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Logistic映射 - 版本历史
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<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>本中文词条由[[用户:趣木木|趣木木]]<del class="diffchange diffchange-inline">参与编译,</del>[[用户:思无涯咿呀咿呀|思无涯咿呀咿呀]]审校,[[用户:薄荷|薄荷]]、[[用户:费米子|费米子]]编辑,欢迎在讨论页面留言。</div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>本中文词条由[[用户:趣木木|趣木木]]<ins class="diffchange diffchange-inline">,张江参与编译,</ins>[[用户:思无涯咿呀咿呀|思无涯咿呀咿呀]]审校,[[用户:薄荷|薄荷]]、[[用户:费米子|费米子]]编辑,欢迎在讨论页面留言。</div></td></tr>
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18621066378:18621066378移动页面Logistics映射至Logistic映射覆盖重定向
2020-04-30T15:02:23Z
<p>18621066378移动页面<a href="/index.php/Logistics%E6%98%A0%E5%B0%84" class="mw-redirect" title="Logistics映射">Logistics映射</a>至<a href="/index.php/Logistic%E6%98%A0%E5%B0%84" title="Logistic映射">Logistic映射</a>覆盖重定向</p>
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18621066378:恢复18621066378(讨论)的编辑至薄荷的最后版本
2020-04-30T15:00:50Z
<p>恢复<a href="/index.php/%E7%89%B9%E6%AE%8A:%E7%94%A8%E6%88%B7%E8%B4%A1%E7%8C%AE/18621066378" title="特殊:用户贡献/18621066378">18621066378</a>(<a href="/index.php?title=%E7%94%A8%E6%88%B7%E8%AE%A8%E8%AE%BA:18621066378&action=edit&redlink=1" class="new" title="用户讨论:18621066378(页面不存在)">讨论</a>)的编辑至<a href="/index.php/%E7%94%A8%E6%88%B7:%E8%96%84%E8%8D%B7" title="用户:薄荷">薄荷</a>的最后版本</p>
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2020年4月30日 (四) 14:59 18621066378
2020-04-30T14:59:27Z
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</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l3" >第3行:</td>
<td colspan="2" class="diff-lineno">第3行:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|description=logistics映射,logistics模型</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|description=logistics映射,logistics模型</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>}}</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>}}</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[File:<del class="diffchange diffchange-inline">logistics </del>.png|400px|thumb|right|以美国人口为例,使用logistics模型来拟合1800年后的人口数据:<math>x(t)=\frac{K}{1+(\frac{K}{x_0}-1)e^{-rt}}</math>,其中<math>K</math>为最大容纳量]]</div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[File:<ins class="diffchange diffchange-inline">logistic </ins>.png|400px|thumb|right|以美国人口为例,使用logistics模型来拟合1800年后的人口数据:<math>x(t)=\frac{K}{1+(\frac{K}{x_0}-1)e^{-rt}}</math>,其中<math>K</math>为最大容纳量]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''logistics映射''',又称单峰映象,是一个二次多项式映射(递归关系),经常作为典型范例来说明复杂的混沌现象是如何从非常简单的非线性动力学方程中产生的。生物学家[[罗伯特·梅 Robert May]] <ref name="May, Robert M 1976">{{cite journal |last=May |first=Robert M. |year=1976 |title=Simple mathematical models with very complicated dynamics |journal=Nature (journal) |volume=261 |issue=5560 |pages=459–467 |doi=10.1038/261459a0 |bibcode=1976Natur.261..459M |pmid=934280 |hdl=10338.dmlcz/104555 |hdl-access=free }}</ref>在1976年的一篇论文中推广了这一映射,<ref>"{{MathWorld | urlname=logisticsEquation | title= logistics Equation}}</ref>它在一定程度上是一个时间离散的人口统计模型,类似于'''皮埃尔·弗朗索瓦·韦胡斯特 Pierre Francois Verhulst''' 首次提出的方程。</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>'''logistics映射''',又称单峰映象,是一个二次多项式映射(递归关系),经常作为典型范例来说明复杂的混沌现象是如何从非常简单的非线性动力学方程中产生的。生物学家[[罗伯特·梅 Robert May]] <ref name="May, Robert M 1976">{{cite journal |last=May |first=Robert M. |year=1976 |title=Simple mathematical models with very complicated dynamics |journal=Nature (journal) |volume=261 |issue=5560 |pages=459–467 |doi=10.1038/261459a0 |bibcode=1976Natur.261..459M |pmid=934280 |hdl=10338.dmlcz/104555 |hdl-access=free }}</ref>在1976年的一篇论文中推广了这一映射,<ref>"{{MathWorld | urlname=logisticsEquation | title= logistics Equation}}</ref>它在一定程度上是一个时间离散的人口统计模型,类似于'''皮埃尔·弗朗索瓦·韦胡斯特 Pierre Francois Verhulst''' 首次提出的方程。</div></td></tr>
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<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l36" >第36行:</td>
<td colspan="2" class="diff-lineno">第36行:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==数值试验==</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==数值试验==</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[File:<del class="diffchange diffchange-inline">logistics_map_animation</del>.gif|400px|thumb|center|图1]]</div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[File:<ins class="diffchange diffchange-inline">logistic_map_animation</ins>.gif|400px|thumb|center|图1]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>首先,可以用一系列数值试验的方法来探讨这个迭代方程。如果使用Mathematica数学软件,只需要用两句话就能实现这个迭代:</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>首先,可以用一系列数值试验的方法来探讨这个迭代方程。如果使用Mathematica数学软件,只需要用两句话就能实现这个迭代:</div></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l135" >第135行:</td>
<td colspan="2" class="diff-lineno">第135行:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===<math>3.6<μ<4</math>===</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===<math>3.6<μ<4</math>===</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[File:450px-<del class="diffchange diffchange-inline">logistics_map</del>.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)]]</div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[File:450px-<ins class="diffchange diffchange-inline">logistic_map</ins>.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)]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>当参数<math>\mu</math>从大约3.56995变化到3.82843时,logistics映射的混沌行为的发展过程有时被称为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<math>c</math>的倍周期是由一系列子序列组成的<math>\mu</math>值范围。第k个子区间包含了<math>\mu</math>的值,其中有一个<math>2^{k}c</math>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。<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> 在一个稳定周期为<math>2^{k^*}c</math>的子范围内,所有<math>k < k^*</math>都存在周期为<math>2^{k}c</math>的不稳定周期。在无限子区间序列末端的<math>\mu</math>值称为谐波级联的积累点。随着<math>\mu</math>的升高,出现了一系列具有不同<math>c</math>值的新窗口。第一个是<math>c</math> = 1;所有包含奇数<math>c</math>的后续窗口都以<math>c</math>的递减顺序出现,以任意大的<math>c</math>开始。<ref name="May" /><ref>{{cite journal |last1=Baumol |first1=William J. |last2=Benhabib |first2=Jess |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> </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>当参数<math>\mu</math>从大约3.56995变化到3.82843时,logistics映射的混沌行为的发展过程有时被称为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<math>c</math>的倍周期是由一系列子序列组成的<math>\mu</math>值范围。第k个子区间包含了<math>\mu</math>的值,其中有一个<math>2^{k}c</math>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。<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> 在一个稳定周期为<math>2^{k^*}c</math>的子范围内,所有<math>k < k^*</math>都存在周期为<math>2^{k}c</math>的不稳定周期。在无限子区间序列末端的<math>\mu</math>值称为谐波级联的积累点。随着<math>\mu</math>的升高,出现了一系列具有不同<math>c</math>值的新窗口。第一个是<math>c</math> = 1;所有包含奇数<math>c</math>的后续窗口都以<math>c</math>的递减顺序出现,以任意大的<math>c</math>开始。<ref name="May" /><ref>{{cite journal |last1=Baumol |first1=William J. |last2=Benhabib |first2=Jess |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> </div></td></tr>
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<td colspan="2" class="diff-lineno">第235行:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==混沌与logistics映射==</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==混沌与logistics映射==</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>[[File:<del class="diffchange diffchange-inline">Iterated_logistics_functions</del>.svg.png|thumb|400px|图9 logistics映射<math>f</math> (blue)及其迭代版本<math>f^2、f^3、f^4</math>和<math>f^5</math>, <math>\mu</math> = 3.5。例如,对于横轴上的任意初始值,<math>f^4</math>给出后面迭代4次的值|right]]</div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>[[File:<ins class="diffchange diffchange-inline">Iterated_logistic_functions</ins>.svg.png|thumb|400px|图9 logistics映射<math>f</math> (blue)及其迭代版本<math>f^2、f^3、f^4</math>和<math>f^5</math>, <math>\mu</math> = 3.5。例如,对于横轴上的任意初始值,<math>f^4</math>给出后面迭代4次的值|right]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>logistics映射<math> f</math> (blue)在<math>\mu</math>=3.5的条件下进行迭代,得到<math> f^2</math>、<math> f^3</math> 、<math> f^4</math>和 <math> f^5</math>。 例如,对于水平轴上的任何初始值,<math> f^4</math>为四次迭代之后得到的值。和其他混沌系统比较,logistics映射的相对简单性使它成为考虑混沌概念的一个广泛使用的切入点。简单来说,混沌就是对初始条件的高度灵敏度。<math>\mu</math>=3.5是在3.57及4之间的大部分数值都可以使logistics映射出现该特性。<ref name="May, Robert M 1976" /> 由于映射本身对定义域的拉伸及折叠,使得其对初始条件有高度灵敏度,故表现出来了混沌特性。logistics映射的二次差分方程可视为是对于区间(0,1)拉伸及折叠的过程。<ref name="Gleick">{{cite book |last=Gleick |first=James |title=Chaos: Making a New Science |year=1987 |publisher=Penguin Books |location=London |isbn=978-0-14-009250-9 }}</ref></div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>logistics映射<math> f</math> (blue)在<math>\mu</math>=3.5的条件下进行迭代,得到<math> f^2</math>、<math> f^3</math> 、<math> f^4</math>和 <math> f^5</math>。 例如,对于水平轴上的任何初始值,<math> f^4</math>为四次迭代之后得到的值。和其他混沌系统比较,logistics映射的相对简单性使它成为考虑混沌概念的一个广泛使用的切入点。简单来说,混沌就是对初始条件的高度灵敏度。<math>\mu</math>=3.5是在3.57及4之间的大部分数值都可以使logistics映射出现该特性。<ref name="May, Robert M 1976" /> 由于映射本身对定义域的拉伸及折叠,使得其对初始条件有高度灵敏度,故表现出来了混沌特性。logistics映射的二次差分方程可视为是对于区间(0,1)拉伸及折叠的过程。<ref name="Gleick">{{cite book |last=Gleick |first=James |title=Chaos: Making a New Science |year=1987 |publisher=Penguin Books |location=London |isbn=978-0-14-009250-9 }}</ref></div></td></tr>
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18621066378:18621066378移动页面Logistic映射至Logistics映射
2020-04-30T14:57:04Z
<p>18621066378移动页面<a href="/index.php/Logistic%E6%98%A0%E5%B0%84" title="Logistic映射">Logistic映射</a>至<a href="/index.php/Logistics%E6%98%A0%E5%B0%84" class="mw-redirect" title="Logistics映射">Logistics映射</a></p>
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薄荷:/* 数值试验 */
2020-04-30T05:15:48Z
<p><span dir="auto"><span class="autocomment">数值试验</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">2020年4月30日 (四) 05:15的版本</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l132" >第132行:</td>
<td colspan="2" class="diff-lineno">第132行:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>图6中所示为一次模拟试验运行了20000个周期,<math>x(t)</math>在这不同时刻的值在[0,1]区间上的累计分布图。图形中的平台部分表示取值概率基本为0。也就是说<math>x(t)</math>的取值基本集中在0.3~0.6和0.8~0.9这两个区间里。而往右上按照直线倾斜的部分表示<math>x(t)</math>在该区间近似呈现均匀分布。由此可见,这个时候迭代系统表现出随机性,然而整个迭代方程都是确定性的,因此认为产生了确定性的混沌。</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>图6中所示为一次模拟试验运行了20000个周期,<math>x(t)</math>在这不同时刻的值在[0,1]区间上的累计分布图。图形中的平台部分表示取值概率基本为0。也就是说<math>x(t)</math>的取值基本集中在0.3~0.6和0.8~0.9这两个区间里。而往右上按照直线倾斜的部分表示<math>x(t)</math>在该区间近似呈现均匀分布。由此可见,这个时候迭代系统表现出随机性,然而整个迭代方程都是确定性的,因此认为产生了确定性的混沌。</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===<math>3.6<μ<4</math>===</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===<math>3.6<μ<4</math>===</div></td></tr>
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薄荷
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薄荷:/* 分岔行为 */
2020-04-30T05:15:21Z
<p><span dir="auto"><span class="autocomment">分岔行为</span></span></p>
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</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l316" >第316行:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:X1x2.png|400px|thumb|图12]]</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>[[File:X1x2.png|400px|thumb|图12]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>其中,下面蓝色曲线为<math>x_1^*</math>,上面的为<math>x_2^*</math>。与上一小节的系统极限行为随<math>\mu</math>变化图相比,该图显然夸大了二分周期点的存在范围。看来,我们不能仅仅根据满足迭代关系这一个条件来确定二分周期点发生的参数区间。</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>其中,下面蓝色曲线为<math>x_1^*</math>,上面的为<math>x_2^*</math>。与上一小节的系统极限行为随<math>\mu</math>变化图相比,该图显然夸大了二分周期点的存在范围。看来,我们不能仅仅根据满足迭代关系这一个条件来确定二分周期点发生的参数区间。</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===二分周期点的稳定性===</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>===二分周期点的稳定性===</div></td></tr>
</table>
薄荷
https://wiki.swarma.org/index.php?title=Logistic%E6%98%A0%E5%B0%84&diff=6227&oldid=prev
2020年4月30日 (四) 05:14 薄荷
2020-04-30T05:14:20Z
<p></p>
<table class="diff diff-contentalign-left diff-editfont-monospace" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">←上一版本</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">2020年4月30日 (四) 05:14的版本</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l660" >第660行:</td>
<td colspan="2" class="diff-lineno">第660行:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>下面,将不同阶迭代法则之间的相似几何关系作为一个基本要求。即对于任意的迭代法则<math>f(\mu,x)</math>,其中<math>f(\mu,x)</math>可以写为<math>f(\mu,x)=\mu g(x)</math>,<math>g(x)</math>为[0,1]内的单峰函数,要求它在变换R下不变。其中R为对函数<math>f</math>进行一系列操作:</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>下面,将不同阶迭代法则之间的相似几何关系作为一个基本要求。即对于任意的迭代法则<math>f(\mu,x)</math>,其中<math>f(\mu,x)</math>可以写为<math>f(\mu,x)=\mu g(x)</math>,<math>g(x)</math>为[0,1]内的单峰函数,要求它在变换R下不变。其中R为对函数<math>f</math>进行一系列操作:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>(1)<del class="diffchange diffchange-inline">. </del>从<math>f</math>得到<math>f^{(2)}</math>:<math>f^{(2)}=f(f(\hat{\mu_1},x))</math>,其中<math>\hat{\mu_1}</math>为<math>f</math>的超稳定不动点对应的参数;</div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>(1) 从<math>f</math>得到<math>f^{(2)}</math>:<math>f^{(2)}=f(f(\hat{\mu_1},x))</math>,其中<math>\hat{\mu_1}</math>为<math>f</math>的超稳定不动点对应的参数;</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>(2)<del class="diffchange diffchange-inline">. </del>从<math>f^{(2)}</math>可以计算出它的超稳定不动点<math>\hat{\mu_2}</math>对应的参数;</div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>(2) 从<math>f^{(2)}</math>可以计算出它的超稳定不动点<math>\hat{\mu_2}</math>对应的参数;</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>(3)<del class="diffchange diffchange-inline">. </del>对<math>f^{(2)}</math>进行尺度缩放和上下左右翻转:<math>f^{(2)}(\hat{\mu_2},x)\rightarrow -\alpha f^{(2)}(\hat{\mu_2},-\frac{x}{\alpha})</math>(其中<math>\alpha</math>为一个常数,将从重整化方程的求解过程中确定它的值)。</div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>(3) 对<math>f^{(2)}</math>进行尺度缩放和上下左右翻转:<math>f^{(2)}(\hat{\mu_2},x)\rightarrow -\alpha f^{(2)}(\hat{\mu_2},-\frac{x}{\alpha})</math>(其中<math>\alpha</math>为一个常数,将从重整化方程的求解过程中确定它的值)。</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l734" >第734行:</td>
<td colspan="2" class="diff-lineno">第734行:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*[http://www.wolframscience.com/nksonline/page-918c-text “Logistic映射的历史”] ,[[一种新科学 A New Kind of Science]],作者[[史蒂芬·沃尔夫勒姆 Stephen Wolfram]]</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*[http://www.wolframscience.com/nksonline/page-918c-text “Logistic映射的历史”] ,[[一种新科学 A New Kind of Science]],作者[[史蒂芬·沃尔夫勒姆 Stephen Wolfram]]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*[http://chaosbook.org/~predrag/papers/universalFunct.html 关于周期加倍的普遍性简短历史]</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*[http://chaosbook.org/~predrag/papers/universalFunct.html 关于周期加倍的普遍性简短历史]</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>*[https://chaosbook.blogspot.com/1993/05/acceptance-speech-1993-nkt-research.html <del class="diffchange diffchange-inline">普 · 维塔诺维的《宇宙功能并不那么短的历史》</del>]</div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>*[https://chaosbook.blogspot.com/1993/05/acceptance-speech-1993-nkt-research.html <ins class="diffchange diffchange-inline">普·维塔诺维的《宇宙功能并不那么短的历史》</ins>]</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*[http://demonstrations.wolfram.com/OrbitDiagramOfTwoCoupledLogisticMaps/ 乘法耦合2周期的Logistic映射]作者:C. Pellicer-Lostao and R. Lopez-Ruiz after work by Ed Pegg Jr </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*[http://demonstrations.wolfram.com/OrbitDiagramOfTwoCoupledLogisticMaps/ 乘法耦合2周期的Logistic映射]作者:C. Pellicer-Lostao and R. Lopez-Ruiz after work by Ed Pegg Jr </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*[https://en.wikipedia.org/wiki/Wolfram_Demonstrations_Project Wolfram 演示项目]</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>*[https://en.wikipedia.org/wiki/Wolfram_Demonstrations_Project Wolfram 演示项目]</div></td></tr>
</table>
薄荷
https://wiki.swarma.org/index.php?title=Logistic%E6%98%A0%E5%B0%84&diff=6226&oldid=prev
费米子:/* μ = 4 */
2020-04-30T05:12:52Z
<p><span dir="auto"><span class="autocomment">μ = 4</span></span></p>
<table class="diff diff-contentalign-left diff-editfont-monospace" data-mw="interface">
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">←上一版本</td>
<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">2020年4月30日 (四) 05:12的版本</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l177" >第177行:</td>
<td colspan="2" class="diff-lineno">第177行:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>对于<math>x_0 \in [0,1)</math><del class="diffchange diffchange-inline">。此解没有混沌的特性。由于对不包括不稳定固定点0在内的</del><math>x_{0}</math>,当<math>t</math>趋近无限大时,<math>(1-2x_0)^{2^{t}}</math>会趋近于零,因此<math>x_t</math><del class="diffchange diffchange-inline">会趋近稳定的固定点</del><math>\tfrac{1}{2}.</math>。</div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>对于<math>x_0 \in [0,1)</math><ins class="diffchange diffchange-inline">。此解没有混沌的特性。由于对不包括不稳定的不动点点0在内的</ins><math>x_{0}</math>,当<math>t</math>趋近无限大时,<math>(1-2x_0)^{2^{t}}</math>会趋近于零,因此<math>x_t</math><ins class="diffchange diffchange-inline">会趋近稳定的不动点</ins><math>\tfrac{1}{2}.</math>。</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l200" >第200行:</td>
<td colspan="2" class="diff-lineno">第200行:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>对于<math>\mu </math>=4的Logistic映射,此时对应<math>\mu </math>= 2的帐篷映射 Tent map。(最小)长度k = 1,2,3,…的循环数是一个已知的整数序列(OEIS中的序列A001037):2,1 ,2、3、6、9、18、30、56、99、186、335、630、1161…这告诉我们,<math>\mu </math>=<del class="diffchange diffchange-inline">4的Logistic映射具有2个固定点,长度为2时的周期为1,长度为3时的周期为2,依此类推。对于素数k有序列:</del><math>2\frac{2^{k-1}-1}{k}</math></div></td><td class='diff-marker'>+</td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>对于<math>\mu </math>=4的Logistic映射,此时对应<math>\mu </math>= 2的帐篷映射 Tent map。(最小)长度k = 1,2,3,…的循环数是一个已知的整数序列(OEIS中的序列A001037):2,1 ,2、3、6、9、18、30、56、99、186、335、630、1161…这告诉我们,<math>\mu </math>=<ins class="diffchange diffchange-inline">4的Logistic映射具有2个不动点点,长度为2时的周期为1,长度为3时的周期为2,依此类推。对于素数k有序列:</ins><math>2\frac{2^{k-1}-1}{k}</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>例如:<math>2\frac{2^{13-1}-1}{13}</math>是长度为13的循环数。在所有初始条件下,映射都是混乱的,所以这些有限长度的循环都是不稳定的。</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>例如:<math>2\frac{2^{13-1}-1}{13}</math>是长度为13的循环数。在所有初始条件下,映射都是混乱的,所以这些有限长度的循环都是不稳定的。</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"></td></tr>
</table>
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