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此词条暂由彩云小译翻译，未经人工整理和审校，带来阅读不便，请见谅。{{short description|Field of mathematics}}

{{other uses|Chaos theory (disambiguation)|Chaos (disambiguation)}}

[[File:Lorenz attractor yb.svg|thumb|right|A plot of the [[Lorenz attractor]] for values {{nowrap|''r'' {{=}} 28}}, {{nowrap|σ {{=}} 10}}, {{nowrap|''b'' {{=}} 8/3}}]]

A plot of the [[Lorenz attractor for values 28}}, 10}}, 8/3}}]]

[[[洛伦兹值28} ，10} ，8 / 3}]吸引子的一个图

[[File:Double-compound-pendulum.gif|thumb|A animation of a [[double pendulum|double-rod pendulum]] at an intermediate energy showing chaotic behavior. Starting the pendulum from a slightly different [[initial condition]] would result in a vastly different [[trajectory]]. The double-rod pendulum is one of the simplest dynamical systems with chaotic solutions.]]

double-rod pendulum at an intermediate energy showing chaotic behavior. Starting the pendulum from a slightly different initial condition would result in a vastly different trajectory. The double-rod pendulum is one of the simplest dynamical systems with chaotic solutions.]]

中等能量的双杆摆表现出混沌行为。从稍微不同的初始条件开始摆动会导致一个完全不同的轨迹。双杆摆是最简单的具有混沌解的动力系统之一

'''Chaos theory''' is a branch of [[mathematics]] focusing on the study of chaos—states of [[dynamical system]]s whose apparently-random states of disorder and irregularities are often governed by deterministic laws that are highly sensitive to [[initial conditions]].<ref>{{Cite web|url=https://mathvault.ca/math-glossary/#chaos|title=The Definitive Glossary of Higher Mathematical Jargon — Chaos|last=|first=|date=2019-08-01|website=Math Vault|language=en-US|url-status=live|archive-url=|archive-date=|access-date=2019-11-24}}</ref><ref>{{Cite web|url=https://www.britannica.com/science/chaos-theory|title=chaos theory {{!}} Definition & Facts|website=Encyclopedia Britannica|language=en|access-date=2019-11-24}}</ref> Chaos theory is an interdisciplinary theory stating that, within the apparent randomness of [[chaotic complex system]]s, there are underlying patterns, interconnectedness, constant [[feedback loops]], repetition, [[self-similarity]], [[fractals]], and [[self-organization]].<ref name=":1">{{Cite web|url=https://fractalfoundation.org/resources/what-is-chaos-theory/|title=What is Chaos Theory? – Fractal Foundation|language=en-US|access-date=2019-11-24}}</ref> The [[butterfly effect]], an underlying principle of chaos, describes how a small change in one state of a [[Deterministic system|deterministic]] [[Nonlinear system|nonlinear]] system can result in large differences in a later state (meaning that there is sensitive dependence on initial conditions).<ref>{{Cite web|url=http://mathworld.wolfram.com/Chaos.html|title=Chaos|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-11-24}}</ref> A metaphor for this behavior is that a butterfly flapping its wings in China can cause a hurricane in Texas.

Chaos theory is a branch of mathematics focusing on the study of chaos—states of dynamical systems whose apparently-random states of disorder and irregularities are often governed by deterministic laws that are highly sensitive to initial conditions. Chaos theory is an interdisciplinary theory stating that, within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnectedness, constant feedback loops, repetition, self-similarity, fractals, and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning that there is sensitive dependence on initial conditions). A metaphor for this behavior is that a butterfly flapping its wings in China can cause a hurricane in Texas.

混沌理论是数学的一个分支，主要研究动力系统的混沌状态，其表面无序和不规则的随机状态往往受到对初始条件高度敏感的确定性规律的支配。混沌理论是一个跨学科的理论，表明，在混沌复杂系统的明显随机性，有潜在的模式，相互联系，不断的反馈循环，重复，自相似，分形和自我组织。蝴蝶效应，一个基本的混沌原理，描述了如何在一个确定性非线性的一个状态的微小变化可以导致大的差异在后来的状态(意味着有对初始条件的敏感依赖)。这种行为的一个隐喻是，一只蝴蝶在中国扇动它的翅膀可以引起德克萨斯州的飓风。

Small differences in initial conditions, such as those due to rounding errors in numerical computation, can yield widely diverging outcomes for such dynamical systems, rendering long-term prediction of their behavior impossible in general.<ref>{{cite book |last = Kellert |first = Stephen H. |title = In the Wake of Chaos: Unpredictable Order in Dynamical Systems |url = https://archive.org/details/inwakeofchaosunp0000kell |url-access = registration |publisher = University of Chicago Press |year = 1993 |isbn = 978-0-226-42976-2 |page = [https://archive.org/details/inwakeofchaosunp0000kell/page/32 32] |ref = harv }}</ref> This can happen even though these systems are [[deterministic system (mathematics)|deterministic]], meaning that their future behavior follows a unique evolution<ref name=":2">{{Citation|last=Bishop|first=Robert|title=Chaos|date=2017|url=https://plato.stanford.edu/archives/spr2017/entries/chaos/|encyclopedia=The Stanford Encyclopedia of Philosophy|editor-last=Zalta|editor-first=Edward N.|edition=Spring 2017|publisher=Metaphysics Research Lab, Stanford University|access-date=2019-11-24}}</ref> and is fully determined by their initial conditions, with no [[randomness|random]] elements involved.<ref>{{harvnb|Kellert|1993|p=56}}</ref> In other words, the deterministic nature of these systems does not make them predictable.<ref>{{harvnb|Kellert|1993|p=62}}</ref><ref name="WerndlCharlotte">{{cite journal |author = Werndl, Charlotte |title = What are the New Implications of Chaos for Unpredictability? |journal = The British Journal for the Philosophy of Science |volume = 60 |issue = 1 |pages = 195–220 |year = 2009 |doi = 10.1093/bjps/axn053 |arxiv = 1310.1576 }}</ref> This behavior is known as '''deterministic chaos''', or simply '''chaos'''. The theory was summarized by [[Edward Lorenz]] as:<ref>{{cite web |url = http://mpe.dimacs.rutgers.edu/2013/03/17/chaos-in-an-atmosphere-hanging-on-a-wall/ |title = Chaos in an Atmosphere Hanging on a Wall |last1 = Danforth |first1 = Christopher M. |date = April 2013 |work = Mathematics of Planet Earth 2013 |accessdate = 12 June 2018 }}</ref>

Small differences in initial conditions, such as those due to rounding errors in numerical computation, can yield widely diverging outcomes for such dynamical systems, rendering long-term prediction of their behavior impossible in general. This can happen even though these systems are deterministic, meaning that their future behavior follows a unique evolution and is fully determined by their initial conditions, with no random elements involved. In other words, the deterministic nature of these systems does not make them predictable. This behavior is known as deterministic chaos, or simply chaos. The theory was summarized by Edward Lorenz as:

初始条件中的微小差异，例如数值计算中的舍入误差，可能导致此类动力系统的结果差异很大，使得对其行为的长期预测通常是不可能的。即使这些系统是确定性的，这意味着它们未来的行为遵循一个独特的演变，完全由它们的初始条件决定，没有任何随机因素参与。换句话说，这些系统的确定性本质并不能使它们具有可预测性。这种行为被称为确定性混沌，或简单的混沌。爱德华 · 洛伦茨将这一理论总结为:

{{Quote|Chaos: When the present determines the future, but the approximate present does not approximately determine the future.}}

Chaotic behavior exists in many natural systems, including fluid flow, heartbeat irregularities, [[weather and climate]].<ref name=Lorenz1961/><ref>{{cite book |last = Ivancevic |first = Vladimir G. |title = Complex nonlinearity: chaos, phase transitions, topology change, and path integrals |year = 2008 |publisher = Springer |isbn = 978-3-540-79356-4 |author2 = Tijana T. Ivancevic }}</ref><ref name=":2" /> It also occurs spontaneously in some systems with artificial components, such as the [[stock market]] and [[road traffic]].<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><ref name=":1" /> This behavior can be studied through the analysis of a chaotic [[mathematical model]], or through analytical techniques such as [[recurrence plot]]s and [[Poincaré map]]s. Chaos theory has applications in a variety of disciplines, including [[meteorology]],<ref name=":2" /> [[anthropology]],<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> [[sociology]], [[physics]],<ref>{{cite journal|last1=Hubler|first1=A|title=Adaptive control of chaotic systems|journal=Swiss Physical Society. Helvetica Physica Acta 62|date=1989|pages=339–342}}</ref> [[environmental science]], [[computer science]], [[engineering]], [[economics]], [[biology]], [[ecology]], and [[philosophy]]. The theory formed the basis for such fields of study as [[dynamical systems|complex dynamical systems]], [[edge of chaos]] theory, and [[self-assembly]] processes.

Chaotic behavior exists in many natural systems, including fluid flow, heartbeat irregularities, weather and climate. sociology, physics, environmental science, computer science, engineering, economics, biology, ecology, and philosophy. The theory formed the basis for such fields of study as complex dynamical systems, edge of chaos theory, and self-assembly processes.

混沌现象存在于许多自然系统中，包括流体流动、心跳异常、天气和气候。社会学、物理学、环境科学、计算机科学、工程学、经济学、生物学、生态学和哲学。该理论为复杂动力系统、混沌边缘理论和自组装过程等领域的研究奠定了基础。

==Introduction==

==Introduction==

引言

Chaos theory concerns deterministic systems whose behavior can in principle be predicted. Chaotic systems are predictable for a while and then 'appear' to become random. The amount of time that the behavior of a chaotic system can be effectively predicted depends on three things: how much uncertainty can be tolerated in the forecast, how accurately its current state can be measured, and a time scale depending on the dynamics of the system, called the [[Lyapunov time]]. Some examples of Lyapunov times are: chaotic electrical circuits, about 1 millisecond; weather systems, a few days (unproven); the inner solar system, 4 to 5 million years.<ref>{{Cite journal|last=Wisdom|first=Jack|last2=Sussman|first2=Gerald Jay|date=1992-07-03|title=Chaotic Evolution of the Solar System|journal=Science|language=en|volume=257|issue=5066|pages=56–62|doi=10.1126/science.257.5066.56|issn=1095-9203|pmid=17800710|bibcode=1992Sci...257...56S|hdl=1721.1/5961|hdl-access=free}}</ref> In chaotic systems, the uncertainty in a forecast increases [[Exponential growth|exponentially]] with elapsed time. Hence, mathematically, doubling the forecast time more than squares the proportional uncertainty in the forecast. This means, in practice, a meaningful prediction cannot be made over an interval of more than two or three times the Lyapunov time. When meaningful predictions cannot be made, the system appears random.<ref>''Sync: The Emerging Science of Spontaneous Order'', Steven Strogatz, Hyperion, New York, 2003, pages 189–190.</ref>

Chaos theory concerns deterministic systems whose behavior can in principle be predicted. Chaotic systems are predictable for a while and then 'appear' to become random. The amount of time that the behavior of a chaotic system can be effectively predicted depends on three things: how much uncertainty can be tolerated in the forecast, how accurately its current state can be measured, and a time scale depending on the dynamics of the system, called the Lyapunov time. Some examples of Lyapunov times are: chaotic electrical circuits, about 1 millisecond; weather systems, a few days (unproven); the inner solar system, 4 to 5 million years. In chaotic systems, the uncertainty in a forecast increases exponentially with elapsed time. Hence, mathematically, doubling the forecast time more than squares the proportional uncertainty in the forecast. This means, in practice, a meaningful prediction cannot be made over an interval of more than two or three times the Lyapunov time. When meaningful predictions cannot be made, the system appears random.

混沌理论关注确定性系统，其行为原则上可以预测。混沌系统在一段时间内是可以预测的，然后“显现”成为随机的。一个混沌系统的行为能够被有效预测的时间取决于三个因素: 预测中能够容忍的不确定性有多大，其当前状态能够被测量的有多准确，以及一个取决于系统动力学的时间尺度，称为李雅普诺夫时间。李雅普诺夫时间的一些例子是: 混沌电路，大约1毫秒; 天气系统，几天(未经证实) ; 内太阳系，400万到500万年。在混沌系统中，预报中的不确定性随着时间的流逝呈指数增长。因此，从数学上来说，预测时间要比预测中比例不确定性的平方多一倍。这意味着，在实践中，一个有意义的预测不能超过两到三倍的李雅普诺夫时间间隔。当不能做出有意义的预测时，系统就会显得随机。

==Chaotic dynamics==

==Chaotic dynamics==

混沌动力学

[[File:Chaos Sensitive Dependence.svg|thumb|The map defined by ''x'' → 4 ''x'' (1 – ''x'') and ''y'' → (''x'' + ''y)'' [[Modulo operation|mod]] 1 displays sensitivity to initial x positions. Here, two series of ''x'' and ''y'' values diverge markedly over time from a tiny initial difference.]]

mod 1 displays sensitivity to initial x positions. Here, two series of x and y values diverge markedly over time from a tiny initial difference.]]

Mod 1 / span 显示对初始 x 位置的敏感度。在这里，两组 x 和 y 值随着时间的推移从一个微小的初始差异显著分化。]]

In common usage, "chaos" means "a state of disorder".<ref>Definition of {{linktext|chaos}} at [[Wiktionary]];</ref><ref>{{Cite web|url=https://www.dictionary.com/browse/chaos|title=Definition of chaos {{!}} Dictionary.com|website=www.dictionary.com|language=en|access-date=2019-11-24}}</ref> However, in chaos theory, the term is defined more precisely. Although no universally accepted mathematical definition of chaos exists, a commonly used definition, originally formulated by [[Robert L. Devaney]], says that to classify a dynamical system as chaotic, it must have these properties:<ref>{{cite book|title=A First Course in Dynamics: With a Panorama of Recent Developments|last=Hasselblatt|first=Boris|author2=Anatole Katok|year=2003|publisher=Cambridge University Press|isbn=978-0-521-58750-1}}</ref>

In common usage, "chaos" means "a state of disorder". However, in chaos theory, the term is defined more precisely. Although no universally accepted mathematical definition of chaos exists, a commonly used definition, originally formulated by Robert L. Devaney, says that to classify a dynamical system as chaotic, it must have these properties:

在通常的用法中，“混沌”意味着“无序的状态”。然而，在混沌理论中，这个术语的定义更为精确。尽管没有一个被广泛接受的关于混沌的数学定义，一个最初由 Robert l. Devaney 提出的常用定义认为，要把动力系统分类为混沌，它必须具备以下特性:

# it must be [[sensitive dependence on initial conditions|sensitive to initial conditions]],

it must be sensitive to initial conditions,

它必须对初始条件很敏感,

# it must be [[Mixing_(mathematics)#Topological_mixing|topologically transitive]],

it must be topologically transitive,

必须具有拓扑传递性,

# it must have [[dense set|dense]] [[periodic orbit]]s.

it must have dense periodic orbits.

它一定有稠密的周期轨道。

In some cases, the last two properties above have been shown to actually imply sensitivity to initial conditions.<ref>{{cite book |author=Elaydi, Saber N. |title=Discrete Chaos |publisher=Chapman & Hall/CRC |year=1999 |isbn=978-1-58488-002-8 |page=117 }}</ref><ref>{{cite book |author=Basener, William F. |title=Topology and its applications |publisher=Wiley |year=2006 |isbn=978-0-471-68755-9 |page=42 }}</ref> In these cases, while it is often the most practically significant property, "sensitivity to initial conditions" need not be stated in the definition.

In some cases, the last two properties above have been shown to actually imply sensitivity to initial conditions. In these cases, while it is often the most practically significant property, "sensitivity to initial conditions" need not be stated in the definition.

在某些情况下，上面提到的最后两个性质实际上暗示了对初始条件的敏感性。在这些情况下，虽然它往往是最重要的实际特性，但定义中不必说明“对初始条件的敏感性”。

If attention is restricted to [[Interval (mathematics)|intervals]], the second property implies the other two.<ref>{{cite journal |author1=Vellekoop, Michel |author2=Berglund, Raoul |title=On Intervals, Transitivity = Chaos |journal=The American Mathematical Monthly |volume=101 |issue=4 |pages=353–5 |date=April 1994 |jstor=2975629 |doi=10.2307/2975629}}</ref> An alternative and a generally weaker definition of chaos uses only the first two properties in the above list.<ref>{{cite book |author1=Medio, Alfredo |author2=Lines, Marji |title=Nonlinear Dynamics: A Primer |publisher=Cambridge University Press |year=2001 |isbn=978-0-521-55874-7 |page=165 }}</ref>

If attention is restricted to intervals, the second property implies the other two. An alternative and a generally weaker definition of chaos uses only the first two properties in the above list.

如果注意力被限制在时间间隔内，那么第二个性质就意味着另外两个性质。混沌的另一种定义和一般较弱的定义只使用了上面列表中的前两个属性。

=== Chaos as a spontaneous breakdown of topological supersymmetry ===

=== Chaos as a spontaneous breakdown of topological supersymmetry ===

混沌作为拓扑超对称性的自发破坏

{{Main|Supersymmetric theory of stochastic dynamics}}In continuous time dynamical systems, chaos is the phenomenon of the spontaneous breakdown of topological supersymmetry, which is an intrinsic property of evolution operators of all stochastic and deterministic (partial) differential equations.<ref>{{cite journal|date=March 2016|title=Introduction to Supersymmetric Theory of Stochastics|journal=Entropy|volume=18|issue=4|pages=108|doi=10.3390/e18040108|author=Ovchinnikov, I.V.|arxiv = 1511.03393 |bibcode = 2016Entrp..18..108O }}</ref><ref>{{cite journal|year=2016|title=Topological supersymmetry breaking: Definition and stochastic generalization of chaos and the limit of applicability of statistics|journal=Modern Physics Letters B|volume=30|issue=8|pages=1650086|doi=10.1142/S021798491650086X|author1=Ovchinnikov, I.V.|author2=Schwartz, R. N.|author3=Wang, K. L.|bibcode = 2016MPLB...3050086O |arxiv=1404.4076}}</ref> This picture of dynamical chaos works not only for deterministic models, but also for models with external noise which is an important generalization from the physical point of view, since in reality, all dynamical systems experience influence from their stochastic environments. Within this picture, the long-range dynamical behavior associated with chaotic dynamics (e.g., the [[butterfly effect]]) is a consequence of the [[Goldstone's theorem]]—in the application to the spontaneous topological supersymmetry breaking.

In continuous time dynamical systems, chaos is the phenomenon of the spontaneous breakdown of topological supersymmetry, which is an intrinsic property of evolution operators of all stochastic and deterministic (partial) differential equations. This picture of dynamical chaos works not only for deterministic models, but also for models with external noise which is an important generalization from the physical point of view, since in reality, all dynamical systems experience influence from their stochastic environments. Within this picture, the long-range dynamical behavior associated with chaotic dynamics (e.g., the butterfly effect) is a consequence of the Goldstone's theorem—in the application to the spontaneous topological supersymmetry breaking.

在连续时间动力系统中，混沌是拓扑超对称性的自发破坏现象，是所有随机和确定性(偏)微分方程的发展算子的内在性质。这种动态混沌图像不仅适用于确定性模型，也适用于有外部噪声的模型。外部噪声是物理学上的一个重要概括，因为在现实中，所有的动态系统都受到其随机环境的影响。在这幅图中，与混沌动力学相关的远程动力学行为(例如，蝴蝶效应)是戈德斯通定理的结果ーー在自发拓扑超对称破缺中的应用。

===Sensitivity to initial conditions===

===Sensitivity to initial conditions===

对初始条件的敏感性

{{Main|Butterfly effect}}

[[File:SensInitCond.gif|thumb|Lorenz equations used to generate plots for the y variable. The initial conditions for ''x'' and ''z'' were kept the same but those for ''y'' were changed between '''1.001''', '''1.0001''' and '''1.00001'''. The values for <math>\rho</math>, <math>\sigma</math> and <math>\beta</math> were '''45.92''', '''16''' and '''4 ''' respectively. As can be seen from the graph, even the slightest difference in initial values causes significant changes after about 12 seconds of evolution in the three cases. This is an example of sensitive dependence on initial conditions.]]

Lorenz equations used to generate plots for the y variable. The initial conditions for x and z were kept the same but those for y were changed between 1.001, 1.0001 and 1.00001. The values for <math>\rho</math>, <math>\sigma</math> and <math>\beta</math> were 45.92, 16 and 4 respectively. As can be seen from the graph, even the slightest difference in initial values causes significant changes after about 12 seconds of evolution in the three cases. This is an example of sensitive dependence on initial conditions.

用于绘制 y 变量图的洛伦兹方程。X 和 z 的初始条件保持不变，y 的初始条件在1.001、1.0001和1.00001之间变化。数学 / 数学、数学 / 数学和数学 / 测验 / 数学的数值分别为45.92、16和4。从图表中可以看出，在这三种情况下，即使初始值的最细微差别也会在大约12秒的进化后引起重大变化。这是对初始条件敏感依赖的一个例子。

'''Sensitivity to initial conditions''' means that each point in a chaotic system is arbitrarily closely approximated by other points, with significantly different future paths or trajectories. Thus, an arbitrarily small change or perturbation of the current trajectory may lead to significantly different future behavior.<ref name=":1" />

Sensitivity to initial conditions means that each point in a chaotic system is arbitrarily closely approximated by other points, with significantly different future paths or trajectories. Thus, an arbitrarily small change or perturbation of the current trajectory may lead to significantly different future behavior.

对初始条件的敏感性意味着一个混沌系统中的每个点都被其他点任意地近似，具有明显不同的未来路径或轨迹。因此，任意小的改变或者对当前轨迹的扰动都可能导致明显不同的未来行为。

Sensitivity to initial conditions is popularly known as the "[[butterfly effect]]", so-called because of the title of a paper given by [[Edward Lorenz]] in 1972 to the [[American Association for the Advancement of Science]] in Washington, D.C., entitled ''Predictability: Does the Flap of a Butterfly's Wings in Brazil set off a Tornado in Texas?''.<ref>{{Cite web|url=http://news.mit.edu/2008/obit-lorenz-0416|title=Edward Lorenz, father of chaos theory and butterfly effect, dies at 90|website=MIT News|access-date=2019-11-24}}</ref> The flapping wing represents a small change in the initial condition of the system, which causes a chain of events that prevents the predictability of large-scale phenomena. Had the butterfly not flapped its wings, the trajectory of the overall system could have been vastly different.

Sensitivity to initial conditions is popularly known as the "butterfly effect", so-called because of the title of a paper given by Edward Lorenz in 1972 to the American Association for the Advancement of Science in Washington, D.C., entitled Predictability: Does the Flap of a Butterfly's Wings in Brazil set off a Tornado in Texas?. The flapping wing represents a small change in the initial condition of the system, which causes a chain of events that prevents the predictability of large-scale phenomena. Had the butterfly not flapped its wings, the trajectory of the overall system could have been vastly different.

对初始条件的敏感性通常被称为“蝴蝶效应” ，这是因为 Edward Lorenz 在1972年给华盛顿特区的美国美国科学进步协会学会的一篇题为《可预测性: 巴西蝴蝶翅膀的扇动是否会在德克萨斯州引发龙卷风》的论文的标题？ .扑翼代表了系统初始条件的一个小的变化，这导致了一系列的事件，阻止了大规模现象的可预测性。如果蝴蝶没有扇动翅膀，整个系统的轨迹可能会大不相同。

A consequence of sensitivity to initial conditions is that if we start with a limited amount of information about the system (as is usually the case in practice), then beyond a certain time, the system would no longer be predictable. This is most prevalent in the case of weather, which is generally predictable only about a week ahead.<ref name="RGW">{{cite book |author=Watts, Robert G. |title=Global Warming and the Future of the Earth |publisher=Morgan & Claypool |year=2007 |page=17 }}</ref> This does not mean that one cannot assert anything about events far in the future—only that some restrictions on the system are present. For example, we do know with weather that the temperature will not naturally reach 100 °C or fall to −130 °C on earth (during the current [[geologic era]]), but that does not mean that we can predict exactly which day will have the hottest temperature of the year.

A consequence of sensitivity to initial conditions is that if we start with a limited amount of information about the system (as is usually the case in practice), then beyond a certain time, the system would no longer be predictable. This is most prevalent in the case of weather, which is generally predictable only about a week ahead. This does not mean that one cannot assert anything about events far in the future—only that some restrictions on the system are present. For example, we do know with weather that the temperature will not naturally reach 100 °C or fall to −130 °C on earth (during the current geologic era), but that does not mean that we can predict exactly which day will have the hottest temperature of the year.

对初始条件敏感的一个后果是，如果我们从有限数量的系统信息开始(在实践中通常是这样) ，然后超过一定的时间，系统将不再是可预测的。这种情况在天气情况下最为普遍，而天气一般只能预测一周之后的情况。这并不意味着人们不能断言任何关于遥远未来的事件ーー只是对系统存在一些限制。例如，我们确实知道，在当前的地质年代，地球的温度不会自然达到100摄氏度或降至 -130摄氏度，但这并不意味着我们能够准确预测一年中哪一天的温度最高。

In more mathematical terms, the [[Lyapunov exponent]] measures the sensitivity to initial conditions, in the form of rate of exponential divergence from the perturbed initial conditions.<ref>{{Cite web|url=http://mathworld.wolfram.com/LyapunovCharacteristicExponent.html|title=Lyapunov Characteristic Exponent|last=Weisstein|first=Eric W.|website=mathworld.wolfram.com|language=en|access-date=2019-11-24}}</ref> More specifically, given two starting [[trajectory|trajectories]] in the [[phase space]] that are infinitesimally close, with initial separation <math>\delta \mathbf{Z}_0</math>, the two trajectories end up diverging at a rate given by

In more mathematical terms, the Lyapunov exponent measures the sensitivity to initial conditions, in the form of rate of exponential divergence from the perturbed initial conditions. More specifically, given two starting trajectories in the phase space that are infinitesimally close, with initial separation <math>\delta \mathbf{Z}_0</math>, the two trajectories end up diverging at a rate given by

用更精确的数学术语来说，李亚普诺夫指数方法测量了对初始条件的敏感度，以指数发散率的形式从扰动的初始条件。更具体地说，给定相空间中无穷小相近的两个起始轨迹，利用初始分离数学 δz }0 / math，这两个轨迹最终以给定的速率发散

:<math> | \delta\mathbf{Z}(t) | \approx e^{\lambda t} | \delta \mathbf{Z}_0 |,</math>

<math> | \delta\mathbf{Z}(t) | \approx e^{\lambda t} | \delta \mathbf{Z}_0 |,</math>

数学 | delta mathbf { z }(t) | | approx ^ { lambda t } | delta mathbf { z }0 | ，/ math

where <math>t</math> is the time and <math>\lambda</math> is the Lyapunov exponent. The rate of separation depends on the orientation of the initial separation vector, so a whole spectrum of Lyapunov exponents can exist. The number of Lyapunov exponents is equal to the number of dimensions of the phase space, though it is common to just refer to the largest one. For example, the maximal Lyapunov exponent (MLE) is most often used, because it determines the overall predictability of the system. A positive MLE is usually taken as an indication that the system is chaotic.<ref name=":2" />

where <math>t</math> is the time and <math>\lambda</math> is the Lyapunov exponent. The rate of separation depends on the orientation of the initial separation vector, so a whole spectrum of Lyapunov exponents can exist. The number of Lyapunov exponents is equal to the number of dimensions of the phase space, though it is common to just refer to the largest one. For example, the maximal Lyapunov exponent (MLE) is most often used, because it determines the overall predictability of the system. A positive MLE is usually taken as an indication that the system is chaotic.

其中 math t / math 是时间 math lambda / math 是李亚普诺夫指数。分离速率取决于初始分离向量的方向，因此可以存在一个完整的李雅普诺夫指数谱。李雅普诺夫指数的个数等于相空间的维数，尽管通常只是指最大的维数。例如，最大李亚普诺夫指数是最常用的，因为它决定了系统的整体可预测性。正极大似然估计通常被认为是系统混沌的表现。

In addition of the above property, other properties related to sensitivity of initial conditions also exist. These include, for example, [[Measure (mathematics)|measure-theoretical]] [[Mixing (mathematics)|mixing]] (as discussed in [[ergodicity|ergodic]] theory) and properties of a [[Kolmogorov automorphism|K-system]].<ref name="WerndlCharlotte" />

In addition of the above property, other properties related to sensitivity of initial conditions also exist. These include, for example, measure-theoretical mixing (as discussed in ergodic theory) and properties of a K-system.

除了上述性质外，还存在与初始条件敏感性有关的其他性质。这些包括，例如，测量理论混合(如遍历理论中所讨论的)和 k 系统的性质。

===Non-periodicity===

===Non-periodicity===

非周期性

A chaotic system may have sequences of values for the evolving variable that exactly repeat themselves, giving periodic behavior starting from any point in that sequence. However, such periodic sequences are repelling rather than attracting, meaning that if the evolving variable is outside the sequence, however close, it will not enter the sequence and in fact will diverge from it. Thus for [[almost all]] initial conditions, the variable evolves chaotically with non-periodic behavior.

A chaotic system may have sequences of values for the evolving variable that exactly repeat themselves, giving periodic behavior starting from any point in that sequence. However, such periodic sequences are repelling rather than attracting, meaning that if the evolving variable is outside the sequence, however close, it will not enter the sequence and in fact will diverge from it. Thus for almost all initial conditions, the variable evolves chaotically with non-periodic behavior.

一个混沌系统可能具有演化变量的值序列，这些值精确地重复自己，从序列中的任何一点开始给出周期性行为。然而，这样的周期序列是排斥而不是吸引，这意味着如果演化变量在序列之外，无论多么接近，它都不会进入序列，事实上将偏离序列。因此，对于几乎所有的初始条件，变量的演化是混沌的，具有非周期性的行为。

===Topological mixing===

===Topological mixing===

拓扑混合

[[File:LogisticTopMixing1-6.gif|thumb|Six iterations of a set of states <math>[x,y]</math> passed through the logistic map. The first iterate (blue) is the initial condition, which essentially forms a circle. Animation shows the first to the sixth iteration of the circular initial conditions. It can be seen that ''mixing'' occurs as we progress in iterations. The sixth iteration shows that the points are almost completely scattered in the phase space. Had we progressed further in iterations, the mixing would have been homogeneous and irreversible. The logistic map has equation <math>x_{k+1} = 4 x_k (1 - x_k )</math>. To expand the state-space of the logistic map into two dimensions, a second state, <math>y</math>, was created as <math>y_{k+1} = x_k + y_k </math>, if <math>x_k + y_k <1</math> and <math>y_{k+1} = x_k + y_k -1</math> otherwise.]]

Six iterations of a set of states <math>[x,y]</math> passed through the logistic map. The first iterate (blue) is the initial condition, which essentially forms a circle. Animation shows the first to the sixth iteration of the circular initial conditions. It can be seen that mixing occurs as we progress in iterations. The sixth iteration shows that the points are almost completely scattered in the phase space. Had we progressed further in iterations, the mixing would have been homogeneous and irreversible. The logistic map has equation <math>x_{k+1} = 4 x_k (1 - x_k )</math>. To expand the state-space of the logistic map into two dimensions, a second state, <math>y</math>, was created as <math>y_{k+1} = x_k + y_k </math>, if <math>x_k + y_k <1</math> and <math>y_{k+1} = x_k + y_k -1</math> otherwise.

一组状态数学[ x，y ] / math 的六次迭代通过 logistic 映射。第一个迭代(蓝色)是初始条件，它基本上形成一个圆。动画显示了循环初始条件的第一次到第六次迭代。可以看出，混合发生在我们迭代的过程中。第六次迭代结果表明，这些点在相空间中几乎完全分散。如果我们在迭代中取得进一步的进展，混合将是均匀的和不可逆的。Logistic 映射有方程式 x { k + 1}4 x k (1-xk) / math。为了将逻辑映射的状态空间扩展为二维，第二种状态，math y / math，被创建为 math y { k + 1} x k + y k / math，如果 math x k + y k 1 / math 和 math y { k + 1} x k + y k-1 / math 不是这样。

[[File:Chaos Topological Mixing.png|thumb|The map defined by ''x'' → 4 ''x'' (1 – ''x'') and ''y'' → (''x'' + ''y)'' [[Modulo operation|mod]] 1 also displays [[topological mixing]]. Here, the blue region is transformed by the dynamics first to the purple region, then to the pink and red regions, and eventually to a cloud of vertical lines scattered across the space.]]

mod 1 also displays topological mixing. Here, the blue region is transformed by the dynamics first to the purple region, then to the pink and red regions, and eventually to a cloud of vertical lines scattered across the space.]]

Mod 1 / span 也显示了拓扑混合。在这里，蓝色区域首先被动态转化为紫色区域，然后转化为粉红色和红色区域，最终转化为散布在空间中的垂直线条云。]

[[Topological mixing]] (or the weaker condition of topological transitivity) means that the system evolves over time so that any given region or [[open set]] of its [[phase space]] eventually overlaps with any other given region. This mathematical concept of "mixing" corresponds to the standard intuition, and the mixing of colored [[dye]]s or fluids is an example of a chaotic system.

Topological mixing (or the weaker condition of topological transitivity) means that the system evolves over time so that any given region or open set of its phase space eventually overlaps with any other given region. This mathematical concept of "mixing" corresponds to the standard intuition, and the mixing of colored dyes or fluids is an example of a chaotic system.

拓扑混合(或较弱的拓扑传递性条件)是指系统随着时间的推移不断演化，使其相空间的任何给定区域或开集最终与任何其他给定区域重叠。这种“混合”的数学概念符合标准的直觉，有色染料或液体的混合就是混沌系统的一个例子。

Topological mixing is often omitted from popular accounts of chaos, which equate chaos with only sensitivity to initial conditions. However, sensitive dependence on initial conditions alone does not give chaos. For example, consider the simple dynamical system produced by repeatedly doubling an initial value. This system has sensitive dependence on initial conditions everywhere, since any pair of nearby points eventually becomes widely separated. However, this example has no topological mixing, and therefore has no chaos. Indeed, it has extremely simple behavior: all points except 0 tend to positive or negative infinity.

Topological mixing is often omitted from popular accounts of chaos, which equate chaos with only sensitivity to initial conditions. However, sensitive dependence on initial conditions alone does not give chaos. For example, consider the simple dynamical system produced by repeatedly doubling an initial value. This system has sensitive dependence on initial conditions everywhere, since any pair of nearby points eventually becomes widely separated. However, this example has no topological mixing, and therefore has no chaos. Indeed, it has extremely simple behavior: all points except 0 tend to positive or negative infinity.

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。

===Topological transitivity===

===Topological transitivity===

拓扑传递性

A map <math>f:X \to X</math> is said to be topologically transitive if for any pair of [[open set]]s <math>U, V \in X</math>, there exists <math>k > 0</math> such that <math>f^{k}(U) \cap V \neq \emptyset</math>. Topological transitivity is a weaker version of [[topological mixing]]. Intuitively, if a map is topologically transitive then given a point ''x'' and a region ''V'', there exists a point ''y'' near ''x'' whose orbit passes through ''V''. This implies that is impossible to decompose the system into two open sets. <ref name="Devaney">{{harvnb|Devaney|2003}}</ref>

A map <math>f:X \to X</math> is said to be topologically transitive if for any pair of open sets <math>U, V \in X</math>, there exists <math>k > 0</math> such that <math>f^{k}(U) \cap V \neq \emptyset</math>. Topological transitivity is a weaker version of topological mixing. Intuitively, if a map is topologically transitive then given a point x and a region V, there exists a point y near x whose orbit passes through V. This implies that is impossible to decompose the system into two open sets.

如果对于 x / math 中的任意一对开集数学 u，v，存在数学 k0 / math，则称 f: x 到 x / math 是拓扑可传递的。拓扑传递性是拓扑混合的一个较弱的形式。直观上，如果一个地图是拓扑传递的，那么给定一个点 x 和一个区域 v，在 x 附近存在一个点 y，这个点的轨道经过 v。这意味着不可能将系统分解为两个开集。

An important related theorem is the Birkhoff Transitivity Theorem. It is easy to see that the existence of a dense orbit implies in topological transitivity. The Birkhoff Transitivity Theorem states that if ''X'' is a [[Second-countable space|second countable]], [[complete metric space]], then topological transitivity implies the existence of a [[dense set]] of points in ''X'' that have dense orbits. <ref>{{harvnb|Robinson|1995}}</ref>

An important related theorem is the Birkhoff Transitivity Theorem. It is easy to see that the existence of a dense orbit implies in topological transitivity. The Birkhoff Transitivity Theorem states that if X is a second countable, complete metric space, then topological transitivity implies the existence of a dense set of points in X that have dense orbits.

一个重要的相关定理是 Birkhoff 传递性定理。在拓扑传递性中，稠密轨道的存在是显而易见的。Birkhoff 传递性定理指出，如果 x 是第二个可数的完备空间，那么拓扑传递性就意味着 x 中存在一个具有稠密轨道的稠密点集。

===Density of periodic orbits===

===Density of periodic orbits===

周期轨道密度

For a chaotic system to have [[Dense set|dense]] [[periodic orbits]] means that every point in the space is approached arbitrarily closely by periodic orbits.<ref name="Devaney"/> The one-dimensional [[logistic map]] defined by ''x'' → 4 ''x'' (1 – ''x'') is one of the simplest systems with density of periodic orbits. For example, <math>\tfrac{5-\sqrt{5}}{8}</math> → <math>\tfrac{5+\sqrt{5}}{8}</math> → <math>\tfrac{5-\sqrt{5}}{8}</math> (or approximately 0.3454915 → 0.9045085 → 0.3454915) is an (unstable) orbit of period 2, and similar orbits exist for periods 4, 8, 16, etc. (indeed, for all the periods specified by [[Sharkovskii's theorem]]).<ref>{{harvnb|Alligood|Sauer|Yorke|1997}}</ref>

For a chaotic system to have dense periodic orbits means that every point in the space is approached arbitrarily closely by periodic orbits.

对于一个具有稠密周期轨道的混沌系统，意味着空间中的每一点都可以被周期轨道任意逼近。

Sharkovskii's theorem is the basis of the Li and Yorke<ref>{{cite journal|last1=Li |first1=T.Y. |last2=Yorke |first2=J.A. |title=Period Three Implies Chaos |journal=[[American Mathematical Monthly]] |volume=82 |pages=985–92 |year=1975 |url=http://pb.math.univ.gda.pl/chaos/pdf/li-yorke.pdf |author-link=Tien-Yien Li |doi=10.2307/2318254 |issue=10 |author2-link=James A. Yorke |bibcode=1975AmMM...82..985L |url-status=dead |archiveurl=https://web.archive.org/web/20091229042210/http://pb.math.univ.gda.pl/chaos/pdf/li-yorke.pdf |archivedate=2009-12-29 |jstor=2318254 |citeseerx=10.1.1.329.5038 }}</ref> (1975) proof that any continuous one-dimensional system that exhibits a regular cycle of period three will also display regular cycles of every other length, as well as completely chaotic orbits.

Sharkovskii's theorem is the basis of the Li and Yorke (1975) proof that any continuous one-dimensional system that exhibits a regular cycle of period three will also display regular cycles of every other length, as well as completely chaotic orbits.

Sharkovskii 的定理是 Li 和 Yorke (1975)证明的基础，证明了任何一维的连续系统，只要表现出周期为三的规则周期，也会表现出其他长度的规则周期，以及完全混沌的轨道。

===Strange attractors===

===Strange attractors===

奇异吸引子

[[File:TwoLorenzOrbits.jpg|thumb|right|The [[Lorenz attractor]] displays chaotic behavior. These two plots demonstrate sensitive dependence on initial conditions within the region of phase space occupied by the attractor.]]

The [[Lorenz attractor displays chaotic behavior. These two plots demonstrate sensitive dependence on initial conditions within the region of phase space occupied by the attractor.]]

文[1]的洛伦兹吸引子表现出混沌行为。这两个图表明敏感的依赖于初始条件在区域的相空间所占据的吸引子。]

Some dynamical systems, like the one-dimensional [[logistic map]] defined by ''x'' → 4 ''x'' (1 – ''x''), are chaotic everywhere, but in many cases chaotic behavior is found only in a subset of phase space. The cases of most interest arise when the chaotic behavior takes place on an [[attractor]], since then a large set of initial conditions leads to orbits that converge to this chaotic region.<ref>{{cite journal|last1=Strelioff|first1=Christopher|last2=et.|first2=al.|title=Medium-Term Prediction of Chaos|journal=Phys. Rev. Lett.|date=2006|volume=96|issue=4|pages=044101|doi=10.1103/PhysRevLett.96.044101|pmid=16486826|bibcode = 2006PhRvL..96d4101S }}</ref>

Some dynamical systems, like the one-dimensional logistic map defined by x → 4 x (1 – x), are chaotic everywhere, but in many cases chaotic behavior is found only in a subset of phase space. The cases of most interest arise when the chaotic behavior takes place on an attractor, since then a large set of initial conditions leads to orbits that converge to this chaotic region.

一些动力学系统，如一维 logistic 映射所定义的跨度类型"white-space: nowrap;"x →4 x (1-x) ，/ span 到处都是混沌，但在许多情况下混沌行为只存在于相空间的一个子集中。最令人感兴趣的情况是当混沌行为发生在一个吸引子上时，从那时起一大组初始条件导致轨道收敛到这个混沌区域。

An easy way to visualize a chaotic attractor is to start with a point in the [[basin of attraction]] of the attractor, and then simply plot its subsequent orbit. Because of the topological transitivity condition, this is likely to produce a picture of the entire final attractor, and indeed both orbits shown in the figure on the right give a picture of the general shape of the Lorenz attractor. This attractor results from a simple three-dimensional model of the [[Edward Lorenz|Lorenz]] weather system. The Lorenz attractor is perhaps one of the best-known chaotic system diagrams, probably because it is not only one of the first, but it is also one of the most complex, and as such gives rise to a very interesting pattern that, with a little imagination, looks like the wings of a butterfly.

An easy way to visualize a chaotic attractor is to start with a point in the basin of attraction of the attractor, and then simply plot its subsequent orbit. Because of the topological transitivity condition, this is likely to produce a picture of the entire final attractor, and indeed both orbits shown in the figure on the right give a picture of the general shape of the Lorenz attractor. This attractor results from a simple three-dimensional model of the Lorenz weather system. The Lorenz attractor is perhaps one of the best-known chaotic system diagrams, probably because it is not only one of the first, but it is also one of the most complex, and as such gives rise to a very interesting pattern that, with a little imagination, looks like the wings of a butterfly.

可视化混沌吸引子的一个简单方法是从吸引子的吸引盆中的一个点开始，然后简单地绘制它的后续轨道。由于拓扑传递性条件，这很可能产生整个最终吸引子的图像，而事实上右图所示的两个轨道都给出了洛伦兹吸引子的一般形状。这个吸引子来自于洛伦兹天气系统的一个简单的三维模型。洛伦兹吸引子也许是最著名的混沌系统图之一，可能是因为它不仅是最早的一个，而且也是最复杂的一个，因此产生了一个非常有趣的图案，稍加想象，看起来就像一只蝴蝶的翅膀。

Unlike [[Attractor#Fixed point|fixed-point attractors]] and [[limit cycle]]s, the attractors that arise from chaotic systems, known as [[strange attractor]]s, have great detail and complexity. Strange attractors occur in both [[continuous function|continuous]] dynamical systems (such as the Lorenz system) and in some [[discrete mathematics|discrete]] systems (such as the [[Hénon map]]). Other discrete dynamical systems have a repelling structure called a [[Julia set]], which forms at the boundary between basins of attraction of fixed points. Julia sets can be thought of as strange repellers. Both strange attractors and Julia sets typically have a [[fractal]] structure, and the [[fractal dimension]] can be calculated for them.

Unlike fixed-point attractors and limit cycles, the attractors that arise from chaotic systems, known as strange attractors, have great detail and complexity. Strange attractors occur in both continuous dynamical systems (such as the Lorenz system) and in some discrete systems (such as the Hénon map). Other discrete dynamical systems have a repelling structure called a Julia set, which forms at the boundary between basins of attraction of fixed points. Julia sets can be thought of as strange repellers. Both strange attractors and Julia sets typically have a fractal structure, and the fractal dimension can be calculated for them.

与定点吸引子和极限环不同，混沌系统产生的吸引子，即所谓的奇怪吸引子，具有很大的细节和复杂性。奇异吸引子存在于连续动力系统(如 Lorenz 系统)和一些离散系统(如 Hénon 映射)中。其他的离散动力系统有一种叫做 Julia 集的排斥结构，这种结构形成于固定点吸引盆地的边界。茱莉亚套装可以被认为是奇怪的排斥器。奇异吸引子和 Julia 集都具有典型的分形结构，可以计算出它们的分形维数。

===Minimum complexity of a chaotic system===

===Minimum complexity of a chaotic system===

一个混沌系统的最小复杂度

[[File:Logistic Map Bifurcation Diagram, Matplotlib.svg|thumb|right|[[Bifurcation diagram]] of the [[logistic map]] ''x'' → ''r'' ''x'' (1 – ''x''). Each vertical slice shows the attractor for a specific value of ''r''. The diagram displays [[Period-doubling bifurcation|period-doubling]] as ''r'' increases, eventually producing chaos.]]

[[Bifurcation diagram of the logistic map x → r x (1 – x). Each vertical slice shows the attractor for a specific value of r. The diagram displays period-doubling as r increases, eventually producing chaos.]]

[逻辑地图的分枝图跨度风格][留白: nowrap; ][ x → r x (1-x)]。每个垂直切片显示一个特定值 r 的吸引子。该图显示了随着 r 的增加周期翻倍，最终产生混沌。]

Discrete chaotic systems, such as the logistic map, can exhibit strange attractors whatever their [[dimension]]ality. Universality of one-dimensional maps with parabolic maxima and [[Feigenbaum constants]] <math>\delta=4.664201...</math>,<math>\alpha=2.502907...</math> <ref>[http://chaosbook.org/extras/mjf/LA-6816-PR.pdf Feigenbaum, M. J. (1976) "Universality in complex discrete dynamics", Los Alamos Theoretical Division Annual Report 1975-1976]</ref><ref name="Feigenbaum 25–52">{{cite journal |first=Mitchell |last=Feigenbaum |title=Quantitative universality for a class of nonlinear transformations |journal=Journal of Statistical Physics |volume=19 |issue=1 |pages=25–52 |date=July 1978 |doi=10.1007/BF01020332 |bibcode=1978JSP....19...25F|citeseerx=10.1.1.418.9339 }}</ref> is well visible with map proposed as a toy

Discrete chaotic systems, such as the logistic map, can exhibit strange attractors whatever their dimensionality. Universality of one-dimensional maps with parabolic maxima and Feigenbaum constants <math>\delta=4.664201...</math>,<math>\alpha=2.502907...</math> is well visible with map proposed as a toy

离散混沌系统，如 logistic 映射，无论其维数如何，都可以表现出奇怪的吸引子。一维地图的普遍性与抛物线最大值和费根鲍姆常数数学 delta 4.664201... / math，math alpha 2.502907... / math 是很明显的与地图作为一个玩具提出

model for discrete laser dynamics:

model for discrete laser dynamics:

离散激光动力学模型:

<math> x \rightarrow G x (1 - \mathrm{tanh} (x))</math>,

<math> x \rightarrow G x (1 - \mathrm{tanh} (x))</math>,

(1- mathrm { tanh }(x)) / math,

where <math>x</math> stands for electric field amplitude, <math>G</math><ref name="Okulov, A Yu 1986">{{cite journal |title=Space–temporal behavior of a light pulse propagating in a nonlinear nondispersive medium|journal=J. Opt. Soc. Am. B |volume=3 |issue=5 |pages=741–746 |year=1986 |last1= Okulov |first1=A Yu |last2=Oraevskiĭ |first2=A N |doi=10.1364/JOSAB.3.000741

where <math>x</math> stands for electric field amplitude, <math>G</math><ref name="Okulov, A Yu 1986">{{cite journal |title=Space–temporal behavior of a light pulse propagating in a nonlinear nondispersive medium|journal=J. Opt. Soc. Am. B |volume=3 |issue=5 |pages=741–746 |year=1986 |last1= Okulov |first1=A Yu |last2=Oraevskiĭ |first2=A N |doi=10.1364/JOSAB.3.000741

其中 math x / math 代表电场幅度 math g / math ref name"okulov，a yu 1986"{ cite journal | title"在非线性非色散介质中传播的光脉冲的时空行为 | journal j。选择。Soc.上午。3 | issue 5 | pages 741-746 | year 1986 | last1 Okulov | first1 a Yu | last2 oraevski | first2 a n | doi 10.1364 / JOSAB. 3.000741

|bibcode=1986OSAJB...3..741O}}</ref> is laser gain as bifurcation parameter. The gradual increase of <math>G</math> at interval <math>[0, \infty)</math> changes dynamics from regular to chaotic one <ref name="Okulov, A Yu 1984">{{cite journal |doi=10.1070/QE1984v014n09ABEH006171

|bibcode=1986OSAJB...3..741O}}</ref> is laser gain as bifurcation parameter. The gradual increase of <math>G</math> at interval <math>[0, \infty)</math> changes dynamics from regular to chaotic one <ref name="Okulov, A Yu 1984">{{cite journal |doi=10.1070/QE1984v014n09ABEH006171

| bibcode 1986OSAJB... 3. . 741 o } / ref 为激光增益分岔参数。数学 g / math 在区间数学[0，infty ] / math 的逐渐增加使动力学从正规变成了混沌，参考名称“ okulov，a yu 1984”{ cite journal | doi 10.1070 / QE1984v014n09ABEH006171

|title=Regular and stochastic self-modulation in a ring laser with nonlinear element

|title=Regular and stochastic self-modulation in a ring laser with nonlinear element

| 题目带有非线性元件的环形激光器中的规则和随机自调制

|journal=Soviet Journal of Quantum Electronics |volume=14 |issue=2 |pages=1235–1237 |year=1984 |last1= Okulov |first1=A Yu |last2=Oraevskiĭ |first2=A N |bibcode=1984QuEle..14.1235O

|journal=Soviet Journal of Quantum Electronics |volume=14 |issue=2 |pages=1235–1237 |year=1984 |last1= Okulov |first1=A Yu |last2=Oraevskiĭ |first2=A N |bibcode=1984QuEle..14.1235O

苏联量子电子学杂志 | 第14卷 | 第2期 | 第1235-1237页 | 1984年 | 最后一页 Okulov | 第一页 a Yu | 最后2页 oraevski | 第一页 a n | bibcode 1984 quele. 14.1235 o

}}</ref> with qualitatively the same [[bifurcation diagram]] as those for [[logistic map]].

}}</ref> with qualitatively the same bifurcation diagram as those for logistic map.

} / ref 与 logistic 映射的分枝图定性相同。

In contrast, for [[continuous function (topology)|continuous]] dynamical systems, the [[Poincaré–Bendixson theorem]] shows that a strange attractor can only arise in three or more dimensions. [[Dimension (vector space)|Finite-dimensional]] [[linear system]]s are never chaotic; for a dynamical system to display chaotic behavior, it must be either [[nonlinearity|nonlinear]] or infinite-dimensional.

In contrast, for continuous dynamical systems, the Poincaré–Bendixson theorem shows that a strange attractor can only arise in three or more dimensions. Finite-dimensional linear systems are never chaotic; for a dynamical system to display chaotic behavior, it must be either nonlinear or infinite-dimensional.

相比之下，对于连续动力系统，庞加莱-本迪克森定理表明奇异吸引子只能出现在三维或三维以上。有限维线性系统永远不会是混沌的; 一个动力系统要表现出混沌行为，它必须是非线性的或无限维的。

The [[Poincaré–Bendixson theorem]] states that a two-dimensional differential equation has very regular behavior. The Lorenz attractor discussed below is generated by a system of three [[differential equation]]s such as:

The Poincaré–Bendixson theorem states that a two-dimensional differential equation has very regular behavior. The Lorenz attractor discussed below is generated by a system of three differential equations such as:

庞加莱-本迪克森定理指出，二维微分方程具有非常规则的行为。下面讨论的 Lorenz 吸引子是由三个微分方程组组成的，例如:

: <math> \begin{align}

<math> \begin{align}

数学 begin { align }

\frac{\mathrm{d}x}{\mathrm{d}t} &= \sigma y - \sigma x, \\

\frac{\mathrm{d}x}{\mathrm{d}t} &= \sigma y - \sigma x, \\

选择最佳方案 & 西格玛 y-sigma x

\frac{\mathrm{d}y}{\mathrm{d}t} &= \rho x - x z - y, \\

\frac{\mathrm{d}y}{\mathrm{d}t} &= \rho x - x z - y, \\

什么时候开始?-什么时候

\frac{\mathrm{d}z}{\mathrm{d}t} &= x y - \beta z.

\frac{\mathrm{d}z}{\mathrm{d}t} &= x y - \beta z.

[咒语][咒语][咒语]。

\end{align} </math>

\end{align} </math>

End { align } / math

where <math>x</math>, <math>y</math>, and <math>z</math> make up the [[State space representation|system state]], <math>t</math> is time, and <math>\sigma</math>, <math>\rho</math>, <math>\beta</math> are the system [[parameter]]s. Five of the terms on the right hand side are linear, while two are quadratic; a total of seven terms. Another well-known chaotic attractor is generated by the [[Rössler map|Rössler equations]], which have only one nonlinear term out of seven. Sprott<ref>{{cite journal|last=Sprott |first=J.C.|year=1997|title=Simplest dissipative chaotic flow|journal=[[Physics Letters A]]|volume=228|issue=4–5 |pages=271–274|doi=10.1016/S0375-9601(97)00088-1|bibcode = 1997PhLA..228..271S }}</ref> found a three-dimensional system with just five terms, that had only one nonlinear term, which exhibits chaos for certain parameter values. Zhang and Heidel<ref>{{cite journal|last=Fu |first=Z. |last2=Heidel |first2=J.|year=1997|title=Non-chaotic behaviour in three-dimensional quadratic systems|journal=[[Nonlinearity (journal)|Nonlinearity]]|volume=10|issue=5 |pages=1289–1303|doi=10.1088/0951-7715/10/5/014

where <math>x</math>, <math>y</math>, and <math>z</math> make up the system state, <math>t</math> is time, and <math>\sigma</math>, <math>\rho</math>, <math>\beta</math> are the system parameters. Five of the terms on the right hand side are linear, while two are quadratic; a total of seven terms. Another well-known chaotic attractor is generated by the Rössler equations, which have only one nonlinear term out of seven. Sprott found a three-dimensional system with just five terms, that had only one nonlinear term, which exhibits chaos for certain parameter values. Zhang and Heidel<ref>{{cite journal|last=Fu |first=Z. |last2=Heidel |first2=J.|year=1997|title=Non-chaotic behaviour in three-dimensional quadratic systems|journal=Nonlinearity|volume=10|issue=5 |pages=1289–1303|doi=10.1088/0951-7715/10/5/014

数学 x / 数学，数学 y / 数学，数学 z / 数学组成系统状态，数学 t / 数学是时间，数学 t / 数学，数学 ρ / 数学，数学 β / 数学是系统参数。右边的五项是线性项，两项是二次项，总共有七项。另一个著名的混沌吸引子是由 r ssler 方程产生的，它只有七个非线性项中的一个。斯普洛特发现了一个只有五个项的三维系统，其中只有一个非线性项，对于某些参数值呈现混沌。张参考了{引用期刊 | 最后的傅 | 第一个 z。三维二次系统中的非混沌行为 | 期刊非线性 | 第10卷 | 第5页 | 1289-1303 | doi 10.1088 / 0951-7715 / 10 / 5 / 014

|bibcode = 1997Nonli..10.1289F }}</ref><ref>{{cite journal|last=Heidel |first=J. |last2=Fu |first2=Z.|year=1999|title=Nonchaotic behaviour in three-dimensional quadratic systems II. The conservative case|journal=Nonlinearity|volume=12|issue=3 |pages=617–633|doi=10.1088/0951-7715/12/3/012|bibcode = 1999Nonli..12..617H }}</ref> showed that, at least for dissipative and conservative quadratic systems, three-dimensional quadratic systems with only three or four terms on the right-hand side cannot exhibit chaotic behavior. The reason is, simply put, that solutions to such systems are asymptotic to a two-dimensional surface and therefore solutions are well behaved.

|bibcode = 1997Nonli..10.1289F }}</ref> showed that, at least for dissipative and conservative quadratic systems, three-dimensional quadratic systems with only three or four terms on the right-hand side cannot exhibit chaotic behavior. The reason is, simply put, that solutions to such systems are asymptotic to a two-dimensional surface and therefore solutions are well behaved.

1997 / nonli. . 10.1289 f } / ref 表明，至少对于耗散和保守的二次系统，右边只有三个或四个项的三维二次系统不能表现出混沌行为。原因很简单，这类系统的解是渐近于二维表面的，因此解的行为是良好的。

While the Poincaré–Bendixson theorem shows that a continuous dynamical system on the Euclidean [[plane (mathematics)|plane]] cannot be chaotic, two-dimensional continuous systems with [[non-Euclidean geometry]] can exhibit chaotic behavior.<ref name="Rosario 2006">{{cite book|last=Rosario|first=Pedro|title=Underdetermination of Science: Part I|date=2006|publisher=Lulu.com|isbn=978-1411693913}}{{self-published source|date=February 2020}}</ref>{{self-published inline|date=February 2020}} Perhaps surprisingly, chaos may occur also in linear systems, provided they are infinite dimensional.<ref>{{cite journal

While the Poincaré–Bendixson theorem shows that a continuous dynamical system on the Euclidean plane cannot be chaotic, two-dimensional continuous systems with non-Euclidean geometry can exhibit chaotic behavior. Perhaps surprisingly, chaos may occur also in linear systems, provided they are infinite dimensional.<ref>{{cite journal

虽然庞加莱-本迪克森定理表明欧氏平面上的连续动力系统不可能是混沌的，但具有非欧几里得几何的二维连续系统可以表现出混沌行为。也许令人惊讶的是，混沌也可能发生在线性系统中，只要它们是无限维的

|last=Bonet |first=J. |last2=Martínez-Giménez |first2=F. |last3=Peris |first3=A.

|last=Bonet |first=J. |last2=Martínez-Giménez |first2=F. |last3=Peris |first3=A.

最后一个博内特 | 第一个 j。|last2=Martínez-Giménez |first2=F.| 最后3个 | 头3个。

|year=2001

|year=2001

2001年

|title=A Banach space which admits no chaotic operator

|title=A Banach space which admits no chaotic operator

不允许混沌算子存在的巴纳赫空间

|journal=Bulletin of the London Mathematical Society

|journal=Bulletin of the London Mathematical Society

伦敦数学学会期刊

|volume=33

|volume=33

第33卷

|issue=2 |pages=196–8

|issue=2 |pages=196–8

第二期，第196-8页

|doi=10.1112/blms/33.2.196

|doi=10.1112/blms/33.2.196

10.1112 / blms / 33.2.196

}}</ref> A theory of linear chaos is being developed in a branch of mathematical analysis known as [[functional analysis]].

}}</ref> A theory of linear chaos is being developed in a branch of mathematical analysis known as functional analysis.

线性混沌理论正在数学分析的一个分支——泛函分析中得到发展。

===Infinite dimensional maps===

===Infinite dimensional maps===

无限维地图

The straightforward generalization of coupled discrete maps <ref name="Moloney, J V 1986">{{cite journal |title=Solitary waves as fixed points of infinite‐dimensional maps for an optical bistable ring cavity: Analysis

The straightforward generalization of coupled discrete maps <ref name="Moloney, J V 1986">{{cite journal |title=Solitary waves as fixed points of infinite‐dimensional maps for an optical bistable ring cavity: Analysis

耦合离散映射的直接推广，参考名称"moloney，j v 1986"{ cite journal | title 孤立波作为光学双稳环形腔无限维映射不动点: 分析

|journal=Journal of Mathematical Physics|volume=29 |issue=1 |pages=63 |year=1988

|journal=Journal of Mathematical Physics|volume=29 |issue=1 |pages=63 |year=1988

数学物理学杂志 | 第29卷 | 第1期 | 第63页 | 1988年

|last1= Adachihara |first1=H

|last1= Adachihara |first1=H

|last1= Adachihara |first1=H

|last2= McLaughlin |first2=D W

|last2= McLaughlin |first2=D W

2 McLaughlin | first2 d w

|last3= Moloney |first3=J V

|last3= Moloney |first3=J V

3 Moloney | first3 j v

|last4= Newell |first4=A C

|last4= Newell |first4=A C

4 a c

|doi=10.1063/1.528136 |bibcode=1988JMP....29...63A}}</ref> is based upon convolution integral which mediates interaction between spatially distributed maps:

|doi=10.1063/1.528136 |bibcode=1988JMP....29...63A}}</ref> is based upon convolution integral which mediates interaction between spatially distributed maps:

10.1063 / 1.528136 | bibcode 1988JMP... 29... 63A } / ref 基于卷积积分，它调节空间分布映射之间的相互作用:

<math>\psi_{n+1}(\vec r,t) = \int K(\vec r - \vec r^{,},t) f [\psi_{n}(\vec r^{,},t) ]d {\vec r}^{,}</math>,

<math>\psi_{n+1}(\vec r,t) = \int K(\vec r - \vec r^{,},t) f [\psi_{n}(\vec r^{,},t) ]d {\vec r}^{,}</math>,

<math>\psi_{n+1}(\vec r,t) = \int K(\vec r - \vec r^{,},t) f [\psi_{n}(\vec r^{,},t) ]d {\vec r}^{,}</math>,

where kernel <math>K(\vec r - \vec r^{,},t)</math> is propagator derived as Green function of a relevant physical system,<ref name="Okulov, A Yu 1988">{{cite book |chapter=Spatiotemporal dynamics of a wave packet in nonlinear medium and discrete maps

where kernel <math>K(\vec r - \vec r^{,},t)</math> is propagator derived as Green function of a relevant physical system,<ref name="Okulov, A Yu 1988">{{cite book |chapter=Spatiotemporal dynamics of a wave packet in nonlinear medium and discrete maps

其中核数学 k ( vec r ^ { ，} ，t) / math 是作为相关物理系统的格林函数导出的传播子，ref name"okulov，a yu 1988"{ cite book | chapter 时空动力学非线性介质和离散映射中的波包

|title=Proceedings of the Lebedev Physics Institute |language=Russian |editor=N.G. Basov |publisher=Nauka |lccn=88174540 |volume=187 |pages=202–222 |year=1988 |last1= Okulov |first1=A Yu |last2=Oraevskiĭ |first2=A N }}</ref>

|title=Proceedings of the Lebedev Physics Institute |language=Russian |editor=N.G. Basov |publisher=Nauka |lccn=88174540 |volume=187 |pages=202–222 |year=1988 |last1= Okulov |first1=A Yu |last2=Oraevskiĭ |first2=A N }}</ref>

| 标题列别捷夫物理研究所会议记录 | 俄语 | 编辑 n.g。Basov | publisher Nauka | lccn 88174540 | volume 187 | pages 202-222 | year 1988 | last1 Okulov | first1 a Yu | last2 oraevski | first2 a n } / ref

<math> f [\psi_{n}(\vec r,t) ] </math> might be logistic map alike <math> \psi \rightarrow G \psi [1 - \tanh (\psi)]</math> or [[complex map]]. For examples of complex maps the [[Julia set]] <math> f[\psi] = \psi^2</math> or [[Ikeda map]]

<math> f [\psi_{n}(\vec r,t) ] </math> might be logistic map alike <math> \psi \rightarrow G \psi [1 - \tanh (\psi)]</math> or complex map. For examples of complex maps the Julia set <math> f[\psi] = \psi^2</math> or Ikeda map

数学 f [ psi { n }( vec r，t)] / math 可能是逻辑映射，类似于 math psi right tarrow g psi [1- tanh ( psi)] / math 或复合映射。对于复杂地图的例子，可以使用 Julia 集合数学 f [ psi ] psi ^ 2 / math 或 Ikeda 地图

<math> \psi_{n+1} = A + B \psi_n e^{i (|\psi_n|^2 + C)} </math> may serve. When wave propagation problems at distance <math>L=ct</math> with wavelength <math>\lambda=2\pi/k</math> are considered the kernel <math>K</math> may have a form of Green function for [[Schrödinger equation]]:<ref name="Okulov, A Yu 2000">{{cite journal |title=Spatial soliton laser: geometry and stability

<math> \psi_{n+1} = A + B \psi_n e^{i (|\psi_n|^2 + C)} </math> may serve. When wave propagation problems at distance <math>L=ct</math> with wavelength <math>\lambda=2\pi/k</math> are considered the kernel <math>K</math> may have a form of Green function for Schrödinger equation:<ref name="Okulov, A Yu 2000">{{cite journal |title=Spatial soliton laser: geometry and stability

数学{ n + 1} a + b psi n e ^ { i (|-psi n | ^ 2 + c)} / math 可能有用。当波的传播数学 l ct / math 和波长数学 l λ 2 pi / k / math 被认为是核数学 k / math 对薛定谔方程可能有一种格林函数形式: ref name"okulov，a yu 2000"{ cite journal | title Spatial isolus laser: geometry and stability

|journal=Optics and Spectroscopy|volume=89 |issue=1 |pages=145–147 |year=2000 |last1= Okulov |first1=A Yu|doi=10.1134/BF03356001 |bibcode=2000OptSp..89..131O|url=https://www.semanticscholar.org/paper/0bd2d3e9a6912a188f42b50316f4652c165d1b6b}}</ref>

|journal=Optics and Spectroscopy|volume=89 |issue=1 |pages=145–147 |year=2000 |last1= Okulov |first1=A Yu|doi=10.1134/BF03356001 |bibcode=2000OptSp..89..131O|url=https://www.semanticscholar.org/paper/0bd2d3e9a6912a188f42b50316f4652c165d1b6b}}</ref>

1 Okulov | first1 a Yu | doi 10.1134 / BF03356001 | bibcode 2000OptSp. . 89.131 o | url } / ref. /

<ref name="Okulov, A Yu 2020">{{cite journal |title=Structured light entities, chaos and nonlocal maps

<ref name="Okulov, A Yu 2020">{{cite journal |title=Structured light entities, chaos and nonlocal maps

文献名称"okulov，a yu 2020"{ cite journal | title / 结构光实体，混沌和非局部地图

|journal=Chaos,Solitons&Fractals|volume=133 |issue=4|page=109638 |year=2020|last1= Okulov |first1=A Yu|doi=10.1016/j.chaos.2020.109638|arxiv=1901.09274}}</ref>.

|journal=Chaos,Solitons&Fractals|volume=133 |issue=4|page=109638 |year=2020|last1= Okulov |first1=A Yu|doi=10.1016/j.chaos.2020.109638|arxiv=1901.09274}}</ref>.

混沌，孤子和分形 | 第133卷 | 第4期 | 第109638页 | 2020年 | last 1 Okulov | first1 a Yu | doi 10.1016 / j.Chaos. 2020.109638 | arxiv 1901.09274} / ref。

<math> K(\vec r - \vec r^{,},L) = \frac {ik\exp[ikL]}{2\pi L}\exp[\frac {ik|\vec r-\vec r^{,}|^2}{2 L} ]</math>.

<math> K(\vec r - \vec r^{,},L) = \frac {ik\exp[ikL]}{2\pi L}\exp[\frac {ik|\vec r-\vec r^{,}|^2}{2 L} ]</math>.

数学 k ( vec r- vec r ^ { ，} ，l) frac { ik exp [ ikL ]}{2 pi l } exp [ frac | vec r- vec r ^ { ，} | 2}{2 l }] / math。

=== Jerk systems ===

=== Jerk systems ===

挺举系统

In [[physics]], [[Jerk (physics)|jerk]] is the third derivative of [[position (vector)|position]], with respect to time. As such, differential equations of the form

In physics, jerk is the third derivative of position, with respect to time. As such, differential equations of the form

在物理学中，挺度是位置对时间的三阶导数。这样，微分方程的形式

:: <math>J\left(\overset{...}{x},\ddot{x},\dot {x},x\right)=0</math>

<math>J\left(\overset{...}{x},\ddot{x},\dot {x},x\right)=0</math>

数学 j 左( x } ， ddot { x } ， x } ，右)0 / math

are sometimes called ''Jerk equations''. It has been shown that a jerk equation, which is equivalent to a system of three first order, ordinary, non-linear differential equations, is in a certain sense the minimal setting for solutions showing chaotic behaviour. This motivates mathematical interest in jerk systems. Systems involving a fourth or higher derivative are called accordingly hyperjerk systems.<ref>K. E. Chlouverakis and J. C. Sprott, Chaos Solitons & Fractals 28, 739–746 (2005), Chaotic Hyperjerk Systems, http://sprott.physics.wisc.edu/pubs/paper297.htm</ref>

are sometimes called Jerk equations. It has been shown that a jerk equation, which is equivalent to a system of three first order, ordinary, non-linear differential equations, is in a certain sense the minimal setting for solutions showing chaotic behaviour. This motivates mathematical interest in jerk systems. Systems involving a fourth or higher derivative are called accordingly hyperjerk systems.

有时被称为 Jerk 等式。证明了 jerk 方程等价于三个一阶非线性常微分方程组，在某种意义上是表现混沌行为的解的最小设定。这激发了人们对挺举系统的数学兴趣。含有四阶或更高阶导数的系统称为相应的超挺举系统。

A jerk system's behavior is described by a jerk equation, and for certain jerk equations, simple electronic circuits can model solutions. These circuits are known as jerk circuits.

A jerk system's behavior is described by a jerk equation, and for certain jerk equations, simple electronic circuits can model solutions. These circuits are known as jerk circuits.

挺举系统的行为用挺举方程描述，对于某些挺举方程，简单的电子线路可以建立解的模型。这些回路被称为挺举回路。

One of the most interesting properties of jerk circuits is the possibility of chaotic behavior. In fact, certain well-known chaotic systems, such as the Lorenz attractor and the [[Rössler map]], are conventionally described as a system of three first-order differential equations that can combine into a single (although rather complicated) jerk equation. Nonlinear jerk systems are in a sense minimally complex systems to show chaotic behaviour; there is no chaotic system involving only two first-order, ordinary differential equations (the system resulting in an equation of second order only).

One of the most interesting properties of jerk circuits is the possibility of chaotic behavior. In fact, certain well-known chaotic systems, such as the Lorenz attractor and the Rössler map, are conventionally described as a system of three first-order differential equations that can combine into a single (although rather complicated) jerk equation. Nonlinear jerk systems are in a sense minimally complex systems to show chaotic behaviour; there is no chaotic system involving only two first-order, ordinary differential equations (the system resulting in an equation of second order only).

挺举电路最有趣的性质之一是混沌行为的可能性。实际上，某些著名的混沌系统，如 Lorenz 吸引子和 r-ssler 映射，通常被描述为三个一阶微分方程组成的系统，它们可以组合成一个单一的(虽然相当复杂) jerk 方程。非线性 jerk 系统在某种意义上是表现出混沌行为的最小复杂系统，不存在只包含两个一阶常微分方程的混沌系统(只产生一个二阶方程的系统)。

An example of a jerk equation with nonlinearity in the magnitude of <math>x</math> is:

An example of a jerk equation with nonlinearity in the magnitude of <math>x</math> is:

一个在 math x / math 数量级中带有非线性的 jerk 方程的例子是:

:<math>\frac{\mathrm{d}^3 x}{\mathrm{d} t^3}+A\frac{\mathrm{d}^2 x}{\mathrm{d} t^2}+\frac{\mathrm{d} x}{\mathrm{d} t}-|x|+1=0.</math>

<math>\frac{\mathrm{d}^3 x}{\mathrm{d} t^3}+A\frac{\mathrm{d}^2 x}{\mathrm{d} t^2}+\frac{\mathrm{d} x}{\mathrm{d} t}-|x|+1=0.</math>

数学框架 ^ 3 x { mathrm { d } t ^ 3} + a frc { d } ^ 2 x }{ mathrm { d } t ^ 2} + frac { d } t-x | + 10. / math

Here, ''A'' is an adjustable parameter. This equation has a chaotic solution for ''A''=3/5 and can be implemented with the following jerk circuit; the required nonlinearity is brought about by the two diodes:

Here, A is an adjustable parameter. This equation has a chaotic solution for A=3/5 and can be implemented with the following jerk circuit; the required nonlinearity is brought about by the two diodes:

这里，a 是一个可调参数。该方程对 a3 / 5有一个混沌解，可以用下面的冲击电路实现，所需的非线性是由两个二极管带来的:

[[File:JerkCircuit01.png|frameless|upright=1.4|center]]

center

中心

In the above circuit, all resistors are of equal value, except <math>R_A=R/A=5R/3</math>, and all capacitors are of equal size. The dominant frequency is <math>1/2\pi R C</math>. The output of [[operational amplifier|op amp]] 0 will correspond to the x variable, the output of 1 corresponds to the first derivative of x and the output of 2 corresponds to the second derivative.

In the above circuit, all resistors are of equal value, except <math>R_A=R/A=5R/3</math>, and all capacitors are of equal size. The dominant frequency is <math>1/2\pi R C</math>. The output of op amp 0 will correspond to the x variable, the output of 1 corresponds to the first derivative of x and the output of 2 corresponds to the second derivative.

在上述电路中，除数学 r a r / a 5r / 3 / math 外，所有电阻值相等，所有电容值相等。主频是数学1 / 2 pi r c / math。运算放大器0的输出对应于 x 变量，1的输出对应于 x 的一阶导数，2的输出对应于二阶导数。

Similar circuits only require one diode<ref>[http://sprott.physics.wisc.edu/pubs/paper352.htm "A New Chaotic Jerk Circuit"], J. C. Sprott, IEEE Transactions on Circuits and Systems,2011.</ref> or no diodes at all.<ref>[http://sprott.physics.wisc.edu/pubs/paper345.htm "Simple Autonomous Chaotic Circuits"], J. C. Sprott, IEEE Transactions on Circuits and Systems--II: Express Briefs, 2010.</ref>

Similar circuits only require one diode or no diodes at all.

类似的电路只需要一个二极管或根本不需要二极管。

See also the well-known [[Chua's circuit]], one basis for chaotic true random number generators.<ref>[http://www.jestr.org/downloads/Volume6Issue4/fulltext11642013.pdf "Secure Image Encryption Based On a Chua Chaotic Noise Generator"], A. S. Andreatos*, and A. P. Leros, Journal of Engineering Science and Technology Review, 2013.</ref> The ease of construction of the circuit has made it a ubiquitous real-world example of a chaotic system.

See also the well-known Chua's circuit, one basis for chaotic true random number generators. The ease of construction of the circuit has made it a ubiquitous real-world example of a chaotic system.

又见著名的蔡氏电路，混沌真随机数发生器的基础之一。电路结构的简易性使它成为一个无处不在的现实世界中的混沌系统的例子。

==Spontaneous order==

==Spontaneous order==

自发秩序

Under the right conditions, chaos spontaneously evolves into a lockstep pattern. In the [[Kuramoto model]], four conditions suffice to produce synchronization in a chaotic system.

Under the right conditions, chaos spontaneously evolves into a lockstep pattern. In the Kuramoto model, four conditions suffice to produce synchronization in a chaotic system.

在适当的条件下，混沌自然而然地演化成一种步调一致的模式。在 Kuramoto 模型中，四个条件足以产生混沌系统的同步。

Examples include the [[coupled oscillation]] of [[Christiaan Huygens]]' pendulums, fireflies, [[neuron]]s, the [[London Millennium Bridge]] resonance, and large arrays of [[Josephson junctions]].<ref>Steven Strogatz, ''Sync: The Emerging Science of Spontaneous Order'', Hyperion, 2003.</ref>

Examples include the coupled oscillation of Christiaan Huygens' pendulums, fireflies, neurons, the London Millennium Bridge resonance, and large arrays of Josephson junctions.

例子包括克里斯蒂安·惠更斯钟摆的耦合振荡、萤火虫、神经元、伦敦千禧桥共振以及大型约瑟夫森结阵列。

==History==

==History==

历史

[[File:Barnsley fern plotted with VisSim.PNG|thumb|upright|[[Barnsley fern]] created using the [[chaos game]]. Natural forms (ferns, clouds, mountains, etc.) may be recreated through an [[iterated function system]] (IFS).]]

[[Barnsley fern created using the chaos game. Natural forms (ferns, clouds, mountains, etc.) may be recreated through an iterated function system (IFS).]]

[[巴恩斯利蕨用混沌游戏创建。自然形态(蕨类、云、山等)可以通过迭代函数系统来重建。]

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''2)''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)。).

==Applications==

==Applications==

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

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

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]

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。

=== Cryptography ===

=== Cryptography ===

密码学

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 ===

=== Robotics ===

=== Robotics ===

Robotics is another area that has recently benefited from chaos theory. Instead of robots acting in a trial-and-error type of refinement to interact with their environment, chaos theory has been used to build a [[Predictive modelling|predictive model]].<ref>{{cite journal|last=Nehmzow|first=Ulrich|date=Dec 2005|title=Quantitative description of robot–environment interaction using chaos theory|journal=Robotics and Autonomous Systems|volume=53|issue=3–4|pages=177–193|doi=10.1016/j.robot.2005.09.009|author2=Keith Walker|url=http://cswww.essex.ac.uk/staff/udfn/ftp/ecmrw3.pdf|access-date=2017-10-25|archive-url=https://web.archive.org/web/20170812003513/http://cswww.essex.ac.uk/staff/udfn/ftp/ecmrw3.pdf|archive-date=2017-08-12|url-status=dead|citeseerx=10.1.1.105.9178}}</ref>

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。

===Biology===

===Biology===

生物学

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。

===Other areas===

===Other areas===

其他范畴

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.

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。

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。

==See also==

==See also==

参见

{{Portal|Systems science|Mathematics}}

;Examples of chaotic systems

Examples of chaotic systems

混沌系统的例子

{{Div col|colwidth=18em}}

* [[Contour advection|Advected contours]]

* [[Arnold's cat map]]

* [[Bouncing ball dynamics]]

* [[Chua's circuit]]

* [[Cliodynamics]]

* [[Coupled map lattice]]

* [[Double pendulum]]

* [[Duffing equation]]

* [[Dynamical billiards]]

* [[Economic bubble]]

* [[Chaotic scattering#Gaspard–Rice system|Gaspard-Rice system]]

* [[Hénon map]]

* [[Horseshoe map]]

* [[List of chaotic maps]]

* [[Rössler attractor]]

* [[Standard map]]

* [[Swinging Atwood's machine]]

* [[Tilt A Whirl]]

{{Div col end}}

;Other related topics

Other related topics

其他相关话题

{{Div col|colwidth=18em}}

* [[Amplitude death]]

* [[Anosov diffeomorphism]]

* [[Catastrophe theory]]

* [[Causality]]

* [[Chaos theory in organizational development]]

* [[Chaos machine]]

* [[Chaotic mixing]]

* [[Chaotic scattering]]

* [[Control of chaos]]

* [[Determinism]]

* [[Edge of chaos]]

* [[Emergence]]

* [[Mandelbrot set]]

* [[Kolmogorov–Arnold–Moser theorem]]

* [[Ill-conditioning]]

* [[Ill-posedness]]

* [[Nonlinear system]]

* [[Patterns in nature]]

* [[Predictability]]

* [[Quantum chaos]]

* [[Santa Fe Institute]]

* [[Synchronization of chaos]]

* [[Unintended consequence]]

{{Div col end}}

;People

People

人们

{{Div col|colwidth=18em}}

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

* [[Michael Berry (physicist)|Michael Berry]]

* [[Leon O. Chua]]

* [[Ivar Ekeland]]

* [[Doyne Farmer]]

* [[Martin Gutzwiller]]

* [[Brosl Hasslacher]]

* [[Michel Hénon]]

* [[Aleksandr Lyapunov]]

* [[Norman Packard]]

* [[Otto Rössler]]

* [[David Ruelle]]

* [[Oleksandr Mikolaiovich Sharkovsky]]

* [[Robert Shaw (physicist)|Robert Shaw]]

* [[Floris Takens]]

* [[James A. Yorke]]

* [[George M. Zaslavsky]]

{{Div col end}}

==References==

==References==

参考资料

{{Reflist|30em}}

==Further reading==

==Further reading==

进一步阅读

===Articles===

===Articles===

文章

* {{cite journal |first=A.N. |last=Sharkovskii |authorlink=Oleksandr Mykolaiovych Sharkovsky |title=Co-existence of cycles of a continuous mapping of the line into itself |journal=Ukrainian Math. J. |volume=16 |pages=61–71 |year=1964 }}

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

*{{cite journal|last1=Alemansour|first1=Hamed|last2=Miandoab|first2=Ehsan Maani|last3=Pishkenari|first3=Hossein Nejat|title=Effect of size on the chaotic behavior of nano resonators|journal=Communications in Nonlinear Science and Numerical Simulation|date=March 2017|volume=44|pages=495–505|doi=10.1016/j.cnsns.2016.09.010|bibcode=2017CNSNS..44..495A}}

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

* {{cite journal |author=Kolyada, S.F. |title=Li-Yorke sensitivity and other concepts of chaos |journal=Ukrainian Math. J. |volume=56 |pages=1242–57 |year=2004 |doi=10.1007/s11253-005-0055-4 |issue=8 |url=https://www.semanticscholar.org/paper/5834c3cdf41d979850c355b93999e785da446562 }}

* {{cite journal | last1 = Day| first1 = R.H. | last2 = Pavlov| first2 = O.V. | year = 2004| title = Computing Economic Chaos | ssrn = 806124| journal = Computational Economics | volume = 23 | issue = 4 | pages = 289–301 | doi = 10.1023/B:CSEM.0000026787.81469.1f }}

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

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

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

===Textbooks===

===Textbooks===

教科书

* {{cite book |last1=Alligood |first1=K.T. |last2=Sauer |first2=T. |last3=Yorke |first3=J.A. |title=Chaos: an introduction to dynamical systems |publisher=Springer-Verlag |year=1997 |isbn=978-0-387-94677-1 |ref=harv |url=https://books.google.com/books?id=48YHnbHGZAgC}}

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

* {{cite book |author1=Badii, R. |author2=Politi A. |title=Complexity: hierarchical structures and scaling in physics |publisher=Cambridge University Press |year=1997 |isbn=978-0-521-66385-4 |url=http://www.cambridge.org/gb/academic/subjects/physics/statistical-physics/complexity-hierarchical-structures-and-scaling-physics}}

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

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

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

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

* {{cite book |author1=Feldman, D. P. |title=Chaos and Fractals: An Elementary Introduction |publisher=Oxford University Press |year=2012 |isbn=978-0-19-956644-0|url=http://chaos.coa.edu/index.html}}

* {{cite book |author1=Gollub, J. P. |author2=Baker, G. L. |title=Chaotic dynamics |publisher=Cambridge University Press |year=1996 |isbn=978-0-521-47685-0 |url=https://books.google.com/books?id=n1qnekRPKtoC}}

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

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

* {{cite book |author=Gutzwiller, Martin |title=Chaos in Classical and Quantum Mechanics |publisher=Springer-Verlag |year=1990 |isbn=978-0-387-97173-5 |url=https://books.google.com/books?id=fnO3XYYpU54C}}

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

* {{cite book |author=Kautz, Richard |title=Chaos: The Science of Predictable Random Motion |publisher=Oxford University Press |year=2011 |isbn=978-0-19-959458-0 |url=https://books.google.com/books?id=x5YbNZjulN0C}}

* {{cite book |author1=Kiel, L. Douglas |author2=Elliott, Euel W. |title=Chaos Theory in the Social Sciences |publisher=Perseus Publishing |year=1997 |isbn=978-0-472-08472-2 |url=https://books.google.com/books?id=K46kkMXnKfcC}}

* {{cite book |author=Moon, Francis |title=Chaotic and Fractal Dynamics |publisher=Springer-Verlag |year=1990 |isbn=978-0-471-54571-2 |url=https://books.google.com/books?id=Ddz-CI-nSKYC}}

* {{cite book |author=Ott, Edward |title=Chaos in Dynamical Systems |publisher=Cambridge University Press |year=2002 |isbn=978-0-521-01084-9 |url=https://books.google.com/books?id=nOLx--zzHSgC}}

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

* {{cite book |last=Sprott |first=Julien Clinton |title=Chaos and Time-Series Analysis |publisher=Oxford University Press |year=2003 |isbn=978-0-19-850840-3 |ref=harv |url=https://books.google.com/books?id=SEDjdjPZ158C}}

* {{cite book |author1=Tél, Tamás |author2=Gruiz, Márton |title=Chaotic dynamics: An introduction based on classical mechanics |publisher=Cambridge University Press |year=2006 |isbn=978-0-521-83912-9 |url=https://books.google.com/books?id=P2JL7s2IvakC}}

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

* {{cite book|vauthors=Thompson JM, Stewart HB | title=Nonlinear Dynamics And Chaos| publisher=John Wiley and Sons Ltd| year=2001| isbn=978-0-471-87645-8}}

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

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

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

===Semitechnical and popular works===

===Semitechnical and popular works===

半技术和通俗作品

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

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

* {{cite book |authorlink=Michael F. Barnsley |first=Michael F. |last=Barnsley |title=Fractals Everywhere |url=https://books.google.com/books?id=oh7NoePgmOIC |year=2000 |publisher=Morgan Kaufmann |isbn=978-0-12-079069-2}}

* {{cite book |first=Richard J. |last=Bird |title=Chaos and Life: Complexity and Order in Evolution and Thought |url=https://books.google.com/books?id=fv3sltQBS54C |year=2003 |publisher=Columbia University Press |isbn=978-0-231-12662-5}}

* [[John Briggs (author)|John Briggs]] and David Peat, ''Turbulent Mirror: : An Illustrated Guide to Chaos Theory and the Science of Wholeness'', Harper Perennial 1990, 224 pp.

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

* {{cite journal |author=Cunningham, Lawrence A. |title=From Random Walks to Chaotic Crashes: The Linear Genealogy of the Efficient Capital Market Hypothesis |journal=George Washington Law Review |volume=62 |page=546 |year=1994 }}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

==External links==

==External links==

外部链接

{{Commons category|Chaos theory}}{{Library resources box}}

* {{springer|title=Chaos|id=p/c021480}}

* [https://web.archive.org/web/20160310065017/http://lagrange.physics.drexel.edu/ Nonlinear Dynamics Research Group] with Animations in Flash

* [http://www.chaos.umd.edu/ The Chaos group at the University of Maryland]

* [http://hypertextbook.com/chaos/ The Chaos Hypertextbook]. An introductory primer on chaos and fractals

* [http://chaosbook.org/ ChaosBook.org] An advanced graduate textbook on chaos (no fractals)

* [http://www.societyforchaostheory.org/ Society for Chaos Theory in Psychology & Life Sciences]

* [http://www.csdc.unifi.it/mdswitch.html?newlang=eng Nonlinear Dynamics Research Group at CSDC], [[Florence]] [[Italy]]

* [https://web.archive.org/web/20051227123602/http://physics.mercer.edu/pendulum/ Interactive live chaotic pendulum experiment], allows users to interact and sample data from a real working damped driven chaotic pendulum

* [http://www.creatingtechnology.org/papers/chaos.htm Nonlinear dynamics: how science comprehends chaos], talk presented by Sunny Auyang, 1998.

* [http://www.egwald.ca/nonlineardynamics/index.php Nonlinear Dynamics]. Models of bifurcation and chaos by Elmer G. Wiens

* [http://www.around.com/chaos.html Gleick's ''Chaos'' (excerpt)]

* [https://web.archive.org/web/20070428110552/http://www.eng.ox.ac.uk/samp/ Systems Analysis, Modelling and Prediction Group] at the University of Oxford

* [https://web.archive.org/web/20090307094012/http://www.mgix.com/snippets/?MackeyGlass A page about the Mackey-Glass equation]

* [https://www.youtube.com/user/thedebtgeneration?feature=mhum High Anxieties — The Mathematics of Chaos] (2008) BBC documentary directed by [[David Malone (independent filmmaker)|David Malone]]

* [https://www.newscientist.com/article/mg20827821.000-the-chaos-theory-of-evolution.html The chaos theory of evolution] – article published in Newscientist featuring similarities of evolution and non-linear systems including fractal nature of life and chaos.

* Jos Leys, [[Étienne Ghys]] et Aurélien Alvarez, [http://www.chaos-math.org/en ''Chaos, A Mathematical Adventure'']. Nine films about dynamical systems, the butterfly effect and chaos theory, intended for a wide audience.

* [http://www.bbc.co.uk/programmes/p00548f6 "Chaos Theory"], BBC Radio 4 discussion with Susan Greenfield, David Papineau & Neil Johnson (''In Our Time'', May 16, 2002)

{{Systems}}

{{Chaos theory}}

{{Patterns in nature}}

{{DEFAULTSORT:Chaos Theory}}

[[Category:Chaos theory| ]]

[[Category:Complex systems theory]]

Category:Complex systems theory

范畴: 复杂系统理论

[[Category:Computational fields of study]]

Category:Computational fields of study

类别: 研究的计算领域

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