倍分岔周期图

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In mathematics, particularly in dynamical systems, a bifurcation diagram shows the values visited or approached asymptotically (fixed points, periodic orbits, or chaotic attractors) of a system as a function of a bifurcation parameter in the system. It is usual to represent stable values with a solid line and unstable values with a dotted line, although often the unstable points are omitted. Bifurcation diagrams enable the visualization of bifurcation theory.

In mathematics, particularly in dynamical systems, a bifurcation diagram shows the values visited or approached asymptotically (fixed points, periodic orbits, or chaotic attractors) of a system as a function of a bifurcation parameter in the system. It is usual to represent stable values with a solid line and unstable values with a dotted line, although often the unstable points are omitted. Bifurcation diagrams enable the visualization of bifurcation theory.

在数学中,特别是在动力系统中,“分岔图”表示系统作为分岔参数的函数访问或接近渐近值(不动点、周期轨道或混沌吸引子)。通常用实线表示稳定值,用虚线表示不稳定值,而不稳定点通常被省略。分岔图使分岔理论的可视化成为可能。


文件:Diagram bifurkacji anim small.gif
Animation showing the formation of bifurcation diagram

Animation showing the formation of bifurcation diagram

动画展示了分枝图的形成

文件:Circle map bifurcation.jpeg
Bifurcation diagram of the circle map. Black regions correspond to Arnold tongues.

Bifurcation diagram of the circle map. Black regions correspond to Arnold tongues.

分枝图[圆形地图。黑色区域对应于阿诺德舌头。]

Bifurcations in 1D discrete dynamical systems

Logistic map

文件:Logistic Map Bifurcation Diagram, Matplotlib.svg
Bifurcation diagram of the logistic map. The attractor for any value of the parameter r is shown on the vertical line at that r.

Bifurcation diagram of the logistic map. The attractor for any value of the parameter r is shown on the vertical line at that r.

[逻辑地图]的分枝图。参数 r 的任意值的吸引子都在 r 的垂直线上显示出来

An example is the bifurcation diagram of the logistic map:

An example is the bifurcation diagram of the logistic map:

一个例子是逻辑地图的分枝图:


[math]\displaystyle{ x_{n+1}=rx_n(1-x_n). \, }[/math]

[math]\displaystyle{ x_{n+1}=rx_n(1-x_n). \, }[/math]

数学 x { n + 1} rx n (1-xn)。数学


The bifurcation parameter r is shown on the horizontal axis of the plot and the vertical axis shows the set of values of the logistic function visited asymptotically from almost all initial conditions.

The bifurcation parameter r is shown on the horizontal axis of the plot and the vertical axis shows the set of values of the logistic function visited asymptotically from almost all initial conditions.

分叉参数 r 显示在图的水平轴上,而垂直轴显示了所有初始条件下所访问的 Logistic函数的渐近值。


The bifurcation diagram shows the forking of the periods of stable orbits from 1 to 2 to 4 to 8 etc. Each of these bifurcation points is a period-doubling bifurcation.

The bifurcation diagram shows the forking of the periods of stable orbits from 1 to 2 to 4 to 8 etc. Each of these bifurcation points is a period-doubling bifurcation.

分枝图图显示了稳定轨道周期的分叉,从1到2到4到8等等。每一个这样的分叉点都是一个週期加倍分岔。

The ratio of the lengths of successive intervals between values of r for which bifurcation occurs converges to the first Feigenbaum constant.

The ratio of the lengths of successive intervals between values of r for which bifurcation occurs converges to the first Feigenbaum constant.

分岔发生时 r 值之间连续区间长度的比值收敛于第一个 Feigenbaum 常数。


The diagram also shows period doublings from 3 to 6 to 12 etc., from 5 to 10 to 20 etc., and so forth.

The diagram also shows period doublings from 3 to 6 to 12 etc., from 5 to 10 to 20 etc., and so forth.

该图还显示了周期倍增,从3到6到12等等,从5到10到20等等。


Symmetry breaking in bifurcation sets

文件:Asymbif.gif
Symmetry breaking in pitchfork bifurcation as the parameter ε is varied. ε = 0 is the case of symmetric pitchfork bifurcation.

Symmetry breaking in pitchfork bifurcation as the parameter ε is varied. ε = 0 is the case of symmetric pitchfork bifurcation.

对称性破缺的叉式分岔是多种多样的。[ & epsilon; 0是对称叉式分岔的情况]

In a dynamical system such as

In a dynamical system such as

在这样的动力系统里


[math]\displaystyle{ \ddot {x} + f(x;\mu) + \varepsilon g(x) = 0, }[/math]

[math]\displaystyle{ \ddot {x} + f(x;\mu) + \varepsilon g(x) = 0, }[/math]

Math ddot { x } + f (x; mu) + varepsilon g (x)0,/ math


which is structurally stable when [math]\displaystyle{ \mu \neq 0 }[/math], if a bifurcation diagram is plotted, treating [math]\displaystyle{ \mu }[/math] as the bifurcation parameter, but for different values of [math]\displaystyle{ \varepsilon }[/math], the case [math]\displaystyle{ \varepsilon = 0 }[/math] is the symmetric pitchfork bifurcation. When [math]\displaystyle{ \varepsilon \neq 0 }[/math], we say we have a pitchfork with broken symmetry. This is illustrated in the animation on the right.

which is structurally stable when [math]\displaystyle{ \mu \neq 0 }[/math], if a bifurcation diagram is plotted, treating [math]\displaystyle{ \mu }[/math] as the bifurcation parameter, but for different values of [math]\displaystyle{ \varepsilon }[/math], the case [math]\displaystyle{ \varepsilon = 0 }[/math] is the symmetric pitchfork bifurcation. When [math]\displaystyle{ \varepsilon \neq 0 }[/math], we say we have a pitchfork with broken symmetry. This is illustrated in the animation on the right.

当 math mu neq 0 / math 时,它是结构稳定的,如果绘制一个分枝图,把 math mu / math 作为分支参数,但对于 math varepsilon / math 的不同值,case math varepsilon 0 / math 是对称的叉式分岔。当 math varepsilon neq 0 / math 时,我们说我们有一个对称性破缺的干草叉。右边的动画说明了这一点。


See also


References


External links

Category:Chaos theory

范畴: 混沌理论

Category:Bifurcation theory

类别: 分岔理论


de:Bifurkationsdiagramm

de:Bifurkationsdiagramm

分叉分析


This page was moved from wikipedia:en:Bifurcation diagram. Its edit history can be viewed at 倍分岔周期图/edithistory