7,129
个编辑
更改
无编辑摘要
}}
}}
[[File:Lorenz attractor yb.svg.png|thumb|right|值{{nowrap|''r'' {{=}} 28}}, {{nowrap|σ {{=}} 10}}, {{nowrap|''b'' {{=}} 8/3}}的[[Lorenz吸引子]]图]]
[[File:Lorenz attractor yb.svg.png|thumb|right|值<math>r = 28,σ=10, b=8/3</math>的[[Lorenz吸引子]]图]]
[[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]]
[[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]]
==混沌动力学==
==混沌动力学==
[[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。]]
[[File:Chaos Sensitive Dependence.svg.png|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。]]
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>
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>
===混沌系统的最小复杂度 Minimum complexity of a chaotic system===
===混沌系统的最小复杂度 Minimum complexity of a chaotic system===
[[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。]]
[[File:Logistic Map Bifurcation Diagram, Matplotlib.svg.png|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。]]
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::<math> x \rightarrow G x (1 - \mathrm{tanh} (x))</math>,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，<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
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::<math> x \rightarrow G x (1 - \mathrm{tanh} (x))</math>,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，<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