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第49行： − + − − − − One can also consider what happens if one holds ''b'' constant and varies ''a''. In the symmetrical case {{nowrap|''b'' {{=}} 0}}, one observes a [[pitchfork bifurcation]] as ''a'' is reduced, with one stable solution suddenly splitting into two stable solutions and one unstable solution as the physical system passes to {{nowrap|''a'' < 0}} through the cusp point (0,0) (an example of [[spontaneous symmetry breaking]]). Away from the cusp point, there is no sudden change in a physical solution being followed: when passing through the curve of fold bifurcations, all that happens is an alternate second solution becomes available. − − − − − − − − 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.
→尖点灾变
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。
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。
我们也可以考虑，如果一个人保持''b''常数，改变''a''会发生什么。在对称情形{{nowrap|''b'' {{=}} 0}}中，我们观察到''a''被还原时的[[叉式分岔]]，当物理系统通过尖点(0,0)时，一个稳定解突然分裂成两个稳定解和一个不稳定解{{nowrap|''a'' < 0}}(自发对称性破缺的一个例子)。离开尖点，所遵循的物理解没有突然的变化: 当通过折叠分叉曲线时，所发生的是一个备用的第二解变得可用。
我们也可以考虑，如果一个人保持''b''常数，改变''a''会发生什么。在对称情形{{nowrap|''b'' {{=}} 0}}中，我们观察到''a''被还原时的[[叉式分岔]]，当物理系统通过尖点(0,0)时，一个稳定解突然分裂成两个稳定解和一个不稳定解(自发对称性破缺的一个例子)。离开尖点，所遵循的物理解没有突然的变化: 当通过折叠分叉曲线时，所发生的是一个备用的第二解变得可用。
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.<ref>[[E.C. Zeeman]], [http://www.gaianxaos.com/pdf/dynamics/zeeman-catastrophe_theory.pdf Catastrophe Theory], ''[[Scientific American]]'', April 1976; pp. 65–70, 75–83</ref> The suggestion is that at moderate stress ({{nowrap|''a'' > 0}}), 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 ({{nowrap|''a'' < 0}}). 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.<ref>[[E.C. Zeeman]], [http://www.gaianxaos.com/pdf/dynamics/zeeman-catastrophe_theory.pdf Catastrophe Theory], ''[[Scientific American]]'', April 1976; pp. 65–70, 75–83</ref> The suggestion is that at moderate stress ({{nowrap|''a'' > 0}}), 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 ({{nowrap|''a'' < 0}}). 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，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。
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。