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第222行： + − An early proponent of chaos theory was [[Henri Poincaré]]. In the 1880s, while studying the [[three-body problem]], he found that there can be orbits that are nonperiodic, and yet not forever increasing nor approaching a fixed point.<ref>{{cite journal |author=Poincaré, Jules Henri |title=Sur le problème des trois corps et les équations de la dynamique. Divergence des séries de M. Lindstedt |journal=Acta Mathematica |volume=13 |issue=1–2 |pages=1–270 |year=1890 |doi=10.1007/BF02392506 |doi-access=free }}</ref><ref>{{Cite book|title=The three-body problem and the equations of dynamics : Poincaré's foundational work on dynamical systems theory|last=Poincaré|first=J. Henri|publisher=Springer International Publishing|others=Popp, Bruce D. (Translator)|year=2017|isbn=9783319528984|location=Cham, Switzerland|pages=|oclc=987302273}}</ref><ref>{{cite book |author1=Diacu, Florin |author2=Holmes, Philip |title=Celestial Encounters: The Origins of Chaos and Stability |publisher=[[Princeton University Press]] |year=1996 }}</ref> In 1898, [[Jacques Hadamard]] published an influential study of the chaotic motion of a free particle gliding frictionlessly on a surface of constant negative curvature, called "[[Hadamard's billiards]]".<ref>{{cite journal|first = Jacques|last = Hadamard|year = 1898|title = Les surfaces à courbures opposées et leurs lignes géodesiques|journal = Journal de Mathématiques Pures et Appliquées|volume = 4|pages = 27–73}}</ref> Hadamard was able to show that all trajectories are unstable, in that all particle trajectories diverge exponentially from one another, with a positive [[Lyapunov exponent]]. − An early proponent of chaos theory was Henri Poincaré. In the 1880s, while studying the three-body problem, he found that there can be orbits that are nonperiodic, and yet not forever increasing nor approaching a fixed point. In 1898, Jacques Hadamard published an influential study of the chaotic motion of a free particle gliding frictionlessly on a surface of constant negative curvature, called "Hadamard's billiards". Hadamard was able to show that all trajectories are unstable, in that all particle trajectories diverge exponentially from one another, with a positive Lyapunov exponent.+ − 亨利· 庞加莱 Henri Poincaré是混沌理论的早期支持者。19世纪80年代，在研究三体时，他发现有些轨道是非周期性的，但不会永远增加，也不会接近一个固定点。1898年，雅克·阿达马发表了一篇影响深远的论文，研究自由粒子在恒负曲率表面上无摩擦滑行的混沌运动，这篇论文被称为“ Hadamard 台球”。哈达马德能够证明所有的轨道都是不稳定的，所有的粒子轨道都以指数形式彼此分离，李亚普诺夫指数为正。 + + − Chaos theory began in the field of [[ergodic theory]]. Later studies, also on the topic of nonlinear [[differential equations]], were carried out by [[George David Birkhoff]],<ref>George D. Birkhoff, ''Dynamical Systems,'' vol. 9 of the American Mathematical Society Colloquium Publications (Providence, Rhode Island: American Mathematical Society, 1927)</ref> [[Andrey Nikolaevich Kolmogorov]],<ref>{{cite journal| last=Kolmogorov | first=Andrey Nikolaevich | authorlink=Andrey Nikolaevich Kolmogorov | year=1941 | title=Local structure of turbulence in an incompressible fluid for very large Reynolds numbers | journal=[[Doklady Akademii Nauk SSSR]] | volume=30 | issue=4 | pages=301–5 |bibcode = 1941DoSSR..30..301K | title-link=turbulence }} Reprinted in: {{cite journal |journal=Proceedings of the Royal Society A |volume=434 |pages=9–13 |year=1991 |doi=10.1098/rspa.1991.0075 |title=The Local Structure of Turbulence in Incompressible Viscous Fluid for Very Large Reynolds Numbers |last1=Kolmogorov |first1=A. N. |issue=1890 |bibcode=1991RSPSA.434....9K|url=https://www.semanticscholar.org/paper/202870134de1f771f678cb540d2ea082b1ab9c5d }}</ref><ref>{{cite journal| last=Kolmogorov | first=A. N. | year=1941 | title=On degeneration of isotropic turbulence in an incompressible viscous liquid | journal=Doklady Akademii Nauk SSSR | volume=31 | issue=6 | pages=538–540}} Reprinted in: {{cite journal |journal=Proceedings of the Royal Society A |volume=434 |pages=15–17 |year=1991 |doi=10.1098/rspa.1991.0076 |title=Dissipation of Energy in the Locally Isotropic Turbulence |last1=Kolmogorov |first1=A. N. |issue=1890 |bibcode=1991RSPSA.434...15K|url=https://www.semanticscholar.org/paper/5874066f6114b679a74fc8edc9db03e48d22251c }}</ref><ref>{{cite book| last=Kolmogorov | first=A. N. | year=1954 | title=Preservation of conditionally periodic movements with small change in the Hamiltonian function | journal=Doklady Akademii Nauk SSSR | volume=98 | pages=527–530| bibcode=1979LNP....93...51K| doi=10.1007/BFb0021737| series=Lecture Notes in Physics| isbn=978-3-540-09120-2}} See also [[Kolmogorov–Arnold–Moser theorem]]</ref> [[Mary Lucy Cartwright]] and [[John Edensor Littlewood]],<ref>{{cite journal |last1=Cartwright |first1=Mary L. |last2=Littlewood |first2=John E. |title=On non-linear differential equations of the second order, I: The equation ''y''" + ''k''(1−''y''<sup>2</sup>)''y<nowiki>'</nowiki>'' + ''y'' = ''b''λkcos(λ''t'' + ''a''), ''k'' large |journal=Journal of the London Mathematical Society |volume=20 |pages=180–9 |year=1945 |doi=10.1112/jlms/s1-20.3.180 |issue=3 }} See also: [[Van der Pol oscillator]]</ref> and [[Stephen Smale]].<ref>{{cite journal |author=Smale, Stephen |title=Morse inequalities for a dynamical system |journal=Bulletin of the American Mathematical Society |volume=66 |pages=43–49 |date=January 1960 |doi=10.1090/S0002-9904-1960-10386-2 |bibcode=1994BAMaS..30..205W |doi-access=free }}</ref> Except for Smale, these studies were all directly inspired by physics: the three-body problem in the case of Birkhoff, turbulence and astronomical problems in the case of Kolmogorov, and radio engineering in the case of Cartwright and Littlewood.{{Citation needed|date=July 2008}} Although chaotic planetary motion had not been observed, experimentalists had encountered turbulence in fluid motion and nonperiodic oscillation in radio circuits without the benefit of a theory to explain what they were seeing.+ − Chaos theory began in the field of ergodic theory. Later studies, also on the topic of nonlinear differential equations, were carried out by George David Birkhoff, Andrey Nikolaevich Kolmogorov, Mary Lucy Cartwright and John Edensor Littlewood, and Stephen Smale. Except for Smale, these studies were all directly inspired by physics: the three-body problem in the case of Birkhoff, turbulence and astronomical problems in the case of Kolmogorov, and radio engineering in the case of Cartwright and Littlewood. Although chaotic planetary motion had not been observed, experimentalists had encountered turbulence in fluid motion and nonperiodic oscillation in radio circuits without the benefit of a theory to explain what they were seeing. − 混沌理论起源于遍历理论。后来的研究，也是关于非线性微分方程的主题，由乔治·戴维·伯克霍夫，安德雷·柯尔莫哥洛夫，Mary Lucy Cartwright 和约翰·恩瑟·李特尔伍德，和 Stephen Smale 进行。除了斯梅尔，这些研究都直接受到物理学的启发: 伯克霍夫的三体，Kolmogorov 的湍流和天文学问题，卡特赖特和利特伍德的无线电工程。虽然还没有观察到混沌的行星运动，但实验人员已经遇到了流体运动中的湍流和无线电电路中的非周期性振荡，而没有一个理论来解释他们所看到的。+ + + − Despite initial insights in the first half of the twentieth century, chaos theory became formalized as such only after mid-century, when it first became evident to some scientists that [[linear theory]], the prevailing system theory at that time, simply could not explain the observed behavior of certain experiments like that of the [[logistic map]]. What had been attributed to measure imprecision and simple "[[Pseudorandom noise|noise]]" was considered by chaos theorists as a full component of the studied systems. − Despite initial insights in the first half of the twentieth century, chaos theory became formalized as such only after mid-century, when it first became evident to some scientists that linear theory, the prevailing system theory at that time, simply could not explain the observed behavior of certain experiments like that of the logistic map. What had been attributed to measure imprecision and simple "noise" was considered by chaos theorists as a full component of the studied systems.+ − 尽管在20世纪上半叶有了初步的认识，但混沌理论直到本世纪中叶才正式形成，当时一些科学家首次发现，线性理论，当时流行的系统理论，根本无法解释某些实验的观察行为，比如逻辑地图。混沌理论家认为测量不精确和简单的“噪声”是所研究系统的完整组成部分。 + + − The main catalyst for the development of chaos theory was the electronic computer. Much of the mathematics of chaos theory involves the repeated [[iteration]] of simple mathematical formulas, which would be impractical to do by hand. Electronic computers made these repeated calculations practical, while figures and images made it possible to visualize these systems. As a graduate student in Chihiro Hayashi's laboratory at Kyoto University, Yoshisuke Ueda was experimenting with analog computers and noticed, on November 27, 1961, what he called "randomly transitional phenomena". Yet his advisor did not agree with his conclusions at the time, and did not allow him to report his findings until 1970.<ref>{{harvnb|Abraham|Ueda|2001|loc=See Chapters 3 and 4}}</ref><ref>{{harvnb|Sprott|2003|p=[https://books.google.com/books?id=SEDjdjPZ158C&pg=PA89 89]}}</ref>+ − − The main catalyst for the development of chaos theory was the electronic computer. Much of the mathematics of chaos theory involves the repeated iteration of simple mathematical formulas, which would be impractical to do by hand. Electronic computers made these repeated calculations practical, while figures and images made it possible to visualize these systems. As a graduate student in Chihiro Hayashi's laboratory at Kyoto University, Yoshisuke Ueda was experimenting with analog computers and noticed, on November 27, 1961, what he called "randomly transitional phenomena". Yet his advisor did not agree with his conclusions at the time, and did not allow him to report his findings until 1970. − − 电子计算机是混沌理论发展的主要催化剂。许多混沌理论的数学涉及简单数学公式的重复迭代，手工操作是不切实际的。电子计算机使这些重复的计算变得可行，而图形和图像使这些系统可视化成为可能。1961年11月27日，作为林千寻在京都大学实验室的研究生，Yoshisuke Ueda 正在用模拟计算机进行实验，他注意到了他所说的“随机过渡现象”。然而他的顾问当时并不同意他的结论，并且直到1970年才允许他报告他的发现。 − − − − − − [[File:Airplane vortex edit.jpg|thumb|left|[[Turbulence]] in the [[Wingtip vortices|tip vortex]] from an [[airplane]] wing. Studies of the critical point beyond which a system creates turbulence were important for chaos theory, analyzed for example by the [[Soviet physicists|Soviet physicist]] [[Lev Landau]], who developed the [[Landau-Hopf theory of turbulence]]. [[David Ruelle]] and [[Floris Takens]] later predicted, against Landau, that [[fluid turbulence]] could develop through a [[strange attractor]], a main concept of chaos theory.]] − − [[Turbulence in the tip vortex from an airplane wing. Studies of the critical point beyond which a system creates turbulence were important for chaos theory, analyzed for example by the Soviet physicist Lev Landau, who developed the Landau-Hopf theory of turbulence. David Ruelle and Floris Takens later predicted, against Landau, that fluid turbulence could develop through a strange attractor, a main concept of chaos theory.]] − − 来自飞机机翼的尖涡中的湍流。研究系统产生湍流的临界点对混沌理论很重要，例如苏联物理学家 Lev Landau 分析了这个临界点，他发展了湍流的 Landau-hopf 理论。后来 David Ruelle 和 Floris Takens 针对 Landau 预测，湍流可以通过一个奇怪的吸引子发展，这是混沌理论的一个主要概念 − − − − − − [[Edward Lorenz]] was an early pioneer of the theory. His interest in chaos came about accidentally through his work on [[meteorology|weather prediction]] in 1961.<ref name=Lorenz1961>{{cite journal |author=Lorenz, Edward N. |title=Deterministic non-periodic flow |journal=Journal of the Atmospheric Sciences |volume=20 |pages=130–141 |year=1963 |doi=10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2 |issue=2 |bibcode=1963JAtS...20..130L|doi-access=free }}</ref> Lorenz was using a simple digital computer, a [[Royal McBee]] [[LGP-30]], to run his weather simulation. He wanted to see a sequence of data again, and to save time he started the simulation in the middle of its course. He did this by entering a printout of the data that corresponded to conditions in the middle of the original simulation. To his surprise, the weather the machine began to predict was completely different from the previous calculation. Lorenz tracked this down to the computer printout. The computer worked with 6-digit precision, but the printout rounded variables off to a 3-digit number, so a value like 0.506127 printed as 0.506. This difference is tiny, and the consensus at the time would have been that it should have no practical effect. However, Lorenz discovered that small changes in initial conditions produced large changes in long-term outcome.<ref>{{cite book|title=Chaos: Making a New Science |last=Gleick |first=James |year=1987 |publisher=Cardinal |location=London|page=17|isbn=978-0-434-29554-8|title-link=Chaos: Making a New Science }}</ref> Lorenz's discovery, which gave its name to [[Lorenz attractor]]s, showed that even detailed atmospheric modelling cannot, in general, make precise long-term weather predictions. − − Edward Lorenz was an early pioneer of the theory. His interest in chaos came about accidentally through his work on weather prediction in 1961. Lorenz was using a simple digital computer, a Royal McBee LGP-30, to run his weather simulation. He wanted to see a sequence of data again, and to save time he started the simulation in the middle of its course. He did this by entering a printout of the data that corresponded to conditions in the middle of the original simulation. To his surprise, the weather the machine began to predict was completely different from the previous calculation. Lorenz tracked this down to the computer printout. The computer worked with 6-digit precision, but the printout rounded variables off to a 3-digit number, so a value like 0.506127 printed as 0.506. This difference is tiny, and the consensus at the time would have been that it should have no practical effect. However, Lorenz discovered that small changes in initial conditions produced large changes in long-term outcome. Lorenz's discovery, which gave its name to Lorenz attractors, showed that even detailed atmospheric modelling cannot, in general, make precise long-term weather predictions. − − 爱德华 · 洛伦茨是这一理论的早期开拓者。他对混沌的兴趣来源于1961年他在天气预报方面的工作。洛伦茨正在使用一台简单的数字计算机，一台 Royal McBee LGP-30运行他的天气模拟。他想再看一次数据序列，为了节省时间，他在模拟过程中间开始了模拟。他通过输入一个打印输出的数据，这些数据对应于原始模拟中的条件。令他惊讶的是，机器开始预测的天气与以前的计算完全不同。洛伦茨通过计算机打印出来的资料查到了这一点。计算机以6位数的精度工作，但打印输出的变量四舍五入到一个3位数字，所以像0.506127这样的值打印为0.506。这种差异是微小的，当时的共识是它不应该有任何实际效果。然而，洛伦茨发现，初始条件的微小变化会导致长期结果的巨大变化。洛伦茨的发现，这给它的名字洛伦茨吸引子，表明即使详细的大气模型，一般来说，不能作出精确的长期天气预报。 − − − − − − In 1963, [[Benoit Mandelbrot]] found recurring patterns at every scale in data on cotton prices.<ref>{{cite journal|first = Benoît|last = Mandelbrot|year = 1963|title = The variation of certain speculative prices|journal = Journal of Business|volume = 36|pages = 394–419|doi = 10.1086/294632|issue = 4|jstor=2350970}}</ref> Beforehand he had studied [[information theory]] and concluded noise was patterned like a [[Cantor set]]: on any scale the proportion of noise-containing periods to error-free periods was a constant – thus errors were inevitable and must be planned for by incorporating redundancy.<ref>{{cite journal |author1=Berger J.M. |author2=Mandelbrot B. | year = 1963 | title = A new model for error clustering in telephone circuits | journal = IBM Journal of Research and Development | volume = 7 |issue=3 | pages = 224–236 | doi=10.1147/rd.73.0224}}</ref> Mandelbrot described both the "Noah effect" (in which sudden discontinuous changes can occur) and the "Joseph effect" (in which persistence of a value can occur for a while, yet suddenly change afterwards).<ref>{{cite book |author=Mandelbrot, B. |title=The Fractal Geometry of Nature |publisher=Freeman |location=New York |year=1977 |page=248 }}</ref><ref>See also: {{cite book |last1=Mandelbrot |first1=Benoît B. |last2=Hudson |first2=Richard L. |title=The (Mis)behavior of Markets: A Fractal View of Risk, Ruin, and Reward |url=https://archive.org/details/misbehaviorofmar00beno |url-access=registration |publisher=Basic Books |location=New York |year=2004 |page=[https://archive.org/details/misbehaviorofmar00beno/page/201 201] }}</ref> This challenged the idea that changes in price were [[normal distribution|normally distributed]]. In 1967, he published "[[How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension|How long is the coast of Britain? Statistical self-similarity and fractional dimension]]", showing that a coastline's length varies with the scale of the measuring instrument, resembles itself at all scales, and is infinite in length for an [[infinitesimal]]ly small measuring device.<ref>{{cite journal |last=Mandelbrot |first=Benoît |title=How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension |journal=Science |volume=156 |issue=3775 |pages=636–8 |date=5 May 1967 |doi=10.1126/science.156.3775.636 |pmid=17837158 |bibcode = 1967Sci...156..636M }}</ref> Arguing that a ball of twine appears as a point when viewed from far away (0-dimensional), a ball when viewed from fairly near (3-dimensional), or a curved strand (1-dimensional), he argued that the dimensions of an object are relative to the observer and may be fractional. An object whose irregularity is constant over different scales ("self-similarity") is a [[fractal]] (examples include the [[Menger sponge]], the [[Sierpiński gasket]], and the [[Koch curve]] or ''snowflake'', which is infinitely long yet encloses a finite space and has a [[fractal dimension]] of circa 1.2619). In 1982, Mandelbrot published ''[[The Fractal Geometry of Nature]]'', which became a classic of chaos theory.<ref>{{cite book|author=Mandelbrot, B.|title=The Fractal Geometry of Nature|date=1982|place=New York|publisher=Macmillan|isbn=978-0716711865|url=https://archive.org/details/fractalgeometryo00beno}}</ref> Biological systems such as the branching of the circulatory and bronchial systems proved to fit a fractal model.<ref>{{cite book |last1=Buldyrev |first1=S.V. |last2=Goldberger |first2=A.L. | last3=Havlin |first3=S. | authorlink3=Shlomo Havlin|last4=Peng |first4=C.K. | last5=Stanley | first5=H.E. |authorlink5=H. Eugene Stanley|editor1-first=Armin |editor1-last=Bunde |editor2-first=Shlomo | editor2-last=Havlin | editor2-link=Shlomo Havlin |title=Fractals in Science |publisher=Springer |year=1994 |pages=49–89 |chapter=Fractals in Biology and Medicine: From DNA to the Heartbeat |isbn=978-3-540-56220-7}}</ref> − − In 1963, Benoit Mandelbrot found recurring patterns at every scale in data on cotton prices. Beforehand he had studied information theory and concluded noise was patterned like a Cantor set: on any scale the proportion of noise-containing periods to error-free periods was a constant – thus errors were inevitable and must be planned for by incorporating redundancy. Mandelbrot described both the "Noah effect" (in which sudden discontinuous changes can occur) and the "Joseph effect" (in which persistence of a value can occur for a while, yet suddenly change afterwards). This challenged the idea that changes in price were normally distributed. In 1967, he published "How long is the coast of Britain? Statistical self-similarity and fractional dimension", showing that a coastline's length varies with the scale of the measuring instrument, resembles itself at all scales, and is infinite in length for an infinitesimally small measuring device. Arguing that a ball of twine appears as a point when viewed from far away (0-dimensional), a ball when viewed from fairly near (3-dimensional), or a curved strand (1-dimensional), he argued that the dimensions of an object are relative to the observer and may be fractional. An object whose irregularity is constant over different scales ("self-similarity") is a fractal (examples include the Menger sponge, the Sierpiński gasket, and the Koch curve or snowflake, which is infinitely long yet encloses a finite space and has a fractal dimension of circa 1.2619). In 1982, Mandelbrot published The Fractal Geometry of Nature, which became a classic of chaos theory. Biological systems such as the branching of the circulatory and bronchial systems proved to fit a fractal model. − − 1963年，本华·曼德博在棉花价格数据中发现了各种规模的循环模式。事先他研究过信息理论，并得出结论，噪音的模式类似于康托集: 在任何尺度上，包含噪音的周期与无误差的周期的比例是一个常数——因此误差是不可避免的，必须通过引入冗余来计划。曼德布洛特既描述了“诺亚效应”(可能会发生突然的不连续变化) ，也描述了“约瑟夫效应”(可能会持续一段时间，然后突然改变)。这挑战了价格变化是正态分布的观点。1967年，他出版了《英国的海岸线有多长？统计自相似性和分维数” ，显示海岸线的长度随测量仪器的规模而变化，在所有规模上都与仪器相似，而且对于一个极小的测量装置来说，长度是无限的。他认为，从远处(0维)看，一个线球似乎是一个点，从相当接近(3维)看，一个球，或一个曲线(一维) ，他认为，一个物体的尺寸是相对于观察者，可能是分数。如果一个物体的不规则性在不同尺度上是常数(“自相似”) ，那么这个物体就是一个分形(例如门格尔海绵、赛尔皮滑雪垫圈和科氏曲线或雪花，它们是无限长的，但覆盖着一个有限的空间，分形维数约为1.2619)。1982年，曼德布洛特发表了《自然的分形几何》 ，成为混沌理论的经典之作。生物学系统，如循环系统和支气管系统的分支证明符合一个分形模型。 − − − − − − In December 1977, the [[New York Academy of Sciences]] organized the first symposium on chaos, attended by David Ruelle, [[Robert May, Baron May of Oxford|Robert May]], [[James A. Yorke]] (coiner of the term "chaos" as used in mathematics), [[Robert Shaw (physicist)|Robert Shaw]], and the meteorologist Edward Lorenz. The following year Pierre Coullet and Charles Tresser published "Iterations d'endomorphismes et groupe de renormalisation", and [[Mitchell Feigenbaum]]'s article "Quantitative Universality for a Class of Nonlinear Transformations" finally appeared in a journal, after 3 years of referee rejections.<ref name="Feigenbaum 25–52"/><ref>Coullet, Pierre, and Charles Tresser. "Iterations d'endomorphismes et groupe de renormalisation." Le Journal de Physique Colloques 39.C5 (1978): C5-25</ref> Thus Feigenbaum (1975) and Coullet & Tresser (1978) discovered the [[Universality (dynamical systems)|universality]] in chaos, permitting the application of chaos theory to many different phenomena. − − In December 1977, the New York Academy of Sciences organized the first symposium on chaos, attended by David Ruelle, Robert May, James A. Yorke (coiner of the term "chaos" as used in mathematics), Robert Shaw, and the meteorologist Edward Lorenz. The following year Pierre Coullet and Charles Tresser published "Iterations d'endomorphismes et groupe de renormalisation", and Mitchell Feigenbaum's article "Quantitative Universality for a Class of Nonlinear Transformations" finally appeared in a journal, after 3 years of referee rejections. Thus Feigenbaum (1975) and Coullet & Tresser (1978) discovered the universality in chaos, permitting the application of chaos theory to many different phenomena. − − 1977年12月，纽约科学院组织了第一次关于混沌的研讨会，出席的有大卫 · 鲁尔、罗伯特 · 梅、詹姆斯 · a · 约克(数学中“混沌”一词的创始人)、罗伯特 · 肖和气象学家爱德华 · 洛伦茨。第二年 Pierre Coullet 和 Charles Tresser 发表了《迭代与重整化群体》 ，米切尔·费根鲍姆的文章《一类非线性变换的定量普遍性》最终发表在一本杂志上，经过三年的裁判拒绝。因此 Feigenbaum (1975)和 Coullet & Tresser (1978)发现了混沌中的普遍性，允许混沌理论应用于许多不同的现象。 − − − − − − In 1979, [[Albert J. Libchaber]], during a symposium organized in Aspen by [[Pierre Hohenberg]], presented his experimental observation of the [[Bifurcation theory|bifurcation]] cascade that leads to chaos and turbulence in [[Rayleigh–Bénard convection]] systems. He was awarded the [[Wolf Prize in Physics]] in 1986 along with [[Mitchell J. Feigenbaum]] for their inspiring achievements.<ref>{{cite web|url = http://www.wolffund.org.il/cat.asp?id=25&cat_title=PHYSICS|title = The Wolf Prize in Physics in 1986.}}</ref> − − In 1979, Albert J. Libchaber, during a symposium organized in Aspen by Pierre Hohenberg, presented his experimental observation of the bifurcation cascade that leads to chaos and turbulence in Rayleigh–Bénard convection systems. He was awarded the Wolf Prize in Physics in 1986 along with Mitchell J. Feigenbaum for their inspiring achievements. − − 1979年，Albert j. Libchaber 在皮埃尔·奥昂贝格在阿斯彭组织的一次研讨会上，提出了他对瑞利-b 纳德对流系统中导致混沌和湍流的分叉级联的实验观察。1986年，由于他们令人鼓舞的成就，他和米切尔 · 费根鲍姆一起被授予沃尔夫物理学奖。 − − − − − − In 1986, the New York Academy of Sciences co-organized with the [[National Institute of Mental Health]] and the [[Office of Naval Research]] the first important conference on chaos in biology and medicine. There, [[Bernardo Huberman]] presented a mathematical model of the [[eye tracking disorder]] among [[schizophrenics]].<ref>{{cite journal |authorlink=Bernardo Huberman |author=Huberman, B.A. |title=A Model for Dysfunctions in Smooth Pursuit Eye Movement |journal=Annals of the New York Academy of Sciences |volume=504 Perspectives in Biological Dynamics and Theoretical Medicine |issue=1 |pages=260–273 |date=July 1987 |doi=10.1111/j.1749-6632.1987.tb48737.x |pmid=3477120 |bibcode = 1987NYASA.504..260H }}</ref> This led to a renewal of [[physiology]] in the 1980s through the application of chaos theory, for example, in the study of pathological [[cardiac cycle]]s. − − In 1986, the New York Academy of Sciences co-organized with the National Institute of Mental Health and the Office of Naval Research the first important conference on chaos in biology and medicine. There, Bernardo Huberman presented a mathematical model of the eye tracking disorder among schizophrenics. This led to a renewal of physiology in the 1980s through the application of chaos theory, for example, in the study of pathological cardiac cycles. − − 1986年，纽约科学院与国家心理健康研究所和海军研究办公室共同组织了第一次关于生物学和医学中的混沌的重要会议。在那里，贝尔纳多 · 休伯曼提出了一个精神分裂症患者眼球追踪障碍的数学模型。这导致了生理学的更新在20世纪80年代通过应用混沌理论，例如，在病理心脏周期的研究。 − − − − − − In 1987, [[Per Bak]], [[Chao Tang]] and [[Kurt Wiesenfeld]] published a paper in ''[[Physical Review Letters]]''<ref>{{cite journal |author1=Bak, Per |author2=Tang, Chao |author3=Wiesenfeld, Kurt |title=Self-organized criticality: An explanation of the 1/f noise |journal=Physical Review Letters |volume=59 |issue=4 |pages=381–4 |date=27 July 1987 |doi=10.1103/PhysRevLett.59.381 |pmid=10035754 |bibcode=1987PhRvL..59..381B}} However, the conclusions of this article have been subject to dispute. {{cite web|url=http://www.nslij-genetics.org/wli/1fnoise/1fnoise_square.html |title=? |url-status=dead |archiveurl=https://web.archive.org/web/20071214033929/https://www.nslij-genetics.org/wli/1fnoise/1fnoise_square.html |archivedate=2007-12-14 }}. See especially: {{cite journal |author1=Laurson, Lasse |author2=Alava, Mikko J. |author3=Zapperi, Stefano |title=Letter: Power spectra of self-organized critical sand piles |journal=Journal of Statistical Mechanics: Theory and Experiment |volume=0511 |id=L001 |date=15 September 2005 }}</ref> describing for the first time [[self-organized criticality]] (SOC), considered one of the mechanisms by which [[complexity]] arises in nature. − − In 1987, Per Bak, Chao Tang and Kurt Wiesenfeld published a paper in Physical Review Letters describing for the first time self-organized criticality (SOC), considered one of the mechanisms by which complexity arises in nature. − − 1987年，Per Bak，Chao Tang 和 Kurt Wiesenfeld 在《物理评论快报》上发表了一篇论文，首次描述了自组织临界性，认为自然界复杂性产生的机制之一。 − − − − − − Alongside largely lab-based approaches such as the [[Bak–Tang–Wiesenfeld sandpile]], many other investigations have focused on large-scale natural or social systems that are known (or suspected) to display [[scale invariance|scale-invariant]] behavior. Although these approaches were not always welcomed (at least initially) by specialists in the subjects examined, SOC has nevertheless become established as a strong candidate for explaining a number of natural phenomena, including [[earthquake]]s, (which, long before SOC was discovered, were known as a source of scale-invariant behavior such as the [[Gutenberg–Richter law]] describing the statistical distribution of earthquake sizes, and the [[Aftershock|Omori law]]<ref>{{cite journal |author=Omori, F. |title=On the aftershocks of earthquakes |journal=Journal of the College of Science, Imperial University of Tokyo |volume=7 |pages=111–200 |year=1894 }}</ref> describing the frequency of aftershocks), [[solar flare]]s, fluctuations in economic systems such as [[financial market]]s (references to SOC are common in [[econophysics]]), landscape formation, [[forest fire]]s, [[landslide]]s, [[epidemic]]s, and [[biological evolution]] (where SOC has been invoked, for example, as the dynamical mechanism behind the theory of "[[punctuated equilibrium|punctuated equilibria]]" put forward by [[Niles Eldredge]] and [[Stephen Jay Gould]]). Given the implications of a scale-free distribution of event sizes, some researchers have suggested that another phenomenon that should be considered an example of SOC is the occurrence of [[war]]s. These investigations of SOC have included both attempts at modelling (either developing new models or adapting existing ones to the specifics of a given natural system), and extensive data analysis to determine the existence and/or characteristics of natural scaling laws. − − Alongside largely lab-based approaches such as the Bak–Tang–Wiesenfeld sandpile, many other investigations have focused on large-scale natural or social systems that are known (or suspected) to display scale-invariant behavior. Although these approaches were not always welcomed (at least initially) by specialists in the subjects examined, SOC has nevertheless become established as a strong candidate for explaining a number of natural phenomena, including earthquakes, (which, long before SOC was discovered, were known as a source of scale-invariant behavior such as the Gutenberg–Richter law describing the statistical distribution of earthquake sizes, and the Omori law describing the frequency of aftershocks), solar flares, fluctuations in economic systems such as financial markets (references to SOC are common in econophysics), landscape formation, forest fires, landslides, epidemics, and biological evolution (where SOC has been invoked, for example, as the dynamical mechanism behind the theory of "punctuated equilibria" put forward by Niles Eldredge and Stephen Jay Gould). Given the implications of a scale-free distribution of event sizes, some researchers have suggested that another phenomenon that should be considered an example of SOC is the occurrence of wars. These investigations of SOC have included both attempts at modelling (either developing new models or adapting existing ones to the specifics of a given natural system), and extensive data analysis to determine the existence and/or characteristics of natural scaling laws. − − 除了大量基于实验室的方法，如 Bak-Tang-Wiesenfeld 沙堆法，许多其他的研究都集中在大规模的自然或社会系统上，这些系统已知(或怀疑)表现出尺度不变的行为。虽然这些方法并不总是受到研究对象专家的欢迎(至少最初是这样) ，但 SOC 已经成为解释一些自然现象的有力候选者，包括地震(早在 SOC 被发现之前，它就被认为是尺度不变行为的来源，例如描述地震大小统计分布的古腾堡-里克特定律，描述余震频率的 Omori 定律) ，太阳，经济系统的波动，例如金融市场(SOC 在经济物理学中很常见) ，地貌形成，森林火灾，流行病和生物进化 (在 SOC 被调用的地方，例如 Niles Eldredge 和史蒂芬·古尔德提出的“间断平衡”理论背后的动力机制) .考虑到事件规模无标度分布的含义，一些研究人员提出，另一个应该被视为 SOC 的例子的现象是战争的发生。对土壤有机碳的这些研究既包括建立模型的尝试(开发新模型或使现有模型适应特定自然系统的具体情况) ，也包括广泛的数据分析，以确定自然定标法的存在和 / 或特点。 − − − − − − In the same year, [[James Gleick]] published ''[[Chaos: Making a New Science]]'', which became a best-seller and introduced the general principles of chaos theory as well as its history to the broad public, though his history under-emphasized important Soviet contributions.{{citation needed|reason=No reason given for soviet under-emphasis|date=May 2016}}<ref>{{cite book|last=Gleick|first=James|title=Chaos: Making a New Science|date=August 26, 2008|publisher=Penguin Books|isbn=978-0143113454}}</ref> Initially the domain of a few, isolated individuals, chaos theory progressively emerged as a transdisciplinary and institutional discipline, mainly under the name of [[nonlinear system]]s analysis. Alluding to [[Thomas Kuhn]]'s concept of a [[paradigm shift]] exposed in ''[[The Structure of Scientific Revolutions]]'' (1962), many "chaologists" (as some described themselves) claimed that this new theory was an example of such a shift, a thesis upheld by Gleick. − − In the same year, James Gleick published Chaos: Making a New Science, which became a best-seller and introduced the general principles of chaos theory as well as its history to the broad public, though his history under-emphasized important Soviet contributions. Initially the domain of a few, isolated individuals, chaos theory progressively emerged as a transdisciplinary and institutional discipline, mainly under the name of nonlinear systems analysis. Alluding to Thomas Kuhn's concept of a paradigm shift exposed in The Structure of Scientific Revolutions (1962), many "chaologists" (as some described themselves) claimed that this new theory was an example of such a shift, a thesis upheld by Gleick. − − 同年，詹姆斯•格莱克(James Gleick)出版了《混沌: 创造新科学》(Chaos: Making a New Science)一书，成为畅销书，并向广大公众介绍了混沌理论的一般原理及其历史，但他对苏联的重要贡献重视不够。混沌理论最初只是少数孤立个体的领域，逐渐成为一门跨学科和制度学科，主要以非线性系统分析的名义出现。许多“混沌论者”(正如一些人自己描述的那样)提到了 Thomas Kuhn 在《科学革命的结构(1962)提出的范式转换的概念，声称这个新理论就是这种转换的一个例子，这是 Gleick 支持的一个论点。 − − − − − − The availability of cheaper, more powerful computers broadens the applicability of chaos theory. Currently, chaos theory remains an active area of research,<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> involving many different disciplines ([[mathematics]], [[topology]], [[physics]],<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> [[social systems]],<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> [[population model]]ing, [[biology]], [[meteorology]], [[astrophysics]], [[information theory]], [[computational neuroscience]], etc.). − − The availability of cheaper, more powerful computers broadens the applicability of chaos theory. Currently, chaos theory remains an active area of research, involving many different disciplines (mathematics, topology, physics, social systems, population modeling, biology, meteorology, astrophysics, information theory, computational neuroscience, etc.). − − 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，涉及许多不同的学科(数学、拓扑学、物理学、社会系统、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)。). + +
→历史
[[File:Barnsley fern plotted with VisSim.PNG|thumb|upright|巴恩斯利蕨类植物是用混沌游戏创造出来的。可以通过迭代函数系统（IFS）重建自然形式（蕨类植物，云朵，山脉等）。]]
[[File:Barnsley fern plotted with VisSim.PNG|thumb|upright|巴恩斯利蕨类植物是用混沌游戏创造出来的。可以通过迭代函数系统（IFS）重建自然形式（蕨类植物，云朵，山脉等）。]]
亨利·庞加莱 Henri Poincaré是混沌理论的早期支持者。19世纪80年代，在研究三体时，他发现有些轨道是非周期性的，但不会永远增加，也不会接近一个固定点。<ref>{{cite journal |author=Poincaré, Jules Henri |title=Sur le problème des trois corps et les équations de la dynamique. Divergence des séries de M. Lindstedt |journal=Acta Mathematica |volume=13 |issue=1–2 |pages=1–270 |year=1890 |doi=10.1007/BF02392506 |doi-access=free }}</ref><ref>{{Cite book|title=The three-body problem and the equations of dynamics : Poincaré's foundational work on dynamical systems theory|last=Poincaré|first=J. Henri|publisher=Springer International Publishing|others=Popp, Bruce D. (Translator)|year=2017|isbn=9783319528984|location=Cham, Switzerland|pages=|oclc=987302273}}</ref><ref>{{cite book |author1=Diacu, Florin |author2=Holmes, Philip |title=Celestial Encounters: The Origins of Chaos and Stability |publisher=[[Princeton University Press]] |year=1996 }}</ref>1898年，雅克·哈达马德 Jacques Hadamard发表了一篇影响深远的论文，研究自由粒子在恒负曲率表面上无摩擦滑行的混沌运动，这篇论文被称为“ Hadamard 台球”。<ref>{{cite journal|first = Jacques|last = Hadamard|year = 1898|title = Les surfaces à courbures opposées et leurs lignes géodesiques|journal = Journal de Mathématiques Pures et Appliquées|volume = 4|pages = 27–73}}</ref>Hadamard能够证明所有的轨道都是不稳定的，所有的粒子轨道都以指数形式彼此分离，李亚普诺夫指数为正。
混沌理论起源于遍历理论。后来的研究，也是关于非线性微分方程的主题，由乔治·戴维·伯克霍夫 George David Birkhoff，<ref>George D. Birkhoff, ''Dynamical Systems,'' vol. 9 of the American Mathematical Society Colloquium Publications (Providence, Rhode Island: American Mathematical Society, 1927)</ref>安德雷·柯尔莫哥洛夫 Andrey Nikolaevich Kolmogorov，<ref>{{cite journal| last=Kolmogorov | first=Andrey Nikolaevich | authorlink=Andrey Nikolaevich Kolmogorov | year=1941 | title=Local structure of turbulence in an incompressible fluid for very large Reynolds numbers | journal=[[Doklady Akademii Nauk SSSR]] | volume=30 | issue=4 | pages=301–5 |bibcode = 1941DoSSR..30..301K | title-link=turbulence }} Reprinted in: {{cite journal |journal=Proceedings of the Royal Society A |volume=434 |pages=9–13 |year=1991 |doi=10.1098/rspa.1991.0075 |title=The Local Structure of Turbulence in Incompressible Viscous Fluid for Very Large Reynolds Numbers |last1=Kolmogorov |first1=A. N. |issue=1890 |bibcode=1991RSPSA.434....9K|url=https://www.semanticscholar.org/paper/202870134de1f771f678cb540d2ea082b1ab9c5d }}</ref><ref>{{cite journal| last=Kolmogorov | first=A. N. | year=1941 | title=On degeneration of isotropic turbulence in an incompressible viscous liquid | journal=Doklady Akademii Nauk SSSR | volume=31 | issue=6 | pages=538–540}} Reprinted in: {{cite journal |journal=Proceedings of the Royal Society A |volume=434 |pages=15–17 |year=1991 |doi=10.1098/rspa.1991.0076 |title=Dissipation of Energy in the Locally Isotropic Turbulence |last1=Kolmogorov |first1=A. N. |issue=1890 |bibcode=1991RSPSA.434...15K|url=https://www.semanticscholar.org/paper/5874066f6114b679a74fc8edc9db03e48d22251c }}</ref><ref>{{cite book| last=Kolmogorov | first=A. N. | year=1954 | title=Preservation of conditionally periodic movements with small change in the Hamiltonian function | journal=Doklady Akademii Nauk SSSR | volume=98 | pages=527–530| bibcode=1979LNP....93...51K| doi=10.1007/BFb0021737| series=Lecture Notes in Physics| isbn=978-3-540-09120-2}} See also [[Kolmogorov–Arnold–Moser theorem]]</ref>玛丽·露西·卡特赖特 Mary Lucy Cartwright 和约翰·恩瑟·李特尔伍德 John Edensor Littlewood，<ref>{{cite journal |last1=Cartwright |first1=Mary L. |last2=Littlewood |first2=John E. |title=On non-linear differential equations of the second order, I: The equation ''y''" + ''k''(1−''y''<sup>2</sup>)''y<nowiki>'</nowiki>'' + ''y'' = ''b''λkcos(λ''t'' + ''a''), ''k'' large |journal=Journal of the London Mathematical Society |volume=20 |pages=180–9 |year=1945 |doi=10.1112/jlms/s1-20.3.180 |issue=3 }} See also: [[Van der Pol oscillator]]</ref>和[[斯蒂芬·斯梅尔 Stephen Smale]]进行。<ref>{{cite journal |author=Smale, Stephen |title=Morse inequalities for a dynamical system |journal=Bulletin of the American Mathematical Society |volume=66 |pages=43–49 |date=January 1960 |doi=10.1090/S0002-9904-1960-10386-2 |bibcode=1994BAMaS..30..205W |doi-access=free }}</ref>除了Smale，这些研究都直接受到物理学的启发: Birkhoff的三体，Kolmogorov 的湍流和天文学问题，Cartwright和Littlewood的无线电工程。虽然还没有观察到混沌的行星运动，但实验人员已经遇到了流体运动中的湍流和无线电电路中的非周期性振荡，而没有一个理论来解释他们所看到的。
尽管在20世纪上半叶有了初步的认识，但混沌理论直到本世纪中叶才正式形成，当时一些科学家首次发现，线性理论，当时流行的系统理论，根本无法解释某些实验的观察行为，比如logistic映射。混沌理论家认为测量不精确和简单的“噪声”是所研究系统的完整组成部分。
电子计算机是混沌理论发展的主要催化剂。许多混沌理论的数学涉及简单数学公式的重复迭代，手工操作是不切实际的。电子计算机使这些重复的计算变得可行，而图形和图像使这些系统可视化成为可能。1961年11月27日，作为林千寻在京都大学实验室的研究生，Yoshisuke Ueda 正在用模拟计算机进行实验，他注意到了他所说的“随机过渡现象”。然而他的顾问当时并不同意他的结论，并且直到1970年才允许他报告他的发现。<ref>{{harvnb|Abraham|Ueda|2001|loc=See Chapters 3 and 4}}</ref><ref>{{harvnb|Sprott|2003|p=[https://books.google.com/books?id=SEDjdjPZ158C&pg=PA89 89]}}</ref>
[[File:Airplane vortex edit.jpg|thumb|left|来自飞机机翼的尖涡中的湍流。研究系统产生湍流的临界点对混沌理论很重要，例如苏联物理学家 Lev Landau 分析了这个临界点，他发展了湍流的 Landau-hopf 理论。后来 David Ruelle 和 Floris Takens 针对 Landau 预测，湍流可以通过一个奇异吸引子发展，这是混沌理论的一个主要概念]]
爱德华·洛伦茨 Edward Lorenz是这一理论的早期开拓者。他对混沌的兴趣来源于1961年他在天气预报方面的工作。<ref name=Lorenz1961>{{cite journal |author=Lorenz, Edward N. |title=Deterministic non-periodic flow |journal=Journal of the Atmospheric Sciences |volume=20 |pages=130–141 |year=1963 |doi=10.1175/1520-0469(1963)020<0130:DNF>2.0.CO;2 |issue=2 |bibcode=1963JAtS...20..130L|doi-access=free }}</ref>Lorenz正在使用一台简单的数字计算机，一台 Royal McBee LGP-30运行他的天气模拟。他想再看一次数据序列，为了节省时间，他在模拟过程中间开始了模拟。他通过输入一个打印输出的数据，这些数据对应于原始模拟中的条件。令他惊讶的是，机器开始预测的天气与以前的计算完全不同。Lorenz通过计算机打印出来的资料查到了这一点。计算机以6位数的精度工作，但打印输出的变量四舍五入到一个3位数字，所以像0.506127这样的值打印为0.506。这种差异是微小的，当时的共识是它不应该有任何实际效果。然而，Lorenz发现，初始条件的微小变化会导致长期结果的巨大变化。<ref>{{cite book|title=Chaos: Making a New Science |last=Gleick |first=James |year=1987 |publisher=Cardinal |location=London|page=17|isbn=978-0-434-29554-8|title-link=Chaos: Making a New Science }}</ref>Lorenz的发现，这给它的名字洛伦茨吸引子，表明即使详细的大气模型，一般来说，不能作出精确的长期天气预报。
1963年，[[伯努·瓦·曼德布洛特 Benoit Mandelbrot]]在棉花价格数据中发现了各种规模的循环模式。<ref>{{cite journal|first = Benoît|last = Mandelbrot|year = 1963|title = The variation of certain speculative prices|journal = Journal of Business|volume = 36|pages = 394–419|doi = 10.1086/294632|issue = 4|jstor=2350970}}</ref>事先他研究过信息理论，并得出结论，噪音的模式类似于康托集: 在任何尺度上，包含噪音的周期与无误差的周期的比例是一个常数——因此误差是不可避免的，必须通过引入冗余来计划。<ref>{{cite journal |author1=Berger J.M. |author2=Mandelbrot B. | year = 1963 | title = A new model for error clustering in telephone circuits | journal = IBM Journal of Research and Development | volume = 7 |issue=3 | pages = 224–236 | doi=10.1147/rd.73.0224}}</ref>Mandelbrot既描述了“诺亚效应”(可能会发生突然的不连续变化) ，也描述了“约瑟夫效应 Joseph effect”(可能会持续一段时间，然后突然改变)。<ref>{{cite book |author=Mandelbrot, B. |title=The Fractal Geometry of Nature |publisher=Freeman |location=New York |year=1977 |page=248 }}</ref><ref>See also: {{cite book |last1=Mandelbrot |first1=Benoît B. |last2=Hudson |first2=Richard L. |title=The (Mis)behavior of Markets: A Fractal View of Risk, Ruin, and Reward |url=https://archive.org/details/misbehaviorofmar00beno |url-access=registration |publisher=Basic Books |location=New York |year=2004 |page=[https://archive.org/details/misbehaviorofmar00beno/page/201 201] }}</ref>这挑战了价格变化是正态分布的观点。1967年，他出版了《英国的海岸线有多长？统计自相似性和分维数”，显示海岸线的长度随测量仪器的规模而变化，在所有规模上都与仪器相似，而且对于一个极小的测量装置来说，长度是无限的。<ref>{{cite journal |last=Mandelbrot |first=Benoît |title=How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension |journal=Science |volume=156 |issue=3775 |pages=636–8 |date=5 May 1967 |doi=10.1126/science.156.3775.636 |pmid=17837158 |bibcode = 1967Sci...156..636M }}</ref>他认为，从远处(0维)看，一个线球似乎是一个点，从相当接近(3维)看，一个球，或一个曲线(一维) ，他认为，一个物体的尺寸是相对于观察者，可能是分数。如果一个物体的不规则性在不同尺度上是常数(“自相似”) ，那么这个物体就是一个分形(例如门格尔海绵 the Menger sponge、赛尔皮滑雪垫圈 the Sierpiński gasket和科氏曲线或雪花 Koch curve or snowflake，它们是无限长的，但覆盖着一个有限的空间，分形维数约为1.2619)。1982年，Mandelbrot发表了《自然的分形几何》 ，成为混沌理论的经典之作。<ref>{{cite book|author=Mandelbrot, B.|title=The Fractal Geometry of Nature|date=1982|place=New York|publisher=Macmillan|isbn=978-0716711865|url=https://archive.org/details/fractalgeometryo00beno}}</ref>生物学系统，如循环系统和支气管系统的分支证明符合一个分形模型。<ref>{{cite book |last1=Buldyrev |first1=S.V. |last2=Goldberger |first2=A.L. | last3=Havlin |first3=S. | authorlink3=Shlomo Havlin|last4=Peng |first4=C.K. | last5=Stanley | first5=H.E. |authorlink5=H. Eugene Stanley|editor1-first=Armin |editor1-last=Bunde |editor2-first=Shlomo | editor2-last=Havlin | editor2-link=Shlomo Havlin |title=Fractals in Science |publisher=Springer |year=1994 |pages=49–89 |chapter=Fractals in Biology and Medicine: From DNA to the Heartbeat |isbn=978-3-540-56220-7}}</ref>
1977年12月，纽约科学院组织了第一次关于混沌的研讨会，出席的有大卫·鲁尔 David Ruelle、[[罗伯特·梅 Robert May]]、詹姆斯·a·约克 James A. Yorke(数学中“混沌”一词的创始人)、罗伯特·肖 Robert Shaw和气象学家爱德华·洛伦茨 Edward Lorenz。第二年 Pierre Coullet 和 Charles Tresser 发表了《迭代与重整化群体 Iterations d'endomorphismes et groupe de renormalisation》 ，米切尔·费根鲍姆 Mitchell Feigenbaum的文章《一类非线性变换的定量普遍性 Quantitative Universality for a Class of Nonlinear Transformations》最终发表在一本杂志上，经过三年的裁判拒绝。<ref name="Feigenbaum 25–52"/><ref>Coullet, Pierre, and Charles Tresser. "Iterations d'endomorphismes et groupe de renormalisation." Le Journal de Physique Colloques 39.C5 (1978): C5-25</ref> 因此 Feigenbaum (1975)和 Coullet & Tresser (1978)发现了混沌中的普遍性，允许混沌理论应用于许多不同的现象。
1979年，Albert j. Libchaber 在皮埃尔·奥昂贝格 Pierre Hohenberg在阿斯彭组织的一次研讨会上，提出了他对瑞利-贝纳德对流系统 Rayleigh–Bénard convection中导致混沌和湍流的分叉理论级联的实验观察。1986年，由于他们令人鼓舞的成就，他和Mitchell Feigenbaum一起被授予沃尔夫物理学奖。
1986年，纽约科学院与国家心理健康研究所和海军研究办公室共同组织了第一次关于生物学和医学中的混沌的重要会议。在那里，贝尔纳多·休伯曼 Bernardo Huberman提出了一个精神分裂症患者眼球追踪障碍的数学模型。<ref>{{cite journal |authorlink=Bernardo Huberman |author=Huberman, B.A. |title=A Model for Dysfunctions in Smooth Pursuit Eye Movement |journal=Annals of the New York Academy of Sciences |volume=504 Perspectives in Biological Dynamics and Theoretical Medicine |issue=1 |pages=260–273 |date=July 1987 |doi=10.1111/j.1749-6632.1987.tb48737.x |pmid=3477120 |bibcode = 1987NYASA.504..260H }}</ref>这导致了生理学的更新在20世纪80年代通过应用混沌理论，例如，在病理心脏周期的研究。
1987年，Per Bak，Chao Tang 和 Kurt Wiesenfeld 在《物理评论快报 Physical Review Letters》上发表了一篇论文，<ref>{{cite journal |author1=Bak, Per |author2=Tang, Chao |author3=Wiesenfeld, Kurt |title=Self-organized criticality: An explanation of the 1/f noise |journal=Physical Review Letters |volume=59 |issue=4 |pages=381–4 |date=27 July 1987 |doi=10.1103/PhysRevLett.59.381 |pmid=10035754 |bibcode=1987PhRvL..59..381B}} However, the conclusions of this article have been subject to dispute. {{cite web|url=http://www.nslij-genetics.org/wli/1fnoise/1fnoise_square.html |title=? |url-status=dead |archiveurl=https://web.archive.org/web/20071214033929/https://www.nslij-genetics.org/wli/1fnoise/1fnoise_square.html |archivedate=2007-12-14 }}. See especially: {{cite journal |author1=Laurson, Lasse |author2=Alava, Mikko J. |author3=Zapperi, Stefano |title=Letter: Power spectra of self-organized critical sand piles |journal=Journal of Statistical Mechanics: Theory and Experiment |volume=0511 |id=L001 |date=15 September 2005 }}</ref>首次描述了自组织临界性，认为自然界复杂性产生的机制之一。
除了大量基于实验室的方法，如 Bak-Tang-Wiesenfeld 沙堆法 Bak–Tang–Wiesenfeld sandpile，许多其他的研究都集中在大规模的自然或社会系统上，这些系统已知(或怀疑)表现出尺度不变的行为。虽然这些方法并不总是受到研究对象专家的欢迎(至少最初是这样) ，但 SOC 已经成为解释一些自然现象的有力候选者，包括地震(早在 SOC 被发现之前，它就被认为是尺度不变行为的来源，例如描述地震大小统计分布的古腾堡-里克特定律 Gutenberg–Richter law，描述余震频率的 Omori 定律) ，<ref>{{cite journal |author=Omori, F. |title=On the aftershocks of earthquakes |journal=Journal of the College of Science, Imperial University of Tokyo |volume=7 |pages=111–200 |year=1894 }}</ref> 太阳，经济系统的波动，例如金融市场(SOC 在经济物理学中很常见) ，地貌形成，森林火灾，流行病和生物进化 (在 SOC 被调用的地方，例如 Niles Eldredge 和史蒂芬·古尔德 Stephen Jay Gould提出的“间断平衡”理论背后的动力机制)。考虑到事件规模无标度分布的含义，一些研究人员提出，另一个应该被视为 SOC 的例子的现象是战争的发生。对土壤有机碳的这些研究既包括建立模型的尝试(开发新模型或使现有模型适应特定自然系统的具体情况) ，也包括广泛的数据分析，以确定自然定标法的存在和 / 或特点。
同年，詹姆斯•格莱克 James Gleick出版了《混沌: 创造新科学 Chaos: Making a New Science》一书，成为畅销书，并向广大公众介绍了混沌理论的一般原理及其历史，但他对苏联的重要贡献重视不够。<ref>{{cite book|last=Gleick|first=James|title=Chaos: Making a New Science|date=August 26, 2008|publisher=Penguin Books|isbn=978-0143113454}}</ref> 混沌理论最初只是少数孤立个体的领域，逐渐成为一门跨学科和制度学科，主要以非线性系统分析的名义出现。许多“混沌论者”(正如一些人自己描述的那样)提到了 Thomas Kuhn 在《科学革命的结构 The Structure of Scientific Revolutions》(1962)提出的范式转换的概念，声称这个新理论就是这种转换的一个例子，这是 Gleick 支持的一个论点。
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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