Catastrophe models

Control variables

Potential function

Bifurcation set

Normalization formulas

Folding catastrophe

1

V ( x ) = x 3 + u 1 x

u 1 = 3 x 2

X u 1 = u 1

Cusp catastrophe

2

V ( x ) = 1 4 x 4 + 1 2 u 1 x 2 + u 2 x

u 1 = 6 x 2 , u 2 = 8 x 3

X u 1 = u 1 , X u 2 = u 2 3

Swallowtail catastrophe

3

V ( x ) = 1 5 x 5 + 1 3 u 1 x 3 + 1 2 u 2 x 2 + u 3 x

u 1 = 6 x 2 , u 2 = 8 x 3 , u 3 = 3 x

X u 1 = u 1 , X u 2 = u 2 3 , X u 3 = u 3 4

Butterfly catastrophe

4

V ( x ) = 1 6 x 6 + 1 4 u 1 x 4 + 1 3 u 2 x 3 + 1 2 u 3 x 2 + u 4 x

u 1 = 10 x 2 , u 2 = 20 x 3 ,

u 3 = 15 x 4 , u 4 = 4 x 5

X u 1 = u 1 , X u 2 = u 2 3 ,

X u 3 = u 3 4 , X u 4 = u 4 5