Catastrophe models

Control variables

Potential function

Bifurcation set

Normalization formulas

Folding catastrophe

1

$V\left(x\right)=x^3+u_{1}x$

$u_1=-3x^2$

$X_{u1}=\sqrt{u_1}$

Cusp catastrophe

2

$V\left(x\right)=\frac{1}{4}{x}^4+\frac{1}{2}{u}_1{x}^2+u_{2}x$

$u_1=-6x^2,u_2=8x^3$

$X_{u1}=\sqrt{u_1}$ , $X_{u2}=\sqrt[3]{u_2}$

Swallowtail catastrophe

3

$V\left(x\right)=\frac{1}{5}{x}^5+\frac{1}{3}{u}_1{x}^3+\frac{1}{2}{u}_2{x}^2+u_{3}x$

$u_1=-6x^2$ , $u_2=8x^3$ , $u_3=-3x$

$X_{u1}=\sqrt{u_1}$ , $X_{u2}=\sqrt[3]{u_2}$ , $X_{u3}=\sqrt[4]{u_3}$

Butterfly catastrophe

4

$\begin{array}{l}V\left(x\right)=\frac{1}{6}{x}^6+\frac{1}{4}{u}_1{x}^4+\frac{1}{3}{u}_2{x}^3\\ +\frac{1}{2}{u}_3{x}^2+u_{4}x\end{array}$

$u_1=-10x^2$ , $u_2=20x^3$ ,

$u_3=-15x^4$ , $u_4=4x^5$

$X_{u1}=\sqrt{u_1}$ , $X_{u2}=\sqrt[3]{u_2}$ ,

$X_{u3}=\sqrt[4]{u_3}$ , $X_{u4}=\sqrt[5]{u_4}$