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第51行： − 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。+ + − + − − − A simple mechanical system, the "Zeeman Catastrophe Machine", nicely illustrates a cusp catastrophe. In this device, smooth variations in the position of the end of a spring can cause sudden changes in the rotational position of an attached wheel.<ref>Cross, Daniel J., [http://lagrange.physics.drexel.edu/flash/zcm/ Zeeman's Catastrophe Machine in Flash] {{webarchive|url=https://archive.is/20121211093251/http://lagrange.physics.drexel.edu/flash/zcm/ |date=2012-12-11 }}</ref> − − A simple mechanical system, the "Zeeman Catastrophe Machine", nicely illustrates a cusp catastrophe. In this device, smooth variations in the position of the end of a spring can cause sudden changes in the rotational position of an attached wheel. − − 一个简单的机械系统，“塞曼灾难机器” ，很好地说明了尖点灾难。在这种装置中，弹簧末端位置的平滑变化可以引起附加轮转动位置的突然变化。 − − − − − − Catastrophic failure of a [[complex system]] with parallel redundancy can be evaluated based on the relationship between local and external stresses. The model of the [[structural fracture mechanics]] is similar to the cusp catastrophe behavior. The model predicts reserve ability of a complex system. − − Catastrophic failure of a complex system with parallel redundancy can be evaluated based on the relationship between local and external stresses. The model of the structural fracture mechanics is similar to the cusp catastrophe behavior. The model predicts reserve ability of a complex system. − − 并联冗余复杂系统的灾变失效可以根据局部应力与外部应力之间的关系进行评估。结构断裂力学模型与尖点突变行为相似。该模型预测了复杂系统的储备能力。 − − − − − − Other applications include the [[outer sphere electron transfer]] frequently encountered in chemical and biological systems<ref>{{cite journal|last=Xu|first=F|title=Application of catastrophe theory to the ∆G<sup>≠</sup> to -∆G relationship in electron transfer reactions.| journal=Zeitschrift für Physikalische Chemie |series=Neue Folge | volume=166 | pages=79–91 | date=1990|doi=10.1524/zpch.1990.166.Part_1.079}}</ref> and modelling real estate prices.<ref>{{cite journal|last=Bełej|first=Mirosław|author2=Kulesza, Sławomir|title=Modeling the Real Estate Prices in Olsztyn under Instability Conditions|journal=Folia Oeconomica Stetinensia|volume=11|issue=1|pages=61–72|doi=10.2478/v10031-012-0008-7|year=2012|doi-access=free}}</ref> − − Other applications include the outer sphere electron transfer frequently encountered in chemical and biological systems and modelling real estate prices. − − 其他应用还包括化学和生物系统中经常遇到的外层电子转移和房地产价格模型。 − − − − − − Fold bifurcations and the cusp geometry are by far the most important practical consequences of catastrophe theory. They are patterns which reoccur again and again in physics, engineering and mathematical modelling. − − Fold bifurcations and the cusp geometry are by far the most important practical consequences of catastrophe theory. They are patterns which reoccur again and again in physics, engineering and mathematical modelling. − They produce the strong gravitational lensing events and provide astronomers with one of the methods used for detecting [[black holes]] and the [[dark matter]] of the universe, via the phenomenon of [[gravitational lensing]] producing multiple images of distant [[quasars]].<ref>A.O. Petters, H. Levine and J. Wambsganss, Singularity Theory and Gravitational Lensing", Birkhäuser Boston (2001)</ref>+ − − They produce the strong gravitational lensing events and provide astronomers with one of the methods used for detecting black holes and the dark matter of the universe, via the phenomenon of gravitational lensing producing multiple images of distant quasars. − − 它们产生强烈的引力透镜效应事件，为天文学家提供了探测黑洞和宇宙暗物质的方法之一---- 通过引力透镜效应现象产生遥远类星体的多幅图像。 − − − − − − The remaining simple catastrophe geometries are very specialised in comparison, and presented here only for curiosity value. − − The remaining simple catastrophe geometries are very specialised in comparison, and presented here only for curiosity value.
→尖点灾变
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{{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。
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。<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> 建议是，在适度的压力({{nowrap|''a'' > 0}}) ，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({{nowrap|''a'' < 0}})。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。
一个简单的机械系统，“塞曼灾难机器” ，很好地说明了尖点灾难。在这种装置中，弹簧末端位置的平滑变化可以引起附加轮转动位置的突然变化。<ref>Cross, Daniel J., [http://lagrange.physics.drexel.edu/flash/zcm/ Zeeman's Catastrophe Machine in Flash] {{webarchive|url=https://archive.is/20121211093251/http://lagrange.physics.drexel.edu/flash/zcm/ |date=2012-12-11 }}</ref>
并联冗余[[复杂系统]]的灾变失效可以根据局部应力与外部应力之间的关系进行评估。[[结构断裂力学]]模型与尖点突变行为相似。该模型预测了复杂系统的储备能力。
其他应用还包括化学和生物系统<ref>{{cite journal|last=Xu|first=F|title=Application of catastrophe theory to the ∆G<sup>≠</sup> to -∆G relationship in electron transfer reactions.| journal=Zeitschrift für Physikalische Chemie |series=Neue Folge | volume=166 | pages=79–91 | date=1990|doi=10.1524/zpch.1990.166.Part_1.079}}</ref> 中经常遇到的外层电子转移和房地产价格模型。<ref>{{cite journal|last=Bełej|first=Mirosław|author2=Kulesza, Sławomir|title=Modeling the Real Estate Prices in Olsztyn under Instability Conditions|journal=Folia Oeconomica Stetinensia|volume=11|issue=1|pages=61–72|doi=10.2478/v10031-012-0008-7|year=2012|doi-access=free}}</ref>
折叠分岔和尖点几何是迄今为止突变理论最重要的实际结果。它们是在物理学、工程学和数学模型中反复出现的模式。
折叠分岔和尖点几何是迄今为止突变理论最重要的实际结果。它们是在物理学、工程学和数学模型中反复出现的模式。
它们产生强烈的引力透镜效应事件，为天文学家提供了探测[[黑洞]]和宇宙[[暗物质]]的方法之一---- 通过[[引力透镜]]效应现象产生遥远类星体的多幅图像。<ref>A.O. Petters, H. Levine and J. Wambsganss, Singularity Theory and Gravitational Lensing", Birkhäuser Boston (2001)</ref>
剩下的简单的灾难几何图形在比较中非常专业，在这里展示只是为了好奇的价值。
剩下的简单的灾难几何图形在比较中非常专业，在这里展示只是为了好奇的价值。