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第82行： − − − − 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. − − − Thom proposed that the hyperbolic umbilic catastrophe modeled the breaking of a wave and the elliptical umbilic modeled the creation of hair-like structures. − − Thom proposed that the hyperbolic umbilic catastrophe modeled the breaking of a wave and the elliptical umbilic modeled the creation of hair-like structures. 第99行：
第88行： − − − − − <math>V = x^3 + y^3 + axy + bx +cy \, </math> − − 数学 v x ^ 3 + y ^ 3 + axy + bx + cy，/ math − − − − − <math>V = \frac{x^3}{3} - xy^2 + a(x^2+y^2) + bx + cy \, </math> − − 3}-xy ^ 2 + a (x ^ 2 + y ^ 2) + bx + cy，/ math − − − − − <math>V = x^2y + y^4 + ax^2 + by^2 + cx + dy \, </math> − − 数学 v x ^ 2 y + y ^ 4 + ax ^ 2 + by ^ 2 + cx + dy ，/ math − − + − − − Vladimir Arnold gave the catastrophes the ADE classification, due to a deep connection with simple Lie groups. − − 弗拉迪米尔 · 阿诺德将这些灾难归为 ADE 类，因为它们与简单的李群有很深的联系。 − − − − − − − − − − − − − − − − − − − − − − − − − − − − − − There are objects in singularity theory which correspond to most of the other simple Lie groups. − − There are objects in singularity theory which correspond to most of the other simple Lie groups. 第225行：
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无编辑摘要
== 两个活动变量的势函数 ==
== 两个活动变量的势函数 ==
脐带灾难是2次灾难的例子。它们可以在光线从三维表面反射而形成的焦面的光学中观察到，并且与近球面的几何形状密切相关。
脐带灾难是2次灾难的例子。它们可以在光线从三维表面反射而形成的焦面的光学中观察到，并且与近球面的几何形状密切相关。
Thom 提出双曲线脐状突变模拟了波的破裂，椭圆脐状突变模拟了毛发状结构的产生。
Thom 提出双曲线脐状突变模拟了波的破裂，椭圆脐状突变模拟了毛发状结构的产生。
=== 双曲线脐点突变 ===
=== 双曲线脐点突变 ===
:<math>V = x^3 + y^3 + axy + bx +cy \, </math>
:<math>V = x^3 + y^3 + axy + bx +cy \, </math>
=== 椭圆脐灾变 ===
=== 椭圆脐灾变 ===
:<math>V = \frac{x^3}{3} - xy^2 + a(x^2+y^2) + bx + cy \, </math>
:<math>V = \frac{x^3}{3} - xy^2 + a(x^2+y^2) + bx + cy \, </math>
=== 抛物线脐点突变 ===
=== 抛物线脐点突变 ===
:<math>V = x^2y + y^4 + ax^2 + by^2 + cx + dy \, </math>
:<math>V = x^2y + y^4 + ax^2 + by^2 + cx + dy \, </math>
==阿诺德记数法==
==阿诺德记数法==
[[弗拉迪米尔 · 阿诺德]]将这些灾难归为 ADE 类，因为它们与[[简单的李群]]有很深的联系。
[[Vladimir Arnold]] gave the catastrophes the [[ADE classification]], due to a deep connection with [[simple Lie group]]s.
*''A''<sub>0</sub> - a non-singular point: <math>V = x</math>.
*''A''<sub>0</sub> - a non-singular point: <math>V = x</math>.
*''A''<sub>1</sub> - a local extremum, either a stable minimum or unstable maximum <math>V = \pm x^2 + a x</math>.
*''A''<sub>1</sub> - a local extremum, either a stable minimum or unstable maximum <math>V = \pm x^2 + a x</math>.
*''A''<sub>2</sub> - the fold
*''A''<sub>2</sub> - the fold
*''A''<sub>3</sub> - the cusp
*''A''<sub>3</sub> - the cusp
*''A''<sub>4</sub> - the swallowtail
*''A''<sub>4</sub> - the swallowtail
*''A''<sub>5</sub> - the butterfly
*''A''<sub>5</sub> - the butterfly
*''A''<sub>k</sub> - a representative of an infinite sequence of one variable forms <math>V=x^{k+1}+\cdots</math>
*''A''<sub>k</sub> - a representative of an infinite sequence of one variable forms <math>V=x^{k+1}+\cdots</math>
*''D''<sub>4</sub><sup>−</sup> - the elliptical umbilic
*''D''<sub>4</sub><sup>−</sup> - the elliptical umbilic
*''D''<sub>4</sub><sup>+</sup> - the hyperbolic umbilic
*''D''<sub>4</sub><sup>+</sup> - the hyperbolic umbilic
*''D''<sub>5</sub> - the parabolic umbilic
*''D''<sub>5</sub> - the parabolic umbilic
*''D''<sub>k</sub> - a representative of an infinite sequence of further umbilic forms
*''D''<sub>k</sub> - a representative of an infinite sequence of further umbilic forms
*''E''<sub>6</sub> - the symbolic umbilic <math>V = x^3+y^4+a x y^2 +bxy+cx+dy+ey^2</math>
*''E''<sub>6</sub> - the symbolic umbilic <math>V = x^3+y^4+a x y^2 +bxy+cx+dy+ey^2</math>
*''E''<sub>7</sub>
*''E''<sub>7</sub>
*''E''<sub>8</sub>
*''E''<sub>8</sub>
在奇点理论中有一些对象，它们对应于大多数其他简单的李群。
在奇点理论中有一些对象，它们对应于大多数其他简单的李群。
*[[Valentin Afraimovich|V. S. Afrajmovich]], V. I. Arnold, et al., Bifurcation Theory And Catastrophe Theory, {{ISBN|3-540-65379-1}}
*[[Valentin Afraimovich|V. S. Afrajmovich]], V. I. Arnold, et al., Bifurcation Theory And Catastrophe Theory, {{ISBN|3-540-65379-1}}
*Bełej,M. Kulesza, S. Modeling the Real Estate Prices in Olsztyn under Instability Conditions. Folia Oeconomica Stetinensia. Volume 11, Issue 1, Pages 61–72, ISSN (Online) 1898-0198, ISSN (Print) 1730-4237, {{doi|10.2478/v10031-012-0008-7}}, 2013
*Bełej,M. Kulesza, S. Modeling the Real Estate Prices in Olsztyn under Instability Conditions. Folia Oeconomica Stetinensia. Volume 11, Issue 1, Pages 61–72, ISSN (Online) 1898-0198, ISSN (Print) 1730-4237, {{doi|10.2478/v10031-012-0008-7}}, 2013
*Castrigiano, Domenico P. L. and Hayes, Sandra A. Catastrophe Theory, 2nd ed. Boulder: Westview, 2004. {{ISBN|0-8133-4126-4}}
*Castrigiano, Domenico P. L. and Hayes, Sandra A. Catastrophe Theory, 2nd ed. Boulder: Westview, 2004. {{ISBN|0-8133-4126-4}}