Catastrophe theory

Diagram of cusp catastrophe, showing curves (brown, red) of x satisfying dV/dx = 0 for parameters (a,b), drawn for parameter b continuously varied, for several values of parameter a.

Outside the cusp locus of bifurcations (blue), for each point (a,b) in parameter space there is only one extremising value of x. Inside the cusp, there are two different values of x giving local minima of V(x) for each (a,b), separated by a value of x giving a local maximum.

Cusp shape in parameter space (a,b) near the catastrophe point, showing the locus of fold bifurcations separating the region with two stable solutions from the region with one.

Pitchfork bifurcation at a = 0 on the surface b = 0

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.

单词	Catastrophe theory
释义	Catastrophe theory 英语例句库 Catastrophe theory has become a useful tool in studying incontinuous phenomena since it was established. 摘要突变理论自问世以来已成为研究现实世界中一些不连续现象的有力工具。英语百科 Catastrophe theory 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.
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