CATASTROPHE THEORY
CATASTROPHE THEORY
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DOI：
10.1038/scientificamerican0476-65
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发表时间：
1976-01-01
影响因子：
3
通讯作者：
ZEEMAN, EC
中科院分区：
文献类型：
--
作者：
ZEEMAN, EC
科学家们经常通过建立数学模型来描述事件。事实上，当这样一个模型特别成功时，它不仅可以描述事件，而且可以“解释”它们;如果模型可以简化为一个简单的方程，它甚至可以被称为自然法则。300年来，牛顿和莱布尼茨发明的微分学一直是建立这种模型的最杰出的方法。牛顿自己用微分方程表达了他的运动定律和引力定律，詹姆斯·克拉克麦克斯韦在他的电磁理论中运用了这些定律。爱因斯坦的广义相对论也在一组微分方程中达到高潮，这些例子可以添加许多不太有名的例子。然而，作为一种描述性语言，微分方程有一个固有的局限性：它们只能描述那些变化是平滑和连续的现象。用数学术语来说，微分方程的解必须是可微分的函数。相对而言，很少有现象是那样的，或者说是那样的缓慢和良好;相反，世界充满了突然的变化和不可预测的发散，这就需要不可微的函数。处理不连续和发散现象的数学方法是最近才发展起来的。该方法具有描述自然界各个方面的形式演变的潜力，因此它体现了一种非常普遍的理论;它可以在格拉德变化的力量或动机导致行为突变的情况下特别有效地应用。由于这个原因，该方法被命名为突变理论。物理学中的许多事件现在都可以被认为是数学悖论的例子。然而，最终，该理论最重要的应用可能是在生物学和社会科学中，在这些领域，不连续和发散现象无处不在，而到目前为止，其他计算机技术被证明是无效的。因此，突变理论可以为迄今为止“不精确”的科学提供一种数学语言。
Scientists often describe events by con structing a mathematical model. In deed, when such a model is particular ly successful, it is said not only to describe the events but also to" explain" them; if the model can be reduced to a simple equation, it may even be called a law of nature. For 300 years the preeminent method in build ing such models has been the differential calculus invented by Newton and Leibniz. Newton himself expressed his laws of mo tion and gravitation in terms of differential equations, and James Clerk Maxwell em ployed them in his theory of electromagnet ism. Einstein's general theory of relativity also culminates in a set of differential equa tions, and to these examples could be added many less celebrated ones. Nevertheless, as a descriptive language differential equations have an inherent limitation: they can de scribe only those phenomena where change is smooth and continuous. In mathematical terms, the solutions to a differential equa tion must be functions that are differenti able. Relatively few phenomena are that or derly and well behaved; on the contrary, the world is full of sudden transformations and unpredictable divergences, which call for functions that are not differentiable. A mathematical method for dealing with discontinuous and divergent phenomena has only recently been developed. The method has the potential for describing the evolution of forms in all aspects of nature, and hence it embodies a theory of great generality; it can be applied with particular effectiveness in those situations where grad ually changing forces or motivations lead to abrupt changes in behavior. For this reason the method has been named catastrophe theory. Many events in physics can now be recognized as examples of mathematical ca tastrophes. Ultimately, however, the most important applications of the theory may be in biology and the social sciences, where discontinuous and divergent phenomena are ubiquitous and where other mathemati cal techniques have so far proved ineffec tive. Catastrophe theory could thus provide a mathematical language for the hitherto" inexact" sciences.