1,068
个编辑
更改
跳到导航
跳到搜索
第19行：
第19行： + − + − Catastrophe theory analyzes ''degenerate critical points'' of the potential function — points where not just the first derivative, but one or more higher derivatives of the potential function are also zero. These are called the [[germ (mathematics)|germs]] of the catastrophe geometries. The degeneracy of these critical points can be ''unfolded'' by expanding the potential function as a [[Taylor series]] in small perturbations of the parameters. − − Catastrophe theory analyzes degenerate critical points of the potential function — points where not just the first derivative, but one or more higher derivatives of the potential function are also zero. These are called the germs of the catastrophe geometries. The degeneracy of these critical points can be unfolded by expanding the potential function as a Taylor series in small perturbations of the parameters. − − 突变论分析势函数的退化临界点（在退化临界点处，势函数不仅一阶导数为零，而且有一个或多个高阶导数也为零），这被视为突变几何的萌芽。临界点的退化可以通过在参数的微小扰动中将势函数按照泰勒级数来展开。 − − − − − When the degenerate points are not merely accidental, but are [[structural stability|structurally stable]], the degenerate points exist as organising centres for particular geometric structures of lower degeneracy, with critical features in the parameter space around them. If the potential function depends on two or fewer active variables, and four or fewer active parameters, then there are only seven generic structures for these bifurcation geometries, with corresponding standard forms into which the Taylor series around the catastrophe germs can be transformed by [[diffeomorphism]] (a smooth transformation whose inverse is also smooth).{{Citation needed|date=May 2010}} These seven fundamental types are now presented, with the names that Thom gave them. − − When the degenerate points are not merely accidental, but are structurally stable, the degenerate points exist as organising centres for particular geometric structures of lower degeneracy, with critical features in the parameter space around them. If the potential function depends on two or fewer active variables, and four or fewer active parameters, then there are only seven generic structures for these bifurcation geometries, with corresponding standard forms into which the Taylor series around the catastrophe germs can be transformed by diffeomorphism (a smooth transformation whose inverse is also smooth). These seven fundamental types are now presented, with the names that Thom gave them. − − 当退化点不只是巧合的，而且是结构性稳定的，退化点作为具有较低退化度的特定几何结构的组织中心存在，其周围的参数空间具有临界特征。如果势函数依赖于两个或更少的活动变量以及四个或更少的活动参数，那么对于这些分支几何形状，只有七个通用结构，具有可以通过微分同胚(一种逆光滑的光滑变换)将灾变芽周围的泰勒级数转化为相应的标准形式。这七种基本类型现在被呈现出来，Thom也给他们取了名字。 − 第43行：
第29行： − − − Stable and unstable pair of extrema disappear at a fold bifurcation 第56行：
第39行： − + − − − − At negative values of ''a'', the potential ''V'' has two extrema - one stable, and one unstable. If the parameter ''a'' is slowly increased, the system can follow the stable minimum point. But at {{nowrap|''a'' {{=}} 0}} the stable and unstable extrema meet, and annihilate. This is the bifurcation point. At {{nowrap|''a'' > 0}} there is no longer a stable solution. If a physical system is followed through a fold bifurcation, one therefore finds that as ''a'' reaches 0, the stability of the {{nowrap|''a'' < 0}} solution is suddenly lost, and the system will make a sudden transition to a new, very different behaviour. {{anchor|Tipping point}}This bifurcation value of the parameter ''a'' is sometimes called the "[[Tipping point (physics)|tipping point]]". − − At negative values of a, the potential V has two extrema - one stable, and one unstable. If the parameter a is slowly increased, the system can follow the stable minimum point. But at 0}} the stable and unstable extrema meet, and annihilate. This is the bifurcation point. At there is no longer a stable solution. If a physical system is followed through a fold bifurcation, one therefore finds that as a reaches 0, the stability of the solution is suddenly lost, and the system will make a sudden transition to a new, very different behaviour. This bifurcation value of the parameter a is sometimes called the "tipping point". − − − − − − 第131行：
第102行： − − − 第277行：
第245行： + − == Potential functions of two active variables == − − == Potential functions of two active variables == − − 两个活动变量的势函数
无编辑摘要
== 基本的灾难 ==
== 基本的灾难 ==
突变论分析势函数的退化临界点（在退化临界点处，势函数不仅一阶导数为零，而且有一个或多个高阶导数也为零），这被视为突变几何的萌芽。临界点的退化可以通过在参数的微小扰动中将势函数按照[[泰勒级数]]来展开。
当退化点不只是巧合的，而且是结构性稳定的，退化点作为具有较低退化度的特定几何结构的组织中心存在，其周围的参数空间具有临界特征。如果势函数依赖于两个或更少的活动变量以及四个或更少的活动参数，那么对于这些分支几何形状，只有七个通用结构，具有可以通过[[微分同胚]](一种逆光滑的光滑变换)将灾变芽周围的泰勒级数转化为相应的标准形式。{{Citation needed|date=May 2010}}这七种基本类型现在被呈现出来，Thom也给他们取了名字。
=== 折叠灾难 ===
=== 折叠灾难 ===
[[File:fold bifurcation.svg|frame|right|160px|Stable and unstable pair of extrema disappear at a fold bifurcation|链接=Special:FilePath/Fold_bifurcation.svg]]
[[File:fold bifurcation.svg|frame|right|160px|Stable and unstable pair of extrema disappear at a fold bifurcation|链接=Special:FilePath/Fold_bifurcation.svg]]
稳定和不稳定的极值对在折叠分叉处消失
稳定和不稳定的极值对在折叠分叉处消失
X ^ 3 + ax，/ math
X ^ 3 + ax，/ math
在负值 a 时，势 v 有两个极值，一个是稳定的，一个是不稳定的。如果参数 a 缓慢增加，系统可以达到稳定的最小点。但在0时，稳定极值与不稳定极值相遇并湮灭。这就是分歧点。这儿不再有一个稳定解。如果一个物理系统经过一个折叠分叉，人们就会发现，当 a 到达0时，解的稳定性会突然丧失，系统也会突然转变为一个新的，非常不同的行为。参数 a 的这个分叉值有时被称为“[[引爆点]]”。
在负值 a 时，势 v 有两个极值，一个是稳定的，一个是不稳定的。如果参数 a 缓慢增加，系统可以达到稳定的最小点。但在0时，稳定极值与不稳定极值相遇并湮灭。这就是分歧点。这儿不再有一个稳定解。如果一个物理系统经过一个折叠分叉，人们就会发现，当 a 到达0时，解的稳定性会突然丧失，系统也会突然转变为一个新的，非常不同的行为。参数 a 的这个分叉值有时被称为“引爆点”。
{{clear}}
{{clear}}
=== 尖点灾变 ===
=== 尖点灾变 ===
:<math>V = x^4 + ax^2 + bx \,</math>
:<math>V = x^4 + ax^2 + bx \,</math>
但在尖点几何中，分叉曲线回到自身，形成第二个分支，在这个分支中，这个交替解本身失去了稳定性，并将跳回到原来的解集。通过反复增加 b，然后减小 b，人们因此可以观察到滞后回路，因为系统交替地遵循一个解决方案，跳到另一个解决方案，沿着另一个解决方案回来，然后跳回到第一个。
但在尖点几何中，分叉曲线回到自身，形成第二个分支，在这个分支中，这个交替解本身失去了稳定性，并将跳回到原来的解集。通过反复增加 b，然后减小 b，人们因此可以观察到滞后回路，因为系统交替地遵循一个解决方案，跳到另一个解决方案，沿着另一个解决方案回来，然后跳回到第一个。
== 两个活动变量的势函数 ==
Umbilic catastrophes are examples of corank 2 catastrophes. They can be observed in [[optics]] in the focal surfaces created by light reflecting off a surface in three dimensions and are intimately connected with the geometry of nearly spherical surfaces.
Umbilic catastrophes are examples of corank 2 catastrophes. They can be observed in [[optics]] in the focal surfaces created by light reflecting off a surface in three dimensions and are intimately connected with the geometry of nearly spherical surfaces.