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第19行：
第19行： − 在数学上，突变理论是分岔理论动力学系统研究的一个分支; 它也是几何学中更一般的奇点理论的一个特殊情况。+ 第29行：
第29行： − 分岔理论研究和分类的现象拥有属性突然变化的行为产生的小的变化情况下，分析如何定性性质的方程解决方案取决于参数出现在方程。这可能导致突然和戏剧性的变化，例如不可预测的时间和规模的滑坡。+ 第39行：
第39行： − 灾变理论起源于20世纪60年代法国数学家任 · 托姆的工作，在70年代克里斯托弗 · 塞曼的努力下变得非常流行。它考虑的特殊情况下，长期稳定的平衡可以确定为一个光滑的，明确定义的势函数(李亚普诺夫函数)的最小值。+ 第49行：
第49行： − 非线性的某些参数的微小变化可以导致均衡的出现或消失，或者从吸引变为排斥，反之亦然，从而导致系统行为的巨大而突然的变化。然而，在较大的参数空间中，突变理论揭示了这种分叉点往往作为定性几何结构的一部分出现。+
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In mathematics, catastrophe theory is a branch of bifurcation theory in the study of dynamical systems; it is also a particular special case of more general singularity theory in geometry.
In mathematics, catastrophe theory is a branch of bifurcation theory in the study of dynamical systems; it is also a particular special case of more general singularity theory in geometry.
在数学上，突变论是动力学系统研究里分岔理论的一个分支；而在几何学中，它也是奇点理论里的一个特殊情形。
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.
分岔理论主要研究因环境中微小变化导致系统动力学行为发生突变的现象，对动力学方程的解如何依赖方程中的参数进行定性分析。这可能会导致突然而剧烈的变化，例如，无法预测时间和规模的滑坡现象。
Catastrophe theory originated with the work of the French mathematician René Thom in the 1960s, and became very popular due to the efforts of Christopher Zeeman in the 1970s. It considers the special case where the long-run stable equilibrium can be identified as the minimum of a smooth, well-defined potential function (Lyapunov function).
Catastrophe theory originated with the work of the French mathematician René Thom in the 1960s, and became very popular due to the efforts of Christopher Zeeman in the 1970s. It considers the special case where the long-run stable equilibrium can be identified as the minimum of a smooth, well-defined potential function (Lyapunov function).
突变论起源于20世纪60年代法国数学家René Thom的一系列工作。得益于Christopher Zeeman的努力，突变论在20世纪70年代变得非常流行。突变论考虑这样一种特殊情况:长期稳定的平衡可以由一个光滑的、定义明确的势函数(李雅普诺夫函数)的最小值确定。
Small changes in certain parameters of a nonlinear system can cause equilibria to appear or disappear, or to change from attracting to repelling and vice versa, leading to large and sudden changes of the behaviour of the system. However, examined in a larger parameter space, catastrophe theory reveals that such bifurcation points tend to occur as part of well-defined qualitative geometrical structures.
Small changes in certain parameters of a nonlinear system can cause equilibria to appear or disappear, or to change from attracting to repelling and vice versa, leading to large and sudden changes of the behaviour of the system. However, examined in a larger parameter space, catastrophe theory reveals that such bifurcation points tend to occur as part of well-defined qualitative geometrical structures.
非线性系统某些参数的微小变化可以导致平衡态的出现或消失，或者从吸引变为排斥，又或相反，从而导致系统行为产生巨大而突然的变化。然而，在较大的参数空间中，突变论揭示了这种分叉点往往作为定性几何结构的一部分出现。