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Bifurcation theory studies and classifies phenomena characterized by sudden shifts in behavior arising from small changes in circumstances, analysing how the qualitative nature of equation solutions depends on the parameters that appear in the equation.  This may lead to sudden and dramatic changes, for example the unpredictable timing and magnitude of a landslide.

Bifurcation theory studies and classifies phenomena characterized by sudden shifts in behavior arising from small changes in circumstances, analysing how the qualitative nature of equation solutions depends on the parameters that appear in the equation.  This may lead to sudden and dramatic changes, for example the unpredictable timing and magnitude of a landslide.

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.

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.

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.

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.

当退化点不仅仅是偶然的，而且是结构稳定的时候，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和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，它们具有相应的标准形式，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，还有 Thom 给他们起的名字。

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。

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]]

[[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]]

Stable and unstable pair of extrema disappear at a fold bifurcation

Stable and unstable pair of extrema disappear at a fold bifurcation

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".

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".

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。在年，不再有一个稳定的解决方案。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，解的稳定性突然丧失，系统就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”。

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[[File:cusp catastrophe.svg|thumb|upright=1.5|Diagram of cusp catastrophe, showing curves (brown, red) of ''x'' satisfying ''dV''/''dx'' = ''0'' for parameters (''a'',''b''), drawn for parameter ''b'' continuously varied, for several values of parameter ''a''.

[[File:cusp catastrophe.svg|thumb|upright=1.5|Diagram of cusp catastrophe, showing curves (brown, red) of ''x'' satisfying ''dV''/''dx'' = ''0'' for parameters (''a'',''b''), drawn for parameter ''b'' continuously varied, for several values of parameter ''a''.

[[File:cusp catastrophe.svg|thumb|upright=1.5|Diagram of cusp catastrophe, showing curves (brown, red) of x satisfying dV/dx = 0 for parameters (a,b), drawn for parameter b continuously varied, for several values of parameter a.

[[File:cusp catastrophe.svg|thumb|upright=1.5| 尖点突变图，显示 x 的曲线(棕色，红色)满足 dv / dx 0的参数(a，b) ，为参数 b 绘制的曲线连续变化，为参数 a 的几个值绘制的曲线(棕色，红色)。

 

[文件: 尖点突变. svg | thumb | upright 1.5 | 尖点突变图，显示 x 的曲线(棕色，红色)满足 dv / dx 0的参数(a，b) ，为参数 b 绘制的曲线连续变化，为参数 a 的几个值绘制的曲线(棕色，红色)。

Outside the cusp locus of bifurcations (blue), for each point (''a'',''b'') in parameter space there is only one extremising value of ''x''. Inside the cusp, there are two different values of ''x'' giving local minima of ''V''(''x'') for each (''a'',''b''), separated by a value of ''x'' giving a local maximum.]]

Outside the cusp locus of bifurcations (blue), for each point (''a'',''b'') in parameter space there is only one extremising value of ''x''. Inside the cusp, there are two different values of ''x'' giving local minima of ''V''(''x'') for each (''a'',''b''), separated by a value of ''x'' giving a local maximum.|链接=Special:FilePath/Cusp_catastrophe.svg]]

Outside the cusp locus of bifurcations (blue), for each point (a,b) in parameter space there is only one extremising value of x. Inside the cusp, there are two different values of x giving local minima of V(x) for each (a,b), separated by a value of x giving a local maximum.]]

Outside the cusp locus of bifurcations (blue), for each point (a,b) in parameter space there is only one extremising value of x. Inside the cusp, there are two different values of x giving local minima of V(x) for each (a,b), separated by a value of x giving a local maximum.|链接=Special:FilePath/Cusp_catastrophe.svg]]

在分岔点(蓝色)的尖点轨迹外，参数空间中的每个点(a，b)只有一个 x 的极值。在尖点内部，有两个不同的 x 值，给出每个(a，b)的局部极小值 v (x) ，中间用 x 值分隔，给出局部极大值。]]

在分岔点(蓝色)的尖点轨迹外，参数空间中的每个点(a，b)只有一个 x 的极值。在尖点内部，有两个不同的 x 值，给出每个(a，b)的局部极小值 v (x) ，中间用 x 值分隔，给出局部极大值。]]

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| [[File:cusp shape.svg|thumb|upright=0.7|Cusp shape in parameter space (''a'',''b'') near the catastrophe point, showing the locus of fold bifurcations separating the region with two stable solutions from the region with one.|链接=Special:FilePath/Cusp_shape.svg]]

 

| [[File:cusp shape.svg|thumb|upright=0.7|Cusp shape in parameter space (''a'',''b'') near the catastrophe point, showing the locus of fold bifurcations separating the region with two stable solutions from the region with one.]]

| Cusp shape in parameter space (a,b) near the catastrophe point, showing the locus of fold bifurcations separating the region with two stable solutions from the region with one.

| Cusp shape in parameter space (a,b) near the catastrophe point, showing the locus of fold bifurcations separating the region with two stable solutions from the region with one.

在突变点附近的参数空间(a，b)中的 | 尖点形状，显示了用两个稳定解从一个突变点区域分离区域的折叠分岔轨迹。

在突变点附近的参数空间(a，b)中的 | 尖点形状，显示了用两个稳定解从一个突变点区域分离区域的折叠分岔轨迹。

| [[File:pitchfork bifurcation left.svg|thumb|upright=0.7|Pitchfork bifurcation at {{nowrap|''a'' {{=}} 0}} on the surface {{nowrap|''b'' {{=}} 0}}]]

| [[File:pitchfork bifurcation left.svg|thumb|upright=0.7|Pitchfork bifurcation at {{nowrap|''a'' {{=}} 0}} on the surface {{nowrap|''b'' {{=}} 0}}|链接=Special:FilePath/Pitchfork_bifurcation_left.svg]]

| Pitchfork bifurcation at  0}} on the surface  0}}

| Pitchfork bifurcation at  0}} on the surface  0}}

在叉式分岔表面上

在叉式分岔表面上

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The cusp geometry is very common, when one explores what happens to a fold bifurcation if a second parameter, b, is added to the control space.  Varying the parameters, one finds that there is now a curve (blue) of points in (a,b) space where stability is lost, where the stable solution will suddenly jump to an alternate outcome.

The cusp geometry is very common, when one explores what happens to a fold bifurcation if a second parameter, b, is added to the control space.  Varying the parameters, one finds that there is now a curve (blue) of points in (a,b) space where stability is lost, where the stable solution will suddenly jump to an alternate outcome.

尖点几何是非常常见的，当人们探索如果在控制空间中加入第二个参数 b，将会发生什么。通过改变参数，我们发现在(a，b)空间中存在一条点的曲线(蓝色) ，在这条曲线中失去了稳定性，稳定解突然跳跃到另一个结果。

当人们探索如果在控制空间中加入第二个参数 b，将会发生什么，尖点几何是非常常见的。通过改变参数，我们发现在(a，b)空间中存在一条点的曲线(蓝色) ，在这条曲线中解失去了稳定性，其突然跳跃到另一个结果。

But in a cusp geometry the bifurcation curve loops back on itself, giving a second branch where this alternate solution itself loses stability, and will make a jump back to the original solution set.  By repeatedly increasing b and then decreasing it, one can therefore observe hysteresis loops, as the system alternately follows one solution, jumps to the other, follows the other back, then jumps back to the first.

But in a cusp geometry the bifurcation curve loops back on itself, giving a second branch where this alternate solution itself loses stability, and will make a jump back to the original solution set.  By repeatedly increasing b and then decreasing it, one can therefore observe hysteresis loops, as the system alternately follows one solution, jumps to the other, follows the other back, then jumps back to the first.

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，跳到另一个解决方案，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。

A famous suggestion is that the cusp catastrophe can be used to model the behaviour of a stressed dog, which may respond by becoming cowed or becoming angry.  The suggestion is that at moderate stress (), the dog will exhibit a smooth transition of response from cowed to angry, depending on how it is provoked.  But higher stress levels correspond to moving to the region ().  Then, if the dog starts cowed, it will remain cowed as it is irritated more and more, until it reaches the 'fold' point, when it will suddenly, discontinuously snap through to angry mode.  Once in 'angry' mode, it will remain angry, even if the direct irritation parameter is considerably reduced.

A famous suggestion is that the cusp catastrophe can be used to model the behaviour of a stressed dog, which may respond by becoming cowed or becoming angry.  The suggestion is that at moderate stress (), the dog will exhibit a smooth transition of response from cowed to angry, depending on how it is provoked.  But higher stress levels correspond to moving to the region ().  Then, if the dog starts cowed, it will remain cowed as it is irritated more and more, until it reaches the 'fold' point, when it will suddenly, discontinuously snap through to angry mode.  Once in 'angry' mode, it will remain angry, even if the direct irritation parameter is considerably reduced.

一个著名的建议是尖点灾难可以用来模拟一只受到压力的狗的行为，它可能会变得胆怯或生气。建议是，在适度的压力() ，狗将展示一个平稳过渡的反应，从吓唬到愤怒，这取决于它是如何挑起的。但是较高的应力水平对应于向该区域的移动()。然后，如果狗开始恐吓，它会继续恐吓，因为它被激怒越来越多，直到它达到’折叠’点，当它会突然，不间断地跳转到愤怒的模式。一旦进入“愤怒”模式，即使直接刺激参数大大降低，它也会继续愤怒。

一个著名的建议是尖点灾难可以用来模拟一只受到压力的狗的行为，它可能会变得胆怯或生气。建议是，在适度的压力() ，狗将展示一个平稳过渡的反应，从吓唬到愤怒，这取决于它是如何挑起的。但是较高的应力水平对应于向该区域的移动()。然后，如果狗开始恐吓，它会继续恐吓，因为它被激怒越来越多，直到它达到’折叠’点，其会突然，不间断地跳转到愤怒的模式。一旦进入“愤怒”模式，即使直接刺激参数大大降低，它也会继续愤怒。

燕尾蝶灾难

燕尾蝶灾难

[[File:Smallow tail.jpg|thumb|right|160px|Swallowtail catastrophe surface]]

[[File:Smallow tail.jpg|thumb|right|160px|Swallowtail catastrophe surface|链接=Special:FilePath/Smallow_tail.jpg]]

Swallowtail catastrophe surface

Swallowtail catastrophe surface

Depending on the parameter values, the potential function may have three, two, or one different local minima, separated by the loci of fold bifurcations.  At the butterfly point, the different 3-surfaces of fold bifurcations, the 2-surfaces of cusp bifurcations, and the lines of swallowtail bifurcations all meet up and disappear, leaving a single cusp structure remaining when .

Depending on the parameter values, the potential function may have three, two, or one different local minima, separated by the loci of fold bifurcations.  At the butterfly point, the different 3-surfaces of fold bifurcations, the 2-surfaces of cusp bifurcations, and the lines of swallowtail bifurcations all meet up and disappear, leaving a single cusp structure remaining when .