghahn Books|year=2005|isbn=|location=Oxford|pages=}}</ref> 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">{{cite journal |last1 = Safonov |first1 = Leonid A. |last2 = Tomer |first2 = Elad |last3 = Strygin |first3 = Vadim V. |last4 = Ashkenazy |first4 = Yosef |last5 = Havlin |first5 = Shlomo |title = Multifractal chaotic attractors in a system of delay-differential equations modeling road traffic |journal = Chaos: An Interdisciplinary Journal of Nonlinear Science |volume = 12 |issue = 4 |pages = 1006–1014 |year = 2002 |issn = 1054-1500 |doi = 10.1063/1.1507903 |pmid = 12779624 |bibcode = 2002Chaos..12.1006S }}</ref>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。

===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>

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，其种群增长存在混沌行为。<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>混沌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>

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沌的趋势，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>

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>

雷丁顿 Redington和 Reidbord (1992)试图证明人类的心脏可以表现出混沌的特征。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。

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沌的模式。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。

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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>

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沌解释，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，受到可能导致混沌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>

[[File:复杂性阶梯1.png|400px|right|thumb|[https://campus.swarma.org/course/1112 复杂性思维2020]]]

[[File:复杂性阶梯1.png|400px|right|thumb|[https://campus.swarma.org/course/1112 复杂性思维2020]]]

====[https://campus.swarma.org/course/697 非线性动力学与混沌]====

====[https://campus.swarma.org/course/697 非线性动力学与混沌]====