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第84行： − + − The [[codimension]] of a bifurcation is the number of parameters which must be varied for the bifurcation to occur. This corresponds to the codimension of the parameter set for which the bifurcation occurs within the full space of parameters. Saddle-node bifurcations and Hopf bifurcations are the only generic local bifurcations which are really codimension-one (the others all having higher codimension). However, transcritical and pitchfork bifurcations are also often thought of as codimension-one, because the normal forms can be written with only one parameter. − The codimension of a bifurcation is the number of parameters which must be varied for the bifurcation to occur. This corresponds to the codimension of the parameter set for which the bifurcation occurs within the full space of parameters. Saddle-node bifurcations and Hopf bifurcations are the only generic local bifurcations which are really codimension-one (the others all having higher codimension). However, transcritical and pitchfork bifurcations are also often thought of as codimension-one, because the normal forms can be written with only one parameter.+ − − 分岔的余维是分岔发生时必须改变的参数个数。这对应于参数集的余维数，对于这个余维数，分岔发生在参数的整个空间。鞍结分支和 Hopf 分支是唯一真余维一的一般局部分支(其他分支均具有较高余维)。然而，跨临界分岔和干音叉分岔也经常被认为是余维数 -1，因为正规形可以只用一个参数来写。 − − − − An example of a well-studied codimension-two bifurcation is the [[Bogdanov-Takens bifurcation|Bogdanov–Takens bifurcation]]. − − An example of a well-studied codimension-two bifurcation is the Bogdanov–Takens bifurcation. − − Bogdanov-Takens 分岔是余维 -2分岔研究的一个很好的例子。 +
→Codimension of a bifurcation
全局分岔还涉及到更复杂的集合，例如[[混沌吸引子]](如[[危机]])。
全局分岔还涉及到更复杂的集合，例如[[混沌吸引子]](如[[危机]])。
==Codimension of a bifurcation==
==分岔的余维数==
分岔的[[余维数]]是分岔发生时必须改变的参数个数。这对应于参数集的余维数，对于该余维数，分岔发生在参数的整个空间中。鞍结分岔和霍普夫分岔是局部分岔中真正的余维数为一的分岔(其他分岔均具有较高的余维数)。然而，跨临界分岔和岔式分岔的正规形可以只用一个参数来表示，因此它们的的余维数也常被认为是一。
[[Bogdanov-Takens分岔]]是研究余维数为2的分岔的一个很好的例子。
==Applications in semiclassical and quantum physics==
==Applications in semiclassical and quantum physics==