Catastrophies

a non-singular point a local extremum – a stable minimum or an unstable maximum the fold, the cusp, the swallowtail, the butterfly, and more.

the elliptical umbilic, the hyperbolic umbilic, the parabolic umbilic, and more.

the symbolic umbilic.

and more.

_A_0 - a non-singular point: V = x $V = x$ .
_A_1 - a local extremum, either a stable minimum or unstable maximum V = ± x 2 + a x $V = \pm x^2 + a x$ .
_A_2 - the fold
_A_3 - the cusp
_A_4 - the swallowtail
_A_5 - the butterfly
_A_k - a representative of an infinite sequence of one variable forms V = x k + 1 + ⋯ $V=x^{k+1}+\cdots$
_D_4− - the elliptical umbilic
_D_4+ - the hyperbolic umbilic
_D_5 - the parabolic umbilic
_D_k - a representative of an infinite sequence of further umbilic forms
_E_6 - the symbolic umbilic V = x 3 + y 4 + a x y 2 + b x y + c x + d y + e y 2 $V = x^3+y^4+a x y^2 +bxy+cx+dy+ey^2$
_E_7
_E_8

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