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Logistic映射 (查看源代码)

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===3.6<μ<4===

[[File:450px-Logistic_map.gif|400px|thumb|图7 不同的初始条件下关于μ的函数（横轴的r为μ）]]

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* The development of the chaotic behavior of the logistic sequence as the parameter {{mvar|r}} varies from approximately 3.56995 to approximately 3.82843 is sometimes called the [[Pomeau–Manneville scenario]], characterized by a periodic (laminar) phase interrupted by bursts of aperiodic behavior. Such a scenario has an application in semiconductor devices.<ref name="carson82">{{cite journal|first1=Carson |last1=Jeffries|first2=José |last2=Pérez |journal=[[Physical Review A]]|year=1982|title=Observation of a Pomeau–Manneville intermittent route to chaos in a nonlinear oscillator|volume=26 |issue=4 |pages=2117–2122|doi=10.1103/PhysRevA.26.2117|bibcode = 1982PhRvA..26.2117J |url=http://www.escholarship.org/uc/item/2dm2k8mm}}</ref> There are other ranges that yield oscillation among 5 values etc.; all oscillation periods occur for some values of {{mvar|r}}. A ''period-doubling window'' with parameter {{mvar|c}} is a range of {{mvar|r}}-values consisting of a succession of subranges. The {{mvar|k}}th subrange contains the values of {{mvar|r}} for which there is a stable cycle (a cycle that attracts a set of initial points of unit measure) of period {{math|2''k''''c''}}. This sequence of sub-ranges is called a ''cascade of harmonics''.<ref name="May">{{cite journal | first = R. M. |last=May | title = Simple mathematical models with very complicated dynamics | journal = Nature | year = 1976 | volume = 261 | issue = 5560 | pages = 459–67 | doi = 10.1038/261459a0 | pmid = 934280|bibcode = 1976Natur.261..459M | hdl = 10338.dmlcz/104555 | hdl-access = free }}</ref> In a sub-range with a stable cycle of period {{math|2''k''*''c''}}, there are unstable cycles of period {{math|2''k''''c''}} for all {{math|''k'' < ''k''*}}. The {{mvar|r}} value at the end of the infinite sequence of sub-ranges is called the ''point of accumulation'' of the cascade of harmonics. As {{mvar|r}} rises there is a succession of new windows with different {{mvar|c}} values. The first one is for {{math|''c'' {{=}} 1}}; all subsequent windows involving odd {{mvar|c}} occur in decreasing order of {{mvar|c}} starting with arbitrarily large {{mvar|c}}.<ref name="May" /><ref>{{cite journal |last1=Baumol |first1=William J. |authorlink=William Baumol |last2=Benhabib |first2=Jess |authorlink2=Jess Benhabib |title=Chaos: Significance, Mechanism, and Economic Applications |journal=[[Journal of Economic Perspectives]] |date=February 1989 |volume=3 |issue=1 |pages=77–105 |doi=10.1257/jep.3.1.77 }}</ref>

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当参数<math>μ</math>从大约3.56995变化到3.82843时，Logistic映射的混沌行为的发展过程有时被称为'''Pomeau-Manneville场景 the Pomeau–Manneville scenario'''，其特征是周期性(层流)阶段被非周期性行为突然打断。The Pomeau–Manneville scenario在半导体器件中有应用。<ref name="carson82">{{cite journal|first1=Carson |last1=Jeffries|first2=José |last2=Pérez |journal=Physical Review A|year=1982|title=Observation of a Pomeau–Manneville intermittent route to chaos in a nonlinear oscillator|volume=26 |issue=4 |pages=2117–2122|doi=10.1103/PhysRevA.26.2117|bibcode = 1982PhRvA..26.2117J |url=http://www.escholarship.org/uc/item/2dm2k8mm}}</ref> 此时函数值在5个值之间来回波动;所有的振荡周期都依赖于<math>\mu</math>。带参数c的倍周期是由一系列子序列组成的μ值范围。第k个子区间包含了<math>\mu</math>的值，其中有一个2kc的稳定周期(一个周期吸引了一组单位测度的初始点)。这个子范围的序列称为谐波级联cascade of harmonics。[5]在一个稳定周期为2k*c的子范围内，所有k < k*都存在周期为2kc的不稳定周期。在无限子区间序列末端的<math>\mu</math>值称为谐波级联cascade of harmonics的积累点。随着μ的升高，出现了一系列具有不同c值的新窗口。第一个是c = 1;所有包含奇数c的后续窗口都以c的递减顺序出现，以任意大的c开始。<ref name="May" /><ref>{{cite journal |last1=Baumol |first1=William J. |authorlink=William Baumol |last2=Benhabib |first2=Jess |authorlink2=Jess Benhabib |title=Chaos: Significance, Mechanism, and Economic Applications |journal=Journal of Economic Perspectives]] |date=February 1989 |volume=3 |issue=1 |pages=77–105 |doi=10.1257/jep.3.1.77 }}</ref>

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当参数<math>μ</math>从大约3.56995变化到3.82843时，Logistic映射的混沌行为的发展过程有时被称为'''Pomeau-Manneville场景 the Pomeau–Manneville scenario'''，其特征是周期性(层流)阶段被非周期性行为突然打断。The Pomeau–Manneville scenario在半导体器件中有应用。<ref name="carson82">{{cite journal|first1=Carson |last1=Jeffries|first2=José |last2=Pérez |journal=[[Physical Review A]]|year=1982|title=Observation of a Pomeau–Manneville intermittent route to chaos in a nonlinear oscillator|volume=26 |issue=4 |pages=2117–2122|doi=10.1103/PhysRevA.26.2117|bibcode = 1982PhRvA..26.2117J |url=http://www.escholarship.org/uc/item/2dm2k8mm}}</ref> 此时函数值在5个值之间来回波动;所有的振荡周期都依赖于<math>\mu</math>。带参数c的倍周期是由一系列子序列组成的μ值范围。第k个子区间包含了<math>\mu</math>的值，其中有一个2kc的稳定周期(一个周期吸引了一组单位测度的初始点)。这个子范围的序列称为谐波级联cascade of harmonics。[5]在一个稳定周期为2k*c的子范围内，所有k < k*都存在周期为2kc的不稳定周期。在无限子区间序列末端的<math>\mu</math>值称为谐波级联cascade of harmonics的积累点。随着μ的升高，出现了一系列具有不同c值的新窗口。第一个是c = 1;所有包含奇数c的后续窗口都以c的递减顺序出现，以任意大的c开始。<ref name="May" /><ref>{{cite journal |last1=Baumol |first1=William J. |authorlink=William Baumol |last2=Benhabib |first2=Jess |authorlink2=Jess Benhabib |title=Chaos: Significance, Mechanism, and Economic Applications |journal=[[Journal of Economic Perspectives]] |date=February 1989 |volume=3 |issue=1 |pages=77–105 |doi=10.1257/jep.3.1.77 }}</ref>

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[[category:复杂系统]]

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[[~~category：分形~~]]

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[[category:分形]]

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