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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}}
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| | + | [[File:Lorenz attractor yb.svg|thumb|right|值{{nowrap|''r'' {{=}} 28}}, {{nowrap|σ {{=}} 10}}, {{nowrap|''b'' {{=}} 8/3}}的[[Lorenz吸引子]]图]] |
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| | + | [[File:Double-compound-pendulum.gif|thumb|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]] |
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| − | {{other uses|Chaos theory (disambiguation)|Chaos (disambiguation)}} | + | '''混沌理论 Chaos theory'''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。<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>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。<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> [[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)。<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>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。 |
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| | + | 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。<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>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<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>，完全由它们的初始条件决定，没有任何随机因素参与。<ref>{{harvnb|Kellert|1993|p=56}}</ref>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。<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>这种行为被称为'''确定性混沌'''，或简单的混沌。[[爱德华·洛伦茨 Edward Lorenz]]将这一理论总结为:<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> |
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| − | [[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}}]]
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| − | A plot of the [[Lorenz attractor for values 28}}, 10}}, 8/3}}]]
| + | {{Quote|混沌：现在决定未来，但近似的现在不能近似地决定未来。}} |
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| − | [[[洛伦兹值28} ，10} ，8 / 3}]吸引子的一个图
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| − | [[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.]]
| + | 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。<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" />社会学、物理学、<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> 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织过程等领域的研究奠定了基础。 |
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| − | 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.]]
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| − | 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
| + | ==引言== |
| | + | 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 Lyapunov time]]。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。<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>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。<ref>''Sync: The Emerging Science of Spontaneous Order'', Steven Strogatz, Hyperion, New York, 2003, pages 189–190.</ref> |
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| − | '''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.
| + | ==混沌动力学== |
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| − | 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. | + | [[File:Chaos Sensitive Dependence.svg|thumb|由<span style="white-space: nowrap;">''x'' → 4 ''x'' (1 – ''x'')</span> and <span style="white-space: nowrap;">''y'' → (''x'' + ''y)'' [[Modulo operation|mod]] 1</span> 定义的映射显示对初始 x 位置的灵敏度。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。]] |
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| − | 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。
| + | 在通常的用法中，“混沌”意味着“无序的状态”。<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>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:<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> |
| | + | # 它必须对初始条件很敏感, |
| | + | # 必须具有拓扑传递性, |
| | + | # 它一定有密集的周期轨道。 |
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| | + | 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。<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>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”。 |
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| | + | 如果注意力被限制在时间间隔内，<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>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。 |
| | + | <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> |
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| − | 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>
| + | === 混沌作为拓扑超对称性的自发分解 Chaos as a spontaneous breakdown of topological supersymmetry === |
| | + | {{Main|随机动力学的超对称理论}} |
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| − | 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:
| + | 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属性。<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>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——在自发的拓扑超对称破坏中的应用。 |
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| − | 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:
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| | + | === 对初始条件的敏感性 Sensitivity to initial conditions === |
| | + | {{Main|蝴蝶效应}} |
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| | + | [[File:SensInitCond.gif|thumb|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。<math>\rho</math>, <math>\sigma</math> 和<math>\beta</math> 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。]] |
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| | + | 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。<ref name=":1" /> |
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| − | {{Quote|Chaos: When the present determines the future, but the approximate present does not approximately determine the future.}} | + | 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 Predictability: Does the Flap of a Butterfly's Wings in Brazil set off a Tornado in Texas?》的论文。<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> 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。 |
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| | + | 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。<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>这并不意味着人们不能断言任何关于遥远未来的事件——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。 |
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| − | | + | 用更精确的数学术语来说，[[李亚普诺夫指数 Lyapunov exponent]]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。<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> 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<math>\delta \mathbf{Z}_0</math>，这两个轨迹最终以给定的速率发散。 |
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| − | 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.
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| − | 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.
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| − | 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。
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| − | ==Introduction==
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| − | ==Introduction==
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| − | 引言
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| − | 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>
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| − | 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.
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| − | 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。
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| − | ==Chaotic dynamics==
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| − | ==Chaotic dynamics==
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| − | 混沌动力学
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| − | [[File:Chaos Sensitive Dependence.svg|thumb|The map defined by <span style="white-space: nowrap;">''x'' → 4 ''x'' (1 – ''x'')</span> and <span style="white-space: nowrap;">''y'' → (''x'' + ''y)'' [[Modulo operation|mod]] 1</span> displays sensitivity to initial x positions. Here, two series of ''x'' and ''y'' values diverge markedly over time from a tiny initial difference.]]
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| − | mod 1</span> displays sensitivity to initial x positions. Here, two series of x and y values diverge markedly over time from a tiny initial difference.]]
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| − | 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。]]
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| − | 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>
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| − | 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:
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| − | 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:
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| − | # it must be [[sensitive dependence on initial conditions|sensitive to initial conditions]],
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| − | it must be sensitive to initial conditions,
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| − | 它必须对初始条件很敏感,
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| − | # it must be [[Mixing_(mathematics)#Topological_mixing|topologically transitive]],
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| − | it must be topologically transitive,
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| − | 必须具有拓扑传递性,
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| − | # it must have [[dense set|dense]] [[periodic orbit]]s.
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| − | it must have dense periodic orbits.
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| − | 它一定有稠密的周期轨道。
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| − | 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.
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| − | 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.
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| − | 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”。
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| − | 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>
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| − | 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.
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| − | 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。
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| − | === Chaos as a spontaneous breakdown of topological supersymmetry ===
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| − | === Chaos as a spontaneous breakdown of topological supersymmetry ===
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| − | 混沌作为拓扑超对称性的自发破坏
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| − | {{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.
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| − | 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.
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| − | 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。
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| − | ===Sensitivity to initial conditions===
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| − | ===Sensitivity to initial conditions===
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| − | 对初始条件的敏感性
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| − | {{Main|Butterfly effect}}
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| − | [[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.]]
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| − | 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.
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| − | 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。
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| − | '''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" />
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| − | 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.
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| − | 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。
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| − | 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.
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| − | 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.
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| − | 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。
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| − | 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.
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| − | 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.
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| − | 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。
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| − | 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
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| − | 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
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| − | 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
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| | :<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> |
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| − | <math> | \delta\mathbf{Z}(t) | \approx e^{\lambda t} | \delta \mathbf{Z}_0 |,</math>
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| − | 数学 | delta mathbf { z }(t) | | approx ^ { lambda t } | delta mathbf { z }0 | ，/ math
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| − | 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" />
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| − | 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.
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| − | 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。
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| − | 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" />
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| − | 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.
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| − | 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。
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| − | ===Non-periodicity=== | + | 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(MLE) 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。<ref name=":2" /> |
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| − | ===Non-periodicity===
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| − | 非周期性
| + | 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 Kolmogorov automorphism]]中所讨论的)和K系统 K-system的性质。<ref name="WerndlCharlotte" /> |
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| − | 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.
| + | ===非周期性 Non-periodicity=== |
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| − | 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.
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| | 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。 | | 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。 |
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| − | ===Topological mixing=== | + | ===拓扑混合 Topological mixing=== |
| | + | [[File:LogisticTopMixing1-6.gif|thumb|一组状态数学<math>[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 <math>x_{k+1} = 4 x_k (1 - x_k )</math>。为了将逻辑映射的状态空间扩展为二维，将第二种状态<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> 不是这样的。]] |
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| − | ===Topological mixing===
| + | [[File:Chaos Topological Mixing.png|thumb|由<span style="white-space: nowrap;">''x'' → 4 ''x'' (1 – ''x'')</span>和<span style="white-space: nowrap;">''y'' → (''x'' + ''y)'' [[Modulo operation|mod]] 1</span> 定义的映射还显示拓扑混合。在这里，蓝色区域通过动力学首先转换为紫色区域，然后转换为粉红色和红色区域，最后转换为散布在整个空间中的垂直线条云。 |
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| − | 拓扑混合
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| − | [[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.]]
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| − | 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.
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| − | 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。
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| − | [[File:Chaos Topological Mixing.png|thumb|The map defined by <span style="white-space: nowrap;">''x'' → 4 ''x'' (1 – ''x'')</span> and <span style="white-space: nowrap;">''y'' → (''x'' + ''y)'' [[Modulo operation|mod]] 1</span> 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.]] | |
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| − | mod 1</span> 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.]]
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| − | 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。]
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| − | [[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.
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| − | 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.
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| | 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。 | | 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。 |
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| − | 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.
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| − | 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.
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| | 常见的混沌理论忽略了拓扑混合，认为混沌只是对初始条件敏感。然而，对初始条件的敏感依赖本身并不会造成混乱。例如，考虑一下通过反复将初始值加倍而产生的简单的动力系统。这个系统对任何地方的初始条件都有敏感的依赖关系，因为任何一对邻近的点最终都会变得广泛分离。然而，这个例子没有拓扑混合，因此没有混沌。事实上，它的行为极其简单: 除了0以外的所有点都趋向于正或负无穷。 | | 常见的混沌理论忽略了拓扑混合，认为混沌只是对初始条件敏感。然而，对初始条件的敏感依赖本身并不会造成混乱。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。 |
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| | + | ===拓扑传递性 Topological transitivity=== |
| | + | 如果对于<math>f:X \to X</math>中的任意一对开集数学<math>U, V \in X</math>，存在数学 <math>k > 0</math>，则称<math>f^{k}(U) \cap V \neq \emptyset</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。<ref name="Devaney">{{harvnb|Devaney|2003}}</ref> |
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| | + | 一个重要的相关定理是'''Birkhoff 传递性定理 Birkhoff Transitivity Theorem'''。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。<ref>{{harvnb|Robinson|1995}}</ref> |
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| − | ===Topological transitivity===
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| − | ===Topological transitivity===
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| − | 拓扑传递性
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| − | 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>
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| − | 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.
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| − | 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。
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| − | 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>
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| − | 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.
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| − | 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。
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| − | ===Density of periodic orbits===
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| − | ===Density of periodic orbits===
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| − | 周期轨道密度
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| − | 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 <span style="white-space: nowrap;">''x'' → 4 ''x'' (1 – ''x'')</span> 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>
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| − | For a chaotic system to have dense periodic orbits means that every point in the space is approached arbitrarily closely by periodic orbits.
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| − | 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。
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| − | 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.
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| − | 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.
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| − | 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。
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| − | ===Strange attractors===
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| − | ===Strange attractors===
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| − | 奇异吸引子
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| − | [[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.]]
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| − | 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.]]
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| − | 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。]
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| − | Some dynamical systems, like the one-dimensional [[logistic map]] defined by <span style="white-space: nowrap;">''x'' → 4 ''x'' (1 – ''x''),</span> 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>
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| − | Some dynamical systems, like the one-dimensional logistic map defined by <span style="white-space: nowrap;">x → 4 x (1 – x),</span> 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.
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| − | 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。
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| − | 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.
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| − | 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.
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| − | 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。
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| | + | ===周期轨道密度 Density of periodic orbits=== |
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| | + | 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。<ref name="Devaney"/> |
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| | + | Sharkovskii 的定理是 Li 和 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)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。 |
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| − | 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.
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| − | 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.
| + | ===奇异吸引子 Strange attractors=== |
| | + | [[File:TwoLorenzOrbits.jpg|thumb|right|Lorenz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。]] |
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| − | 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。
| + | 一些动力学系统，如一维 logistic 映射所定义的跨度类型<span style="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。<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> |
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| | + | 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Lorenz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。Lorenz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。 |
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| | + | 与定点吸引子和极限环不同，混沌系统产生的吸引子，即所谓的奇异吸引子，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。 Julia 集可以被认为是奇异的排斥者。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。 |
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| − | ===Minimum complexity of a chaotic system===
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| − | ===Minimum complexity of a chaotic system===
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| − | 一个混沌系统的最小复杂度
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| − | [[File:Logistic Map Bifurcation Diagram, Matplotlib.svg|thumb|right|[[Bifurcation diagram]] of the [[logistic map]] <span style="white-space: nowrap;">''x'' → ''r'' ''x'' (1 – ''x'').</span> 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.]]
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| − | [[Bifurcation diagram of the logistic map <span style="white-space: nowrap;">x → r x (1 – x).</span> Each vertical slice shows the attractor for a specific value of r. The diagram displays period-doubling as r increases, eventually producing chaos.]]
| + | ===混沌系统的最小复杂度 Minimum complexity of a chaotic system=== |
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| − | [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。] | + | [[File:Logistic Map Bifurcation Diagram, Matplotlib.svg|thumb|right|分叉图的的逻辑映射<span style="white-space: nowrap;">''x'' → ''r'' ''x'' (1 – ''x'').</span>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。]] |
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| − | | + | 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 [[]]is well visible with map proposed as a toy |
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| − | 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 | |
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| | 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 | | 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 |
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| − | 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 | + | 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。具有抛物线最大值和[[费根鲍姆常数 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>的一维映射的普适性是显而易见的,将映射作为离散激光动力学的玩具模型提出: |
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| − | model for discrete laser dynamics:
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| − | model for discrete laser dynamics:
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| − | 离散激光动力学模型:
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| − | <math> x \rightarrow G x (1 - \mathrm{tanh} (x))</math>,
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| − | <math> x \rightarrow G x (1 - \mathrm{tanh} (x))</math>, | + | :<math> x \rightarrow G x (1 - \mathrm{tanh} (x))</math>, |
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| − | (1- mathrm { tanh }(x)) / math,
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| − | 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
| + | :<math> x \rightarrow G x (1 - \mathrm{tanh} (x))</math>, |
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| − | 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
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| − | 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
| + | :(1- mathrm { tanh }(x)) / math, |
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| − | |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
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| | + | 其中，<math>x</math>代表电场幅度 <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|bibcode=1986OSAJB...3..741O}}</ref>为激光增益分岔参数。<math>G</math>在区间<math>[0, \infty)</math>的逐渐增加使动力学从正规变成了混沌，参考名称<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 |
| | |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 |
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| − | | 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
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| | |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 |
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| − | |title=Regular and stochastic self-modulation in a ring laser with nonlinear element
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| − | | 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
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| − | |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
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| | |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 |
| | + | }}</ref>，其定性分支图与逻辑图相同。 |
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| − | 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
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| − | }}</ref> with qualitatively the same [[bifurcation diagram]] as those for [[logistic map]].
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| − | }}</ref> with qualitatively the same bifurcation diagram as those for logistic map.
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| − | } / ref 与 logistic 映射的分枝图定性相同。
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| − | 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.
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| − | 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 Poincaré–Bendixson theorem]]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。 |
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| − | 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。
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| − | | + | 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: |
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| − | 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:
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| − | 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:
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| − | 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: | |
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| | : <math> \begin{align} | | : <math> \begin{align} |
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| − | 数学 begin { align }
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| | \frac{\mathrm{d}x}{\mathrm{d}t} &= \sigma y - \sigma x, \\ | | \frac{\mathrm{d}x}{\mathrm{d}t} &= \sigma y - \sigma x, \\ |
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| − | \frac{\mathrm{d}x}{\mathrm{d}t} &= \sigma y - \sigma x, \\
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| − | 选择最佳方案 & 西格玛 y-sigma x
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| | \frac{\mathrm{d}y}{\mathrm{d}t} &= \rho x - x z - y, \\ | | \frac{\mathrm{d}y}{\mathrm{d}t} &= \rho x - x z - y, \\ |
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| − | \frac{\mathrm{d}y}{\mathrm{d}t} &= \rho x - x z - y, \\
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| − | 什么时候开始?-什么时候
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| − | \frac{\mathrm{d}z}{\mathrm{d}t} &= x y - \beta z.
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| | \frac{\mathrm{d}z}{\mathrm{d}t} &= x y - \beta z. | | \frac{\mathrm{d}z}{\mathrm{d}t} &= x y - \beta z. |
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| − | [咒语][咒语][咒语]。
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| − | \end{align} </math>
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| | \end{align} </math> | | \end{align} </math> |
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| − | End { align } / math
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| − | 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
| + | <math>x</math>，<math>y</math>，<math>z</math>组成系统状态，<math>t</math>是时间，<math>\sigma</math>，<math>\rho</math>，<math>\beta</math>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 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>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。张和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|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> 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。 |
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| − | 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
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| − | 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
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| − | |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.
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| − | |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.
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| − | 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。
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| − | 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
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| − | 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
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| − | 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
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| | + | 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。<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>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。.<ref>{{cite journal |
| | |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. |
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| − | 最后一个博内特 | 第一个 j。|last2=Martínez-Giménez |first2=F.| 最后3个 | 头3个。
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| − | 不允许混沌算子存在的巴纳赫空间
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| | + | }}</ref> 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 |
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| − | }}</ref> A theory of linear chaos is being developed in a branch of mathematical analysis known as [[functional analysis]].
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| − | }}</ref> A theory of linear chaos is being developed in a branch of mathematical analysis known as functional analysis.
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| − | 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。
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| − | ===Infinite dimensional maps===
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| − | 无限维地图
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| − | 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
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| − | 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
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| − | 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
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| − | |journal=Journal of Mathematical Physics|volume=29 |issue=1 |pages=63 |year=1988
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| − | 数学物理学杂志 | 第29卷 | 第1期 | 第63页 | 1988年
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| − | |doi=10.1063/1.528136 |bibcode=1988JMP....29...63A}}</ref> is based upon convolution integral which mediates interaction between spatially distributed maps:
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| − | |doi=10.1063/1.528136 |bibcode=1988JMP....29...63A}}</ref> is based upon convolution integral which mediates interaction between spatially distributed maps:
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| − | 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:
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| − | <math>\psi_{n+1}(\vec r,t) = \int K(\vec r - \vec r^{,},t) f [\psi_{n}(\vec r^{,},t) ]d {\vec r}^{,}</math>,
| + | ===无限维地图 Infinite dimensional maps=== |
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| − | <math>\psi_{n+1}(\vec r,t) = \int K(\vec r - \vec r^{,},t) f [\psi_{n}(\vec r^{,},t) ]d {\vec r}^{,}</math>, | + | 耦合离散映射的直接推广<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|journal=Journal of Mathematical Physics|volume=29 |issue=1 |pages=63 |year=1988 |last1= Adachihara |first1=H|last2= McLaughlin |first2=D W|last3= Moloney |first3=J V|last4= Newell |first4=A C|doi=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: |
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| − | <math>\psi_{n+1}(\vec r,t) = \int K(\vec r - \vec r^{,},t) f [\psi_{n}(\vec r^{,},t) ]d {\vec r}^{,}</math>,
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| | + | :<math>\psi_{n+1}(\vec r,t) = \int K(\vec r - \vec r^{,},t) f [\psi_{n}(\vec r^{,},t) ]d {\vec r}^{,}</math>, |
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| | + | 其中，核心<math>K(\vec r - \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=Spatiotemporal dynamics of a wave packet in nonlinear medium and discrete maps |
| | + | |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> 。<math> f [\psi_{n}(\vec r,t) ] </math> 可能是逻辑映射，类似于 <math> \psi \rightarrow G \psi [1 - \tanh (\psi)]</math>或复合映射，例如Julia 集合 <math> f[\psi] = \psi^2</math>或Ikeda映射<math> \psi_{n+1} = A + B \psi_n e^{i (|\psi_n|^2 + C)} </math> 当波的传播距离 <math>L=ct</math>和波长<math>\lambda=2\pi/k</math>认为是核<math>K</math> 可以具有用于格林函数的形式薛定谔方程：<ref name="Okulov, A Yu 2000">{{cite journal |title=Spatial soliton 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>.<ref name="Okulov, A Yu 2020">{{cite journal |title=Structured light entities, chaos and nonlocal maps |
| | + | |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> |
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| − | 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
| + | :<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>. |
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| − | 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
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| − | 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
| + | ===挺举系统 Jerk systems === |
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| − | |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>
| + | 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: |
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| − | |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>
| + | :<math>J\left(\overset{...}{x},\ddot{x},\dot {x},x\right)=0</math> |
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| − | | 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
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| − | <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]] | + | 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。<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> |
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| − | <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
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| − | 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
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| − | <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
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| − | <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
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| − | 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
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| − | |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>
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| − | |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>
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| − | 1 Okulov | first1 a Yu | doi 10.1134 / BF03356001 | bibcode 2000OptSp. . 89.131 o | url } / ref. /
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| − | <ref name="Okulov, A Yu 2020">{{cite journal |title=Structured light entities, chaos and nonlocal maps
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| − | <ref name="Okulov, A Yu 2020">{{cite journal |title=Structured light entities, chaos and nonlocal maps
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| − | 文献名称"okulov，a yu 2020"{ cite journal | title / 结构光实体，混沌和非局部地图
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| − | |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>.
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| − | |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>.
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| − | 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。
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| − | <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>.
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| − | <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>.
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| − | 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。
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| − | === Jerk systems ===
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| − | === Jerk systems ===
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| − | 挺举系统
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| − | In [[physics]], [[Jerk (physics)|jerk]] is the third derivative of [[position (vector)|position]], with respect to time. As such, differential equations of the form
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| − | In physics, jerk is the third derivative of position, with respect to time. As such, differential equations of the form
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| − | 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
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| − | :: <math>J\left(\overset{...}{x},\ddot{x},\dot {x},x\right)=0</math>
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| − | <math>J\left(\overset{...}{x},\ddot{x},\dot {x},x\right)=0</math>
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| − | 数学 j 左( x } ， ddot { x } ， x } ，右)0 / math
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| − | 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>
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| − | 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.
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| − | 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。
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| − | 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.
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| − | 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.
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| | 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。 | | 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。 |
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| | + | 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)。 |
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| | + | 一个在<math>x</math>数量级中带有非线性的 jerk 方程的例子是: |
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| − | 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).
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| − | 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).
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| − | 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)。
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| − | An example of a jerk equation with nonlinearity in the magnitude of <math>x</math> is:
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| − | An example of a jerk equation with nonlinearity in the magnitude of <math>x</math> is:
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| − | 一个在 math x / math 数量级中带有非线性的 jerk 方程的例子是:
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| | :<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> |
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| − | <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>
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| − | 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
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| − | 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:
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| − | 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:
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| − | 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:
| + | 这里，''A''是一个可调参数。该方程对''A''=3/5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: |
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