# 信息生产：逻辑斯谛克映射

The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popularized in a 1976 paper by the biologist Robert May,[1] in part as a discrete-time demographic model analogous to the logistic equation first created by Pierre François Verhulst.[2] Mathematically, the logistic map is written

The logistic map is a polynomial mapping (equivalently, recurrence relation) of degree 2, often cited as an archetypal example of how complex, chaotic behaviour can arise from very simple non-linear dynamical equations. The map was popularized in a 1976 paper by the biologist Robert May,[1] in part as a discrete-time demographic model analogous to the logistic equation first created by Pierre François Verhulst.[2] Mathematically, the logistic map is written Logistic映射是一个二次多项式映射(等价地，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 在1976年的一篇论文中推广了这一映射，[1]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 首次提出的逻辑方程。Logistic映射的数学表达式表示为:

$\displaystyle{ x(t+1)=\mu x(t)(1-x(t)) }$

where xn is a number between zero and one that represents the ratio of existing population to the maximum possible population. The values of interest for the parameter r (sometimes also denoted μ) are those in the interval [0,4]. This nonlinear difference equation is intended to capture two effects:

where xn is a number between zero and one that represents the ratio of existing population to the maximum possible population. The values of interest for the parameter r (sometimes also denoted μ) are those in the interval [0,4]. This nonlinear difference equation is intended to capture two effects:

 --趣木木（讨论）文中规定Logistic映射中的系数为μ  帐篷映射所对应的系数为r  （由于旧版集智中使用的参数是μ wiki中使用的参数是r，最后统一为μ，下会对英文原文中的r进行替换，变为μ） xn换为xt，因为表示人口模型时参数xt中的t可表时间 更加符合表达


 --趣木木（讨论）其中，t为迭代时间步，对于任意的t，$\displaystyle{ x(t)\in [0,1] }$，$\displaystyle{ \mu }$为一可调参数，为了保证映射得到的$\displaystyle{ x(t) }$始终位于[0,1]内，则$\displaystyle{ \mu\in [0,4] }$。当变化不同的参数$\displaystyle{ μ }$的时候，该方程会展现出不同的动力学极限行为（即当t趋于无穷大，x(t)的变化情况），包括：稳定点（即最终x(t)始终为同一个数值）、周期（x(t)会在2个或者多个数值之间跳跃，以及混沌：x(t)的终态不会重复，而会等概率地取遍某区间）。为补充


• reproduction where the population will increase at a rate proportional to the current population when the population size is small.
• starvation (density-dependent mortality) where the growth rate will decrease at a rate proportional to the value obtained by taking the theoretical "carrying capacity" of the environment less the current population.
• reproduction where the population will increase at a rate proportional to the current population when the population size is small.

• starvation 饥饿 (density-dependent mortality) where the growth rate will decrease at a rate proportional to the value obtained by taking the theoretical "carrying capacity" of the environment less the current population.
• 饥饿(与密度有关的死亡率) ，其增长率将以与环境的”承受能力”减去当前人口所得值成正比的速度下降

However, as a demographic model the logistic map has the pathological problem that some initial conditions and parameter values (for example, if r > 4) lead to negative population sizes. This problem does not appear in the older Ricker model, which also exhibits chaotic dynamics.

However, as a demographic model the logistic map has the pathological problem that some initial conditions and parameter values (for example, if μ > 4) lead to negative population sizes. This problem does not appear in the older Ricker model, which also exhibits chaotic dynamics. 然而，Logistic映射作为一种人口统计模型，存在着一些初始条件和参数值(如$\displaystyle{ μ\gt 4 }$ )为某值时所导致的混沌问题。这个问题在较老的瑞克模型中没有出现，该模型也展示了混沌动力学。

The r = 4 case of the logistic map is a nonlinear transformation of both the bit-shift map and the μ = 2 case of the tent map.

The r = 4 case of the logistic map is a nonlinear transformation of both the bit-shift map and the μ = 2 case of the tent map. Logistic映射的 $\displaystyle{ μ=4 }$时，其进行了移位映射和 $\displaystyle{ r = 2 }$时的帐篷映射的非线性变换。帐篷映射在数学中是指一种分段的线性映射，因其函数图像类似帐篷而得名。

 --趣木木（讨论）由于进行旧词条和原词条的合并后，遵循了系数为 μ,将原文中的参数进行对换 以保持统一


$\displaystyle{ x(t+1)-x(t)=(\mu-1) x(t) - \mu x(t)^2 }$

## 数值试验

 μ=0.9;x0=0.1;
NestList[μ # (1 - #) &, x0, 100]


### 1<μ<3

• With r between 1 and 2, the population will quickly approach the value 模板:Sfrac, independent of the initial population.

• With μ between 1 and 2, the population will quickly approach the value independent of the initial population.
• $\displaystyle{ μ }$在1到2之间，种群数量会很快接近$\displaystyle{ \frac{\mu-1}{\mu} }$,不论最初种群为何值

• With r between 2 and 3, the population will also eventually approach the same value 模板:Sfrac, but first will fluctuate around that value for some time. The rate of convergence is linear, except for r = 3, when it is dramatically slow, less than linear (see Bifurcation memory).

• With $\displaystyle{ μ }$ between 2 and 3, the population will also eventually approach the same value but first will fluctuate around that value for some time.The rate of convergence is linear, except for $\displaystyle{ μ }$ = 3, when it is dramatically slow, less than linear (see Bifurcation memory).
• $\displaystyle{ μ }$在2到3之间，人口数在经历一段时间的波动后会趋于稳定值$\displaystyle{ \frac{\mu-1}{\mu} }$。其收敛速度满足线性变化，但当$\displaystyle{ μ = 3 }$时，比线性收敛还要缓慢。（详情见 分岔记忆Bifurcation memory）
 --趣木木（讨论） 关键字Bifurcation memory 分岔记忆没有找到准确释义 添加到专有名词翻译


$\displaystyle{ x^*=\mu x^*(1-x^*) }$

$\displaystyle{ x^*_1=0,x^*_2=\frac{\mu-1}{\mu} }$

### 3<μ<3.6

• With r between 3 and 1 + 模板:Sqrt ≈ 3.44949, from almost all initial conditions the population will approach permanent oscillations between two values. These two values are dependent on r.

With μ between 3 and 1 + √6 ≈ 3.44949, from almost all initial conditions the population will approach permanent oscillations between two values. These two values are dependent on r.

• $\displaystyle{ μ }$在3到1+√6 ≈ 3.44949，在所有的初始条件下，人口数会持续在两个依赖$\displaystyle{ μ }$的值之间近似永久振荡。

• With r between 3.44949 and 3.54409 (approximately), from almost all initial conditions the population will approach permanent oscillations among four values. The latter number is a root of a 12th degree polynomial 模板:OEIS.
• With r between 3.44949 and 3.54409 (approximately), from almost all initial conditions the population will approach permanent oscillations among four values. The latter number is a root of a 12th degree polynomial (sequence A086181 in the OEIS).
• $\displaystyle{ \mu }$大约在3.44949和3.54409之间，在所有的初始条件下，人口数将在四个值之间近似永久振荡。后一个数字是一个12次多项式的根。在OEIS中查看A086181).

### μ=3.6

 --趣木木（讨论）该句摘自旧版的集智百科logistic映射的词条。With r increasing beyond 3.54409, from almost all initial conditions the population will approach oscillations among 8 values, then 16, 32, etc. The lengths of the parameter intervals that yield oscillations of a given length decrease rapidly; the ratio between the lengths of two successive bifurcation intervals approaches the Feigenbaum constant δ ≈ 4.66920. This behavior is an example of a period-doubling cascade. r increasing beyond 3.54409给定长度引起振荡的参数区间的长度迅速减小;由维基上的翻译对比  似乎有错误

• At r ≈ 3.56995 模板:OEIS is the onset of chaos, at the end of the period-doubling cascade. From almost all initial conditions, we no longer see oscillations of finite period. Slight variations in the initial population yield dramatically different results over time, a prime characteristic of chaos.
• At μ≈ 3.56995 3.56995 (sequence A098587 in the 在OEIS) is the onset of chaos, at the end of the period-doubling cascade. From almost all initial conditions, we no longer see oscillations of finite period. Slight variations in the initial population yield dramatically different results over time, a prime characteristic of chaos.
• $\displaystyle{ μ≈ 3.56995 }$（在OEIS中的A098587）,出现混沌现象，在倍周期级联的末端。在所有的初始条件下，不再观察到有限周期内的振动。 随着时间的推移，初始种群数的微小变化会产生明显不同的结果，这是混沌的主要特征。

• Most values of r beyond 3.56995 exhibit chaotic behaviour, but there are still certain isolated ranges of r that show non-chaotic behavior; these are sometimes called islands of stability. For instance, beginning at 1 + 模板:Sqrt[3] (approximately 3.82843) there is a range of parameters r that show oscillation among three values, and for slightly higher values of r oscillation among 6 values, then 12 etc.
• Most values of μ beyond 3.56995 exhibit chaotic behaviour, but there are still certain isolated ranges of μ that show non-chaotic behavior; these are sometimes called islands of stability. For instance, beginning at 1 + √8[3] (approximately 3.82843) there is a range of parameters r that show oscillation among three values, and for slightly higher values of μ oscillation among 6 values, then 12 etc.
• 3.56995以上的$\displaystyle{ μ }$值大多表现出混沌行为，但仍有一定的孤立范围表现出非混沌行为;这些岛屿有时被称为稳定岛。例如，从1 +√8[3](约3.82843)开始，有一个参数$\displaystyle{ μ }$的范围，在3个值之间显示$\displaystyle{ μ }$振荡，在6个值之间显示稍高的$\displaystyle{ μ }$振荡，然后在12个值之间显示稍高的$\displaystyle{ μ }$振荡，等等

### 3.6<μ<4

• The development of the chaotic behavior of the logistic sequence as the parameter 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.[4] There are other ranges that yield oscillation among 5 values etc.; all oscillation periods occur for some values of r. A period-doubling window with parameter c is a range of r-values consisting of a succession of subranges. The kth subrange contains the values of r for which there is a stable cycle (a cycle that attracts a set of initial points of unit measure) of period 2kc. This sequence of sub-ranges is called a cascade of harmonics.[5] In a sub-range with a stable cycle of period 2k*c, there are unstable cycles of period 2kc for all k < k*. The r value at the end of the infinite sequence of sub-ranges is called the point of accumulation of the cascade of harmonics. As r rises there is a succession of new windows with different c values. The first one is for c = 1; all subsequent windows involving odd c occur in decreasing order of c starting with arbitrarily large c.[5][6]
• The development of the chaotic behavior of the logistic sequence as the parameter 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.[4] There are other ranges that yield oscillation among 5 values etc.; all oscillation periods occur for some values of r. A period-doubling window with parameter c is a range of r-values consisting of a succession of subranges. The kth subrange contains the values of r for which there is a stable cycle (a cycle that attracts a set of initial points of unit measure) of period 2kc. This sequence of sub-ranges is called a cascade of harmonics.[5] In a sub-range with a stable cycle of period 2k*c, there are unstable cycles of period 2kc for all k < k*. The r value at the end of the infinite sequence of sub-ranges is called the point of accumulation of the cascade of harmonics. As r rises there is a succession of new windows with different c values. The first one is for c = 1; all subsequent windows involving odd c occur in decreasing order of c starting with arbitrarily large c.[5][6]

  --~~关键字the Pomeau–Manneville scenario，cascade of harmonics未找到合适释义，添加在专有名词查询列表  As r rises there is a succession of new windows with different c values. The first one is for c = 1; all subsequent windows involving odd c occur in decreasing order of c starting with arbitrarily large c.[5][6]该句由于只查到级联谐波是一种将交替的谐波和开弦相结合以产生类似竖琴效果的技术。不太确定 谐波级联cascade of harmonics与级联谐波的区别 不知道其中的windows的确切意义；其中的windows不太理解怎么翻译 故进行了直译


$\displaystyle{ \mu }$持续增大的时候，迭代运行的轨道就会在周期类型和混沌类型之间来回切换。直到$\displaystyle{ \mu=4 }$，系统处于完全混沌的状态，最终的长期行为会在[0,1]区间上均匀分布。

### $\displaystyle{ μ = 4 }$

The special case of r = 4 can in fact be solved exactly, as can the case with r = 2;[7] however, the general case can only be predicted statistically.[8]

The special case of $\displaystyle{ μ = 4 }$ can in fact be solved exactly, as can the case with $\displaystyle{ μ = 2 }$; however, the general case can only be predicted statistically. $\displaystyle{ μ = 4 }$的特殊之处与$\displaystyle{ μ = 2 }$相同，其可以获得精确的解，虽然，一般情况只能通过统计来预测。

The solution when r = 4 is,[7][9] The solution when$\displaystyle{ μ = 4 }$ is, 当$\displaystyle{ μ = 4 }$

$\displaystyle{ x_{n}=\sin^{2}\left(2^{n} \theta \pi\right), }$

where the initial condition parameter θ is given by

where the initial condition parameter θ is given by 给出初始参数 θ

$\displaystyle{ \theta = \tfrac{1}{\pi}\sin^{-1}\left(\sqrt{x_0}\right). }$

For rational θ, after a finite number of iterations xn maps into a periodic sequence. But almost all θ are irrational, and, for irrational θ, xn never repeats itself – it is non-periodic. This solution equation clearly demonstrates the two key features of chaos – stretching and folding: the factor 2n shows the exponential growth of stretching, which results in sensitive dependence on initial conditions, while the squared sine function keeps xn folded within the range [0,1].

For rational θ, after a finite number of iterations $\displaystyle{ x_n }$ maps into a periodic sequence.But almost all θ are irrational, and, for irrational θ, $\displaystyle{ x_n }$ never repeats itself – it is non-periodic. This solution equation clearly demonstrates the two key features of chaos– stretching and folding: the factor $\displaystyle{ 2^n }$shows the exponential growth of stretching, which results in sensitive dependence on initial conditions, while the squared sine function keeps $\displaystyle{ x_t }$ folded within the range $\displaystyle{ [0,1] }$. 针对有理数的 θ，有限次数的迭代后$\displaystyle{ x_t }$就会变成一个周期性的数列。不过几乎所有的θ都是无理数，此时$\displaystyle{ x_t }$不会重复，因此没有周期解。此解可以清楚的看出混沌的二个重要特征：拉伸及折叠。系数$\displaystyle{ 2^t }$表示拉伸的指数增长，因此造成蝴蝶效应，也就是对初始值的高度依赖性，而解中包括正弦函数的平方，使解折叠在[0, 1]的范围内。

For r = 4 an equivalent solution in terms of complex numbers instead of trigonometric functions is[10]

For $\displaystyle{ μ= 4 }$an equivalent solution in terms of complex numbers instead of trigonometric functions is 对于$\displaystyle{ μ= 4 }$，用复数代替三角函数的等价解为：

$\displaystyle{ x_t=\frac{-\alpha^{2^t} -\alpha^{-2^t} +2}{4} }$

where α is either of the complex numbers

where $\displaystyle{ α }$ is either of the complex numbers α也是复数

with modulus equal to 1. Just as the squared sine function in the trigonometric solution leads to neither shrinkage nor expansion of the set of points visited, in the latter solution this effect is accomplished by the unit modulus of α.

with modulus equal to 1. Just as the squared sine function in the trigonometric solution leads to neither shrinkage nor expansion of the set of points visited, in the latter solution this effect is accomplished by the unit modulus of $\displaystyle{ α }$ 其模量=1，正如就像三角函数中的平方正弦函数不会导致点集的缩小或扩大，在后者的解决方案中，可通过单位模量来实现该种结果。

By contrast, the solution when r = 2 is[10]

By contrast, the solution when r = 2 is[14] 相比之下，μ=2的解为：[14] $\displaystyle{ x_t = \frac{1}{2} - \frac{1}{2}(1-2x_0)^{2^{t}} }$

for x0 ∈ [0,1). Since (1 − 2x0) ∈ (−1,1) for any value of x0 other than the unstable fixed point 0, the term (1 − 2x0)2n goes to 0 as n goes to infinity, so xn goes to the stable fixed point 模板:Sfrac.

For the r = 4 case, from almost all initial conditions the iterate sequence is chaotic. Nevertheless, there exist an infinite number of initial conditions that lead to cycles, and indeed there exist cycles of length k for all integers k ≥ 1. We can exploit the relationship of the logistic map to the dyadic transformation (also known as the bit-shift map) to find cycles of any length. If x follows the logistic map xn + 1 = 4xn(1 − xn) and y follows the dyadic transformation

For the r = 4 case, from almost all initial conditions the iterate sequence is chaotic. Nevertheless, there exist an infinite number of initial conditions that lead to cycles, and indeed there exist cycles of length k for all integers k ≥ 1. We can exploit the relationship of the logistic map to the dyadic transformation (also known as the bit-shift map) to find cycles of any length. If x follows the logistic map xn + 1 = 4xn(1 − xn) and y follows the dyadic transformation μ= 4时，几乎所有的初值都会使Logistic映射出现混沌特性，不过也存在无限个初值会使Logistic映射最后呈周期性变化。而且对于所有正整数，都存在一初值使Logistic映射的周期为正整数。可以利用Logistic映射和移位映射之间的关系来找出任何周期的循环。若x依照Logistic映射$\displaystyle{ x_{t+1} = 4 x_t(1-x_t) \, }$,而y依照移位映射 $\displaystyle{ y_{t+1}=\begin{cases}2y_t & 0 \le y_t \lt 0.5 \\2y_t -1 & 0.5 \le y_t \lt 1, \end{cases} }$

then the two are related by a homeomorphism then the two are related by a homeomorphism 则二个变量的关系如下： $\displaystyle{ x_{t}=\sin^{2}(2 \pi y_{t}) }$.

The reason that the dyadic transformation is also called the bit-shift map is that when y is written in binary notation, the map moves the binary point one place to the right (and if the bit to the left of the binary point has become a "1", this "1" is changed to a "0"). A cycle of length 3, for example, occurs if an iterate has a 3-bit repeating sequence in its binary expansion (which is not also a one-bit repeating sequence): 001, 010, 100, 110, 101, or 011. The iterate 001001001… maps into 010010010..., which maps into 100100100..., which in turn maps into the original 001001001...; so this is a 3-cycle of the bit shift map. And the other three binary-expansion repeating sequences give the 3-cycle 110110110… → 101101101… → 011011011… → 110110110.… Either of these 3-cycles can be converted to fraction form: for example, the first-given 3-cycle can be written as 模板:Sfrac模板:Sfrac模板:Sfrac模板:Sfrac.

The reason that the dyadic transformation is also called the bit-shift map is that when y is written in binary notation, the map moves the binary point one place to the right (and if the bit to the left of the binary point has become a "1", this "1" is changed to a "0"). A cycle of length 3, for example, occurs if an iterate has a 3-bit repeating sequence in its binary expansion (which is not also a one-bit repeating sequence): 001, 010, 100, 110, 101, or 011. The iterate 001001001… maps into 010010010..., which maps into 100100100..., which in turn maps into the original 001001001...; so this is a 3-cycle of the bit shift map. And the other three binary-expansion repeating sequences give the 3-cycle 110110110… → 101101101… → 011011011… → 110110110.… Either of these 3-cycles can be converted to fraction form: for example, the first-given 3-cycle can be written as 1/7→ 2/7→ 4/7→ 1/7. 当y以二进制表示时，映射会将二进制的数字左移一位（若左边的二进制点为1，则这个1会变为0），该种映射也称为二元转换或移位映射。例如，如果迭代的二进制扩展中有一个3位重复序列(并不是一个1位重复序列)，则会发生长度为3的循环。迭代001001001…映射到010010010…，映射到100100100…，反过来又映射成原来的001001001…;这是一个3循环的移位图。循环节为010, 011, 100, 101, 110 时也会有类似情形。

Using the above translation from the bit-shift map to the $\displaystyle{ r = 4 }$ logistic map gives the corresponding logistic cycle 0.611260467… → 0.950484434… → 0.188255099… → 0.611260467.… We could similarly translate the other bit-shift 3-cycle into its corresponding logistic cycle. Likewise, cycles of any length k can be found in the bit-shift map and then translated into the corresponding logistic cycles.

Using the above translation from the bit-shift map to the {\displaystyle r=4}{\displaystyle r=4} logistic map gives the corresponding logistic cycle 0.611260467… → 0.950484434… → 0.188255099… → 0.611260467.… We could similarly translate the other bit-shift 3-cycle into its corresponding logistic cycle. Likewise, cycles of any length k can be found in the bit-shift map and then translated into the corresponding logistic cycles. 将其转换到μ=4的Logistic映射后，所得到的逻辑循环为611260467... → 950484434... → 188255099... → 611260467...。其他周期为3的循环也可以转换为Logistic映射。同样地，任何长度为k的循环都可以在移位映射中找到，然后转换成相应的Logistic映射。

The number of cycles of (minimal) length k = 1, 2, 3,… for the logistic map with r = 4 (tent map with μ = 2) is a known integer sequence 模板:OEIS: 2, 1, 2, 3, 6, 9, 18, 30, 56, 99, 186, 335, 630, 1161…. This tells us that the logistic map with r = 4 has 2 fixed points, 1 cycle of length 2, 2 cycles of length 3 and so on. This sequence takes a particularly simple form for prime k: 2 ⋅ 模板:Sfrac. For example: 2 ⋅ 模板:Sfrac = 630 is the number of cycles of length 13. Since this case of the logistic map is chaotic for almost all initial conditions, all of these finite-length cycles are unstable.

The number of cycles of (minimal) length k = 1, 2, 3,… for the logistic map with r = 4 (tent map with μ = 2) is a known integer sequence (sequence A001037 in the OEIS): 2, 1, 2, 3, 6, 9, 18, 30, 56, 99, 186, 335, 630, 1161…. This tells us that the logistic map with r = 4 has 2 fixed points, 1 cycle of length 2, 2 cycles of length 3 and so on. This sequence takes a particularly simple form for prime k: 2 ⋅ 2k − 1 − 1/k。For example: 2 ⋅ 213 − 1 − 1/13 = 630 is the number of cycles of length 13. Since this case of the logistic map is chaotic for almost all initial conditions, all of these finite-length cycles are unstable. 对于μ = 4的Logistic映射，此时对应r= 2的帐篷映射。（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，μ = 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：$\displaystyle{ 2\frac{2^{k-1}-1}{k} }$ 例如：$\displaystyle{ 2\frac{2^{13-1}-1}{13} }$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。

## 不同参数$\displaystyle{ \mu }$下的极限行为

For any value of r there is at most one stable cycle. If a stable cycle exists, it is globally stable, attracting almost all points.[11]:13 Some values of r with a stable cycle of some period have infinitely many unstable cycles of various periods.

For any value of r there is at most one stable cycle. If a stable cycle exists, it is globally stable, attracting almost all points.[7]:13 Some values of r with a stable cycle of some period have infinitely many unstable cycles of various periods. 不论$\displaystyle{ \mu }$取任意值，最多有一个稳定的周期。 如果存在一个稳定的周期，那么它是全局稳定的，吸引了几乎所有的点。 [7] : 13 一些具有周期稳定循环的$\displaystyle{ \mu }$在某些时候会有无穷多个不同周期的不稳定循环。

 interval = 0.001;
results = Reverse[Transpose[Table[
logisticValues =
Table[Nest[a # (1 - #) &, RandomReal[], 2000], {1000}];
intervals = Table[i, {i, 0, 1 - interval, interval}];
result = BinCounts[logisticValues, {0, 1, interval}]/1000;
Log[result + 0.001]
, {a, 2.9, 4, 0.001}]]];
ArrayPlot[70 + 10 results, FrameLabel -> {"x(T)", "\[Mu]"},
FrameTicks -> {Table[{i, N[(i - 1)/(Length[results] - 1)]}, {i,
0.1*(Length[results] - 1) + 1,
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## 混沌与Logistic映射

Logistic function f (blue) and its iterated versions f2, f3, f4 and f5 for r = 3.5. For example, for any initial value on the horizontal axis, f4 gives the value of the iterate four iterations later.The relative simplicity of the logistic map makes it a widely used point of entry into a consideration of the concept of chaos. A rough description of chaos is that chaotic systems exhibit a great sensitivity to initial conditions—a property of the logistic map for most values of r between about 3.57 and 4 (as noted above).[1] A common source of such sensitivity to initial conditions is that the map represents a repeated folding and stretching of the space on which it is defined. In the case of the logistic map, the quadratic difference equation describing it may be thought of as a stretching-and-folding operation on the interval (0,1).[12]

Logistic function f (blue) and its iterated versions f2, f3, f4 and f5 for r = 3.5. For example, for any initial value on the horizontal axis, f4 gives the value of the iterate four iterations later.The relative simplicity of the logistic map makes it a widely used point of entry into a consideration of the concept of chaos. A rough description of chaos is that chaotic systems exhibit a great sensitivity to initial conditions—a property of the logistic map for most values of r between about 3.57 and 4 (as noted above).[1] A common source of such sensitivity to initial conditions is that the map represents a repeated folding and stretching of the space on which it is defined. In the case of the logistic map, the quadratic difference equation describing it may be thought of as a stretching-and-folding operation on the interval (0,1).[8] Logistic映射 f (blue)在μ=3.5的条件下进行迭代，得到$\displaystyle{ f^2 }$$\displaystyle{ f^3 }$$\displaystyle{ f^4 }$$\displaystyle{ f^5 }$。 例如，对于水平轴上的任何初始值，$\displaystyle{ f^4 }$为四次迭代之后得到的值。和其他混沌系统比较，Logistic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。μ是在3.57及4之间的大部分数值都可以使Logistic映射出现该特性。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。Logistic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。

The following figure illustrates the stretching and folding over a sequence of iterates of the map. Figure (a), left, shows a two-dimensional Poincaré plot of the logistic map's state space for r = 4, and clearly shows the quadratic curve of the difference equation (1). However, we can embed the same sequence in a three-dimensional state space, in order to investigate the deeper structure of the map. Figure (b), right, demonstrates this, showing how initially nearby points begin to diverge, particularly in those regions of xt corresponding to the steeper sections of the plot.

The following figure illustrates the stretching and folding over a sequence of iterates of the map. Figure (a), left, shows a two-dimensional Poincaré plot of the logistic map's state space for μ = 4, and clearly shows the quadratic curve of the difference equation (1). However, we can embed the same sequence in a three-dimensional state space, in order to investigate the deeper structure of the map. Figure (b), right, demonstrates this, showing how initially nearby points begin to diverge, particularly in those regions of xt corresponding to the steeper sections of the plot. 图10中，右图说明了在Logistic映射的迭代序列上的伸展和折叠。左边 图(a) 显示了Logistic映射在μ=4条件下的二维庞加莱图，并清楚地显示了差分方程的二次曲线。利用二维及三维的相图可以看出一些Logistic映射的特性。以$\displaystyle{ μ=4 }$的Logistic映射为例，二维相图为一抛物线，但是若用$\displaystyle{ x_{n},x_{n+1},x_{n+2})}(x_{t},x_{{t+1}},x_{{t+2}} }$绘制三维相图，可看出进一步的结构，例如几个一开始很接近的点在迭代后开始发散．特别是位在斜率较大位置的点。 此外，以便研究Logistic映射的更深层结构。 图(b)演示了在最初点的附近是如何开始分叉的，特别是在与图中更陡的部分相对应的 $\displaystyle{ x_t }$ 区域。

This stretching-and-folding does not just produce a gradual divergence of the sequences of iterates, but an exponential divergence (see Lyapunov exponents), evidenced also by the complexity and unpredictability of the chaotic logistic map. In fact, exponential divergence of sequences of iterates explains the connection between chaos and unpredictability: a small error in the supposed initial state of the system will tend to correspond to a large error later in its evolution. Hence, predictions about future states become progressively (indeed, exponentially) worse when there are even very small errors in our knowledge of the initial state. This quality of unpredictability and apparent randomness led the logistic map equation to be used as a pseudo-random number generator in early computers.[12]

This stretching-and-folding does not just produce a gradual divergence of the sequences of iterates, but an exponential divergence (see Lyapunov exponents), evidenced also by the complexity and unpredictability of the chaotic logistic map. In fact, exponential divergence of sequences of iterates explains the connection between chaos and unpredictability: a small error in the supposed initial state of the system will tend to correspond to a large error later in its evolution. Hence, predictions about future states become progressively (indeed, exponentially) worse when there are even very small errors in our knowledge of the initial state. This quality of unpredictability and apparent randomness led the logistic map equation to be used as a pseudo-random number generator in early computers.[8] 拉伸及折叠的结果使迭代的数列以指数形式发散（参照李亚普诺夫指数），可以用有混沌特性时的Logistic映射的复杂性及不可预测性加以说明。事实上，数列的指数发散说明了混沌和不可预测性之间的关系：初值微小的误差在迭代过程中会以指数成长的方式增加，导致结果出现很大的误差。因此当对于初始状态的有微小的误差时．对未来状态的预测准确度也会随迭代次数增加而快速变差。这种不可预测性和明显的随机性使得在早期的计算机中利用Logistic方程生成伪随机数。

Since the map is confined to an interval on the real number line, its dimension is less than or equal to unity. Numerical estimates yield a correlation dimension of 模板:Val (Grassberger, 1983), a Hausdorff dimension of about 0.538 (Grassberger 1981), and an information dimension of approximately 0.5170976 (Grassberger 1983) for r ≈ 3.5699456 (onset of chaos). Note: It can be shown that the correlation dimension is certainly between 0.4926 and 0.5024.

Since the map is confined to an interval on the real number line, its dimension is less than or equal to unity. Numerical estimates yield a correlation dimension of 0.500±0.005 (Grassberger, 1983), a Hausdorff dimension of about 0.538 (Grassberger 1981), and an information dimension of approximately 0.5170976 (Grassberger 1983) for μ ≈ 3.5699456 (onset of chaos). Note: It can be shown that the correlation dimension is certainly between 0.4926 and 0.5024. 由于映射的函数值收敛于某一特定值，其维度小于或等于1。依数值分析的结果，在μ=3.5699456...时（刚开始混沌特性时），其关联维度为0.500 ± 0.005[4]（Grassberger,1983）、豪斯多夫维数大约是0.538(Grassberger 1981)[5]，而分形维数为0.5170976...[4]，还可得知关联维度进一步精确在0.4926和0.5024之间。

It is often possible, however, to make precise and accurate statements about the likelihood of a future state in a chaotic system. If a (possibly chaotic) dynamical system has an attractor, then there exists a probability measure that gives the long-run proportion of time spent by the system in the various regions of the attractor. In the case of the logistic map with parameter r = 4 and an initial state in (0,1), the attractor is also the interval (0,1) and the probability measure corresponds to the beta distribution with parameters a = 0.5 and b = 0.5. Specifically,[13] the invariant measure is

It is often possible, however, to make precise and accurate statements about the likelihood of a future state in a chaotic system. If a (possibly chaotic) dynamical system has an attractor, then there exists a probability measure that gives the long-run proportion of time spent by the system in the various regions of the attractor. In the case of the logistic map with parameter r = 4 and an initial state in (0,1), the attractor is also the interval (0,1) and the probability measure corresponds to the beta distribution with parameters a = 0.5 and b = 0.5. Specifically,[9] the invariant measure is 有些混沌系统可对于其未来状态的可能性作准确的描述。若一个可能有混沌特性的动力系统存在吸引子，则存在一概率量测描述系统长期在吸引子各部分所花时间的比例。以$\displaystyle{ μ=4 }$的Logistic映射为例，初始状态在区间(0,1)中，而吸引子也在区间(0,1)中，其概率量测对应参数$\displaystyle{ a=0.5,b=0.5 }$的Β分布[6]，其不变测度为 $\displaystyle{ {\frac {1}{\pi {\sqrt {x(1-x)}}}}.}{\displaystyle {\frac {1}{\pi {\sqrt {x(1-x)}}}} }$

Unpredictability is not randomness, but in some circumstances looks very much like it. Hence, and fortunately, even if we know very little about the initial state of the logistic map (or some other chaotic system), we can still say something about the distribution of states arbitrarily far into the future, and use this knowledge to inform decisions based on the state of the system.

Unpredictability is not randomness, but in some circumstances looks very much like it. Hence, and fortunately, even if we know very little about the initial state of the logistic map (or some other chaotic system), we can still say something about the distribution of states arbitrarily far into the future, and use this knowledge to inform decisions based on the state of the system. 不可预期性和随机并不一样，不过在一些情形下这二者很类似。 因此，幸运的是，即使我们对Logistic映射(或其他混沌系统)的初始状态知之甚少，我们仍然可以说一些关于任意未来状态分布的问题，并参考一些信息来判断系统的状态从而做出决定。

## 分岔行为

Bifurcation diagram for the logistic map. The attractor for any value of the parameter μis shown on the vertical line at that μ.图11 物流图的分岔图。参数μ的任何值的吸引子都显示在那个r的垂直线上。

The bifurcation diagram at right summarizes this. The horizontal axis shows the possible values of the parameter r while the vertical axis shows the set of values of x visited asymptotically from almost all initial conditions by the iterates of the logistic equation with that r value.

The bifurcation diagram at right summarizes this. The horizontal axis shows the possible values of the parameter r while the vertical axis shows the set of values of x visited asymptotically from almost all initial conditions by the iterates of the logistic equation with that r value.

The bifurcation diagram is a self-similar: if we zoom in on the above-mentioned value r ≈ 3.82843 and focus on one arm of the three, the situation nearby looks like a shrunk and slightly distorted version of the whole diagram. The same is true for all other non-chaotic points. This is an example of the deep and ubiquitous connection between chaos and fractals.

The bifurcation diagram is a self-similar: if we zoom in on the above-mentioned value r ≈ 3.82843 and focus on one arm of the three, the situation nearby looks like a shrunk and slightly distorted version of the whole diagram. The same is true for all other non-chaotic points. This is an example of the deep and ubiquitous connection between chaos and fractals. 分岔图具有自相似性:如果我们把上面提到的μ≈3.82843的图像放大，然后把焦点放在这三个值中的一个上，那么附近的情况看起来就像是整个图的缩小和略微扭曲的版本。这同样适用于所有其他非混沌点。这是混沌和分形之间深刻而普遍的联系的一个例子。

### 二分周期点

$\displaystyle{ x^*=\mu x^*(1-x^*) }$

$\displaystyle{ \left\{\begin{array}{ll} x_2=f(x_1),&\\ x_1=f(x_2).& \end{array}\right. }$

$\displaystyle{ x^*=f(f(x^*))= f^{(2)}(x^*)=\mu^2 x^* - \mu^2 (x^*)^2 - \mu^3 (x^*)^2 + 2 \mu^3 (x^*)^3 - \mu^3 (x^*)^4 }$

$\displaystyle{ x_1^*=\frac{1}{2}(1+\frac{1}{\mu}+\frac{\sqrt{-3-2\mu+\mu^2}}{\mu}),x_2^*=\frac{1}{2}(1+\frac{1}{\mu}-\frac{\sqrt{-3-2\mu+\mu^2}} {\mu}),x_3^*=0, x_4^*=\frac{\mu-1}{\mu} }$

$\displaystyle{ \mu\gt 3 }$或者$\displaystyle{ \mu\lt -1 }$

### 二分周期点的稳定性

$\displaystyle{ x^*=f(f(x^*))=f^{(2)}(x^*) }$

$\displaystyle{ x^*+\epsilon(t+1)=f^{(2)}(x^*+\epsilon(t)) }$

$\displaystyle{ f^{(2)}(x^*+\epsilon(t))\approx f^{(2)}(x^*)+\frac{\partial f^{(2)}(x)}{\partial{x}}\mid_{x=x^*}\epsilon(t) }$

$\displaystyle{ x^*+\epsilon(t+1)\approx f^{(2)}(x^*)+\frac{\partial f^{(2)}(x)}{\partial{x}}\mid_{x=x^*}\epsilon(t) }$

$\displaystyle{ \epsilon(t+1)=\frac{\partial f^{(2)}(x)}{\partial{x}}\mid_{x=x^*}\epsilon(t) }$

$\displaystyle{ \frac{\epsilon(t+1)}{\epsilon(t)}=\frac{\partial f^{(2)}(x)}{\partial{x}}\mid_{x=x^*} }$

$\displaystyle{ \begin{vmatrix}\frac{\epsilon(t+1)}{\epsilon(t)}\end{vmatrix}=\begin{vmatrix}\frac{\partial f^{(2)}(x)}{\partial{x}}\mid_{x=x^*}\end{vmatrix}\lt 1 }$

$\displaystyle{ 3\lt \mu\lt 1+\sqrt{6} }$

### 其它的倍分周期点

$\displaystyle{ x^*=f^{(2^p)}(x^*) }$

$\displaystyle{ \begin{vmatrix}\partial{f^{(2^p)}}\end{vmatrix}\lt 1 }$

## 费根鲍姆常数

### $\displaystyle{ δ }$

$\displaystyle{ \delta_n=\frac{\mu_{n-1}-\mu_{n-2}}{\mu_{n}-\mu_{n-1}} }$

n 周期 分岔参数 ($\displaystyle{ \mu_n }$) 比例 $\displaystyle{ \delta_n=\dfrac{\mu_{n-1}-\mu_{n-2}}{\mu_n-\mu_{n-1}} }$
1 2 3 N/A
2 4 3.4494897 N/A
3 8 3.5440903 4.7514
4 16 3.5644073 4.6562
5 32 3.5687594 4.6683
6 64 3.5696916 4.6686
7 128 3.5698913 4.6692
8 256 3.5699340 4.6694

$\displaystyle{ \delta=4.6692016091029906718532038\cdot\cdot\cdot }$

### $\displaystyle{ α }$

$\displaystyle{ (\frac{\partial{f^{(2^p)}(\mu,x)}}{\partial{x}})|_{x=x^*}=0 }$

### 普适性

Universality of one-dimensional maps with parabolic maxima and Feigenbaum constants $\displaystyle{ \delta=4.669201... }$,$\displaystyle{ \alpha=2.502907... }$ [14] [15] is well visible with map proposed as a toy model for discrete laser dynamics: $\displaystyle{ x \rightarrow G x (1 - \tanh (x)) }$, where $\displaystyle{ x }$ stands for electric field amplitude, $\displaystyle{ G }$[16] is laser gain as bifurcation parameter.

Universality of one-dimensional maps with parabolic maxima and Feigenbaum constants $\displaystyle{ \delta=4.669201... }$,$\displaystyle{ \alpha=2.502907... }$ [15] [16] is well visible with map proposed as a toy model for discrete laser dynamics: $\displaystyle{ x \rightarrow G x (1 - \tanh (x)) }$, where {\displaystyle x}x stands for electric field amplitude, $\displaystyle{ G }$G[17] is laser gain as bifurcation parameter. 具有抛物极大值和费根鲍姆常数（包括$\displaystyle{ \delta=4.669201... }$,$\displaystyle{ \alpha=2.502907... }$）的一维映射在Logistic映射作为离散激光动力学的模型时容易察觉其具有的普遍性质。$\displaystyle{ x \rightarrow G x (1 - \tanh (x)) }$，x为电场振幅，$\displaystyle{ G }$[17]为激光增益分岔参数。

The gradual increase of $\displaystyle{ G }$ at interval $\displaystyle{ [0, \infty) }$ changes dynamics from regular to chaotic one [17] with qualitatively the same bifurcation diagram as those for logistic map.

The gradual increase of $\displaystyle{ G }$at interval $\displaystyle{ [0, \infty) }$ changes dynamics from regular to chaotic one [18] with qualitatively the same bifurcation diagram as those for logistic map. 当$\displaystyle{ G }$在区间$\displaystyle{ [0， \infty) }$中逐渐增大时，其动力学由规则型变为混沌型，其分岔图在某些性质上与Logistic映射相同。

$\displaystyle{ f(x)=a-x^2=a(1-\frac{x^2}{a}). }$

n 周期 分岔参数 (an) 比例 $\displaystyle{ \dfrac{a_{n-1}-a_{n-2}}{a_n-a_{n-1}} }$
1 2 0.75 N/A
2 4 1.25 N/A
3 8 1.3680989 4.2337
4 16 1.3940462 4.5515
5 32 1.3996312 4.6458
6 64 1.4008287 4.6639
7 128 1.4010853 4.6682
8 256 1.4011402 4.6689

n 周期 分岔参数 (an) 比例 $\displaystyle{ \dfrac{a_{n-1}-a_{n-2}}{a_n-a_{n-1}} }$
1 2 2.121 N/A
2 4 2.263 N/A
3 8 2.294 4.665
4 16 2.300 4.668
5 32 2.301 4.669
6 64 2.302 4.669

## 重整化群方程与费根鲍姆常数

### 时间上的尺度变换

$\displaystyle{ x(t+1)=f(x(t)) }$

$\displaystyle{ x'(t+1)=f(f(x'(t))=f^{(2)}(x'(t)) }$

### 图形变换与相似性

$\displaystyle{ f^{(2)}\rightarrow -\alpha f^{(4)}(\hat{\mu_2},-\frac{x}{\alpha}) }$

$\displaystyle{ f^{(2^p)}\rightarrow -\alpha f^{(2^{(p+1)})}(\hat{\mu_2},-\frac{x}{\alpha}) }$

### 重整化

#### 重整化群方程

(1). 从f得到f(2)$\displaystyle{ f^{(2)}=f(f(\hat{\mu_1},x)) }$，其中$\displaystyle{ \hat{\mu_1} }$为f的超稳定不动点对应的参数；

(2). 从f(2)可以计算出它的超稳定不动点$\displaystyle{ \hat{\mu_2} }$对应的参数；

(3). 对f(2)进行尺度缩放和上下左右翻转：$\displaystyle{ f^{(2)}(\hat{\mu_2},x)\rightarrow -\alpha f^{(2)}(\hat{\mu_2},-\frac{x}{\alpha}) }$（其中α为一个常数，将从重整化方程的求解过程中确定它的值）。

$\displaystyle{ R(f)= -\alpha f^{(2)}(\hat{\mu_2},-\frac{x}{\alpha}) }$

$\displaystyle{ f_1(\hat{\mu_1},x)\rightarrow f_2(\hat{\mu_2},x)\rightarrow f_3(\hat{\mu_3},x)\cdot\cdot\cdot }$

$\displaystyle{ f_{n+1}(\hat{\mu_{n+1}},x)=R(f_n(\hat{\mu_n},x)) }$

$\displaystyle{ g(\mu,x)=\lim_{n\rightarrow \infty}f_n(\hat{\mu_{n}},x) }$

$\displaystyle{ g(\mu,x)=R(g(\mu,x)) }$

$\displaystyle{ g(\mu,x)=-\alpha g(\mu,g(\mu,-\frac{x}{\alpha})) }$

## 编者推荐

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