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第36行： − − {| style="float:right; clear:right" − | colspan=2 | − − + − 2 | − − [[File:cusp catastrophe.png|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''. − − − − − 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.|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。]] − − + − |- valign="top"+ − − − | 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。 − − − − | Pitchfork bifurcation at 0}} on the surface 0}} − − 在叉式分岔表面上 − − − |} − − 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. + − 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，沿着另一个解决方案回来，然后跳回到第一个。 − − − However, this is only possible in the region of parameter space {{nowrap|''a'' < 0}}. As ''a'' is increased, the hysteresis loops become smaller and smaller, until above {{nowrap|''a'' {{=}} 0}} they disappear altogether (the cusp catastrophe), and there is only one stable solution. − − However, this is only possible in the region of parameter space . As a is increased, the hysteresis loops become smaller and smaller, until above 0}} they disappear altogether (the cusp catastrophe), and there is only one stable solution. − − 然而，这只有在参数空间的区域才可能。随着 a 的增加，磁滞回线变得越来越小，直到大于0}磁滞回线完全消失(尖点突变) ，只有一个稳定的解。
→尖点灾变
{| style="float:right; clear:right"
{| style="float:right; clear:right"
| colspan=2 |
| colspan=2 |
[[File:cusp catastrophe.png|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)。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。]]
[[File:cusp catastrophe.png|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)。
|- valign="top"
|- valign="top"
| [[File:cusp shape.png|thumb|upright=0.7|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.png|thumb|upright=0.7|叉式分岔 a = 0，表面 b = 0]]
| [[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: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]]
|}
|}
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。
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。
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。
然而，这只有在参数空间的区域才可能{{nowrap|''a'' < 0}}。随着''a''的增加，磁滞回线变得越来越小，直到大于{{nowrap|''a'' {{=}} 0}}磁滞回线完全消失(尖点突变) ，只有一个稳定的解。