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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.

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.

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).

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。

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.