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第66行： − − − − − Swallowtail catastrophe surface − − <math>V = x^5 + ax^3 + bx^2 + cx \, </math> − − X ^ 5 + ax ^ 3 + bx ^ 2 + cx ，/ math − − − − − − The control parameter space is three-dimensional. The bifurcation set in parameter space is made up of three surfaces of fold bifurcations, which meet in two lines of cusp bifurcations, which in turn meet at a single swallowtail bifurcation point. − − The control parameter space is three-dimensional. The bifurcation set in parameter space is made up of three surfaces of fold bifurcations, which meet in two lines of cusp bifurcations, which in turn meet at a single swallowtail bifurcation point. − + − − − − As the parameters go through the surface of fold bifurcations, one minimum and one maximum of the potential function disappear. At the cusp bifurcations, two minima and one maximum are replaced by one minimum; beyond them the fold bifurcations disappear. At the swallowtail point, two minima and two maxima all meet at a single value of ''x''. For values of {{nowrap|''a'' > 0}}, beyond the swallowtail, there is either one maximum-minimum pair, or none at all, depending on the values of ''b'' and ''c''. Two of the surfaces of fold bifurcations, and the two lines of cusp bifurcations where they meet for {{nowrap|''a'' < 0}}, therefore disappear at the swallowtail point, to be replaced with only a single surface of fold bifurcations remaining. [[Salvador Dalí|Salvador Dalí's]] last painting, ''[[The Swallow's Tail]]'', was based on this catastrophe. − − As the parameters go through the surface of fold bifurcations, one minimum and one maximum of the potential function disappear. At the cusp bifurcations, two minima and one maximum are replaced by one minimum; beyond them the fold bifurcations disappear. At the swallowtail point, two minima and two maxima all meet at a single value of x. For values of , beyond the swallowtail, there is either one maximum-minimum pair, or none at all, depending on the values of b and c. Two of the surfaces of fold bifurcations, and the two lines of cusp bifurcations where they meet for , therefore disappear at the swallowtail point, to be replaced with only a single surface of fold bifurcations remaining. Salvador Dalí's last painting, The Swallow's Tail, was based on this catastrophe. − − 当参数经过折叠分叉面时，势函数的一个最小值和一个最大值消失。在尖点分岔处，两个最小值和一个最大值被一个最小值所代替; 在它们之后，折点分岔就消失了。在燕尾点，两个极小值和两个极大值都在一个 x 的值上相交。对于燕尾蝶，根据 b 和 c 的值，有一个最大-最小对，或者根本没有，这取决于折叠分叉的两个面，以及它们相遇的两条尖形分叉线，因此在燕尾蝶点消失，只剩下一个单一的折叠分叉面。萨尔瓦多 · 达利的最后一幅画《燕子尾巴》就是以这场灾难为基础的。 −
→燕尾蝶灾难
=== 燕尾蝶灾难 ===
=== 燕尾蝶灾难 ===
[[File:Smallow tail.jpg|thumb|right|160px|Swallowtail catastrophe surface|链接=Special:FilePath/Smallow_tail.jpg]]
[[File:Smallow tail.jpg|thumb|right|160px|Swallowtail catastrophe surface|链接=Special:FilePath/Smallow_tail.jpg]]
燕尾突变曲面
燕尾突变曲面
:<math>V = x^5 + ax^3 + bx^2 + cx \, </math>
:<math>V = x^5 + ax^3 + bx^2 + cx \, </math>
控制参数空间为三维空间。参数空间中的分岔集是由三个折叠分岔曲面组成的，这三个折叠分岔曲面在两条尖点分岔线上相交，而尖点又在一个单燕尾分岔点上相交。
控制参数空间为三维空间。参数空间中的分岔集是由三个折叠分岔曲面组成的，这三个折叠分岔曲面在两条尖点分岔线上相交，而尖点又在一个单燕尾分岔点上相交。
当参数经过折叠分叉面时，势函数的一个最小值和一个最大值消失。在尖点分岔处，两个最小值和一个最大值被一个最小值所代替; 在它们之后，折点分岔就消失了。在燕尾点{{nowrap|''a'' > 0}}，两个极小值和两个极大值都在一个''x''的值上相交。对于燕尾蝶，根据''b''和''c''的值，有一个最大-最小对，或者根本没有，这取决于折叠分叉的两个面，以及它们相遇的两条尖形分叉线{{nowrap|''a'' < 0}}，因此在燕尾蝶点消失，只剩下一个单一的折叠分叉面。萨尔瓦多 · 达利的最后一幅画《燕子尾巴》就是以这场灾难为基础的。
=== 蝴蝶灾难 ===
=== 蝴蝶灾难 ===