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Catastrophe theory
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=== Fold catastrophe === [[File:Fold_catastrophe_animation.gif|thumb|Fold catastrophe, with surface <math>z = -y^2- 0.1 x^4</math>.]] [[File:fold bifurcation.svg|frame|right|Stable and unstable pair of extrema disappear at a fold bifurcation]] :<math>V = x^3 + ax\,</math> When {{nowrap|''a'' < 0}}, the potential ''V'' has two extrema - one stable, and one unstable. If the parameter ''a'' is slowly increased, the system can follow the stable minimum point. But at {{nowrap|''a'' {{=}} 0}} the stable and unstable extrema meet, and annihilate. This is the bifurcation point. At {{nowrap|''a'' > 0}} there is no longer a stable solution. If a physical system is followed through a fold bifurcation, one therefore finds that as ''a'' reaches 0, the stability of the {{nowrap|''a'' < 0}} solution is suddenly lost, and the system will make a sudden transition to a new, very different behaviour. {{anchor|Tipping point}}This bifurcation value of the parameter ''a'' is sometimes called the "[[tipping point (physics)|tipping point]]". {{Clear}}
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